{"id":128,"date":"2021-02-03T21:36:14","date_gmt":"2021-02-03T21:36:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=128"},"modified":"2021-04-19T17:07:38","modified_gmt":"2021-04-19T17:07:38","slug":"problem-set-basic-classes-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-basic-classes-of-functions\/","title":{"raw":"Problem Set: Basic Classes of Functions","rendered":"Problem Set: Basic Classes of Functions"},"content":{"raw":"

For the following exercises (1-8), for each pair of points, (a)<\/strong> find the slope of the line passing through the points and (b)<\/strong> indicate whether the line is increasing, decreasing, horizontal, or vertical.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex](-2,4)[\/latex] and [latex](1,1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573587743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573587743\"]\r\n

a. \u22121 b. Decreasing<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex](-1,4)[\/latex] and [latex](3,-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex](3,5)[\/latex] and [latex](-1,2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573587845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573587845\"]\r\n

a. 3\/4 b. Increasing<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex](6,4)[\/latex] and [latex](4,-3)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex](2,3)[\/latex] and [latex](5,7)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573587945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573587945\"]\r\n

a. 4\/3 b. Increasing<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex](1,9)[\/latex] and [latex](-8,5)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex](2,4)[\/latex] and [latex](1,4)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573588045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573588045\"]\r\n

a. 0 b. Horizontal<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex](1,4)[\/latex] and [latex](1,0)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (9-16), write the equation of the line satisfying the given conditions in slope-intercept form.<\/p>\r\n\r\n

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9.\u00a0<\/strong>Slope [latex]=-6[\/latex], passes through [latex](1,3)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573588140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573588140\"]\r\n

[latex]y=-6x+9[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>Slope [latex]=3[\/latex], passes through [latex](-3,2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>Slope [latex]=\\dfrac{1}{3}[\/latex], passes through [latex](0,4)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573588255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573588255\"]\r\n

[latex]y=\\frac{1}{3}x+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>Slope [latex]=\\dfrac{2}{5}[\/latex],\u00a0 [latex]x[\/latex]-intercept [latex]=8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>Passing through [latex](2,1)[\/latex] and [latex](-2,-1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573588382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573588382\"]\r\n

[latex]y=\\frac{1}{2}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>Passing through [latex](-3,7)[\/latex] and [latex](1,2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]x[\/latex]-intercept [latex]=5[\/latex] and [latex]y[\/latex]-intercept [latex]=-3[\/latex]<\/p>\r\n\r\n

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[latex]y=\\frac{3}{5}x-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]x[\/latex]-intercept [latex]=-6[\/latex] and [latex]y[\/latex]-intercept [latex]=9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (17-24), for each linear equation, (a)<\/strong> give the slope [latex](m)[\/latex], and [latex]y[\/latex]-intercept [latex](b)[\/latex], if any, and (b)<\/strong> graph the line.<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]y=2x-3[\/latex]<\/p>\r\n\r\n

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a. [latex](m=2, \\, b=-3)[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]y=-\\frac{1}{7}x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]f(x)=-6x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573588773\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573588773\"]\r\n

a. [latex](m=-6, \\, b=0)[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]f(x)=-5x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]4y+24=0[\/latex]<\/p>\r\n\r\n

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a. [latex](m=0, \\, b=-6)[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]8x-4=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>[latex]2x+3y=6[\/latex]<\/p>\r\n\r\n

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a. [latex](m=-\\frac{2}{3}, \\, b=2)[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]6x-5y+15=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (25-29), for each polynomial,<\/p>\r\n\r\n

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  1. find the degree<\/li>\r\n \t
  2. find the zeros, if any<\/li>\r\n \t
  3. find the [latex]y[\/latex]-intercept(s), if any<\/li>\r\n \t
  4. use the leading coefficient to determine the graph\u2019s end behavior<\/li>\r\n \t
  5. determine algebraically whether the polynomial is even, odd, or neither.<\/li>\r\n<\/ol>\r\n
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    25.\u00a0<\/strong>[latex]f(x)=2x^2-3x-5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573589210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573589210\"]\r\n

    a. 2; b. [latex]\\frac{5}{2}, \\, -1[\/latex]; c. \u22125; d. Both ends rise; e. Neither<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    26.\u00a0<\/strong>[latex]f(x)=-3x^2+6x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    27.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2}x^2-1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573589323\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573589323\"]\r\n

    a. 2; b. [latex]\\pm \\sqrt{2}[\/latex]; c. \u22121; d. Both ends rise; e. Even<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    28.\u00a0<\/strong>[latex]f(x)=x^3+3x^2-x-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    29.\u00a0<\/strong>[latex]f(x)=3x-x^3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573589438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573589438\"]\r\n

    a. 3; b. 0, [latex]\\pm \\sqrt{3}[\/latex]; c. 0; d. Left end rises, right end falls; e. Odd<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (30-31), use the graph of [latex]f(x)=x^2[\/latex] to graph each transformed function [latex]g[\/latex].<\/p>\r\n\r\n

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    30.\u00a0<\/strong>[latex]g(x)=x^2-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    31.\u00a0<\/strong>[latex]g(x)=(x+3)^2+1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573589577\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573589577\"]\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (32-33), use the graph of [latex]f(x)=\\sqrt{x}[\/latex] to graph each transformed function [latex]g[\/latex].<\/p>\r\n\r\n

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    32.\u00a0<\/strong>[latex]g(x)=\\sqrt{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    33.\u00a0<\/strong>[latex]g(x)=\u2212\\sqrt{x}-1[\/latex]<\/p>\r\n\r\n

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    <\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (34-35), use the graph of [latex]y=f(x)[\/latex] to graph each transformed function [latex]g[\/latex]<\/p>\r\n\"An<\/span>\r\n

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    34.\u00a0<\/strong>[latex]g(x)=f(x)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    35.\u00a0<\/strong>[latex]g(x)=f(x-1)+2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573589867\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573589867\"]\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (36-39), for each of the piecewise-defined functions, (a)<\/strong> evaluate at the given values of the independent variable and (b)<\/strong> sketch the graph.<\/p>\r\n\r\n

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    36.\u00a0<\/strong>[latex]f(x)=\\begin{cases} 4x+3, & x \\le 0 \\\\ -x+1, & x > 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-3); \\, f(0); \\, f(2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    37.\u00a0<\/strong>[latex]f(x)=\\begin{cases}x^2-3, & x < 0 \\\\ 4x-3, & x \\ge 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-4); \\, f(0); \\, f(2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573590152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573590152\"]\r\n

    a. [latex]13, \\, -3, \\, 5[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    38.\u00a0<\/strong>[latex]h(x)=\\begin{cases} x+1, & x \\le 5 \\\\ 4, & x > 5 \\end{cases}[\/latex];\u00a0 \u00a0[latex]h(0); \\, h(\\pi ); \\, h(5)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    39.\u00a0<\/strong>[latex]g(x)=\\begin{cases} \\left(\\dfrac{3}{x-2}\\right), & x \\ne 2 \\\\ 4, & x = 2 \\end{cases}[\/latex];\u00a0 \u00a0[latex]g(0); \\, g(-4); \\, g(2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170573590432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573590432\"]\r\n

    a. [latex]\\frac{-3}{2}, \\, \\frac{-1}{2}, \\, 4[\/latex] b.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (40-44), determine whether the statement is true<\/strong> or false<\/strong><\/em>. Explain why.<\/p>\r\n\r\n

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    40.\u00a0<\/strong>[latex]f(x)=\\dfrac{(4x+1)}{(7x-2)}[\/latex] is a transcendental function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    41.\u00a0<\/strong>[latex]g(x)=\\sqrt[3]{x}[\/latex] is an odd root function<\/p>\r\n[reveal-answer q=\"fs-id1170573595384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595384\"]\r\n

    True, because [latex]n=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    42.\u00a0<\/strong>A logarithmic function is an algebraic function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    43.\u00a0<\/strong>A function of the form [latex]f(x)=x^b[\/latex], where [latex]b[\/latex] is a real valued constant, is an exponential function.<\/p>\r\n[reveal-answer q=\"fs-id1170573595452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595452\"]\r\n

    False, because [latex]f(x)=x^b[\/latex] - where [latex]b[\/latex] is a real-valued constant - is a power function.\u00a0 Exponential functions are of the form [latex]f(x)=b^x[\/latex], where [latex]b[\/latex] is a real-valued constant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    44.\u00a0<\/strong>The domain of an even root function is all real numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    45. [T]<\/strong> A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.<\/p>\r\n\r\n

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    1. Find a function [latex]y=V(t)[\/latex] that determines the value [latex]V[\/latex] of the equipment at the end of [latex]t[\/latex] years.<\/li>\r\n \t
    2. Find and interpret the meaning of the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts for this situation.<\/li>\r\n \t
    3. What is the value of the equipment at the end of 5 years?<\/li>\r\n \t
    4. When will the value of the equipment be $3000?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170573595577\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595577\"]\r\n

      a. [latex]V(t)=-2733t+20500[\/latex] b. [latex](0, 20500)[\/latex] means that the initial purchase price of the equipment is $20,500; [latex](7.5,0)[\/latex] means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      46. [T]<\/strong> Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 [latex](t=0)[\/latex], total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.<\/p>\r\n\r\n

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      1. Find a linear function [latex]S[\/latex] that estimates the total online holiday sales in the year [latex]t[\/latex].<\/li>\r\n \t
      2. Interpret the slope of the graph of [latex]S[\/latex].<\/li>\r\n \t
      3. Use part (a) to predict the year when online shopping during Christmas will reach $60 billion.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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        47. [T]<\/strong> A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost [latex]C[\/latex] as a function of number of cupcakes made.<\/p>\r\n\r\n

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        1. Find a linear function that relates cost [latex]C[\/latex] to [latex]x[\/latex], the number of cupcakes made.<\/li>\r\n \t
        2. Find the cost to bake 160 cupcakes.<\/li>\r\n \t
        3. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint<\/em>: Use the INTERSECTION function on a calculator to find this number.)<\/em><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170573595790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595790\"]\r\n

          a. [latex]C=0.75x+125[\/latex] b. $245 c. 167 cupcakes<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          48. [T]<\/strong> A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.<\/p>\r\n\r\n

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          1. Find a linear function that models the price [latex]P[\/latex] of the house versus the number of years [latex]t[\/latex] since the original purchase.<\/li>\r\n \t
          2. Interpret the slope of the graph of [latex]P[\/latex].<\/li>\r\n \t
          3. Find the price of the house 15 years from when it was originally purchased.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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            49. [T]<\/strong> A car was purchased for $26,000. The value of the car depreciates by $1500 per year.<\/p>\r\n\r\n

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            1. Find a linear function that models the value [latex]V[\/latex] of the car after [latex]t[\/latex] years.<\/li>\r\n \t
            2. Find and interpret [latex]V(4)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170573595958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595958\"]\r\n

              a. [latex]V(t)=-1500t+26,000[\/latex] b. In 4 years, the value of the car is $20,000.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              50. [T]<\/strong> A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              51. [T]<\/strong> The total cost [latex]C[\/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[\/latex], where [latex]x[\/latex] is the number of items produced. Determine the cost to produce 175 items.<\/p>\r\n[reveal-answer q=\"fs-id1170573596072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573596072\"]\r\n

              $30,337.50<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              52. [T]<\/strong> A professor asks her class to report the amount of time [latex]t[\/latex] they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.<\/p>\r\n\r\n

                \r\n \t
              1. Find the linear function [latex]y=N(t)[\/latex] that models this situation, where [latex]N[\/latex] is the number of pages typed and [latex]t[\/latex] is the time in minutes.<\/li>\r\n \t
              2. Use part (a) to determine how many pages can be typed in 2 hours.<\/li>\r\n \t
              3. Use part (a) to determine how long it takes to type a 20-page assignment.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                \r\n
                \r\n

                53. [T]<\/strong> The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[\/latex], where [latex]t[\/latex] is time in years and [latex]t=0[\/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.<\/p>\r\n[reveal-answer q=\"fs-id1170573596252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573596252\"]\r\n

                96% of the total capacity<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                \r\n
                \r\n

                54. [T]<\/strong> The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.<\/p>\r\n\r\n

                  \r\n \t
                1. Find the linear function [latex]y=N(x)[\/latex], where [latex]N[\/latex] is the number of students that actually enroll and [latex]x[\/latex] is the number of all students offered admission to the class of 2019.<\/li>\r\n \t
                2. If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"

                  For the following exercises (1-8), for each pair of points, (a)<\/strong> find the slope of the line passing through the points and (b)<\/strong> indicate whether the line is increasing, decreasing, horizontal, or vertical.<\/p>\n

                  \n
                  \n

                  1.\u00a0<\/strong>[latex](-2,4)[\/latex] and [latex](1,1)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  a. \u22121 b. Decreasing<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  2.\u00a0<\/strong>[latex](-1,4)[\/latex] and [latex](3,-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  3.\u00a0<\/strong>[latex](3,5)[\/latex] and [latex](-1,2)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  a. 3\/4 b. Increasing<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  4.\u00a0<\/strong>[latex](6,4)[\/latex] and [latex](4,-3)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  5.\u00a0<\/strong>[latex](2,3)[\/latex] and [latex](5,7)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  a. 4\/3 b. Increasing<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  6.\u00a0<\/strong>[latex](1,9)[\/latex] and [latex](-8,5)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  7.\u00a0<\/strong>[latex](2,4)[\/latex] and [latex](1,4)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  a. 0 b. Horizontal<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  8.\u00a0<\/strong>[latex](1,4)[\/latex] and [latex](1,0)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  For the following exercises (9-16), write the equation of the line satisfying the given conditions in slope-intercept form.<\/p>\n

                  \n
                  \n

                  9.\u00a0<\/strong>Slope [latex]=-6[\/latex], passes through [latex](1,3)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  [latex]y=-6x+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  10.\u00a0<\/strong>Slope [latex]=3[\/latex], passes through [latex](-3,2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  11.\u00a0<\/strong>Slope [latex]=\\dfrac{1}{3}[\/latex], passes through [latex](0,4)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  [latex]y=\\frac{1}{3}x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  12.\u00a0<\/strong>Slope [latex]=\\dfrac{2}{5}[\/latex],\u00a0 [latex]x[\/latex]-intercept [latex]=8[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  13.\u00a0<\/strong>Passing through [latex](2,1)[\/latex] and [latex](-2,-1)[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  [latex]y=\\frac{1}{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  14.\u00a0<\/strong>Passing through [latex](-3,7)[\/latex] and [latex](1,2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  15.\u00a0<\/strong>[latex]x[\/latex]-intercept [latex]=5[\/latex] and [latex]y[\/latex]-intercept [latex]=-3[\/latex]<\/p>\n

                  \n
                  Show Solution<\/span><\/p>\n
                  \n

                  [latex]y=\\frac{3}{5}x-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  16.\u00a0<\/strong>[latex]x[\/latex]-intercept [latex]=-6[\/latex] and [latex]y[\/latex]-intercept [latex]=9[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  For the following exercises (17-24), for each linear equation, (a)<\/strong> give the slope [latex](m)[\/latex], and [latex]y[\/latex]-intercept [latex](b)[\/latex], if any, and (b)<\/strong> graph the line.<\/p>\n

                  \n
                  \n

                  17.\u00a0<\/strong>[latex]y=2x-3[\/latex]<\/p>\n

                  \n
                  Show Solution<\/span><\/p>\n
                  \n

                  a. [latex](m=2, \\, b=-3)[\/latex] b.<\/p>\n

                  \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  18.\u00a0<\/strong>[latex]y=-\\frac{1}{7}x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  19.\u00a0<\/strong>[latex]f(x)=-6x[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  a. [latex](m=-6, \\, b=0)[\/latex] b.<\/p>\n

                  \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  20.\u00a0<\/strong>[latex]f(x)=-5x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  21.\u00a0<\/strong>[latex]4y+24=0[\/latex]<\/p>\n

                  \n
                  Show Solution<\/span><\/p>\n
                  \n

                  a. [latex](m=0, \\, b=-6)[\/latex] b.<\/p>\n

                  \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  22.\u00a0<\/strong>[latex]8x-4=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  23.\u00a0<\/strong>[latex]2x+3y=6[\/latex]<\/p>\n

                  \n
                  Show Solution<\/span><\/p>\n
                  \n

                  a. [latex](m=-\\frac{2}{3}, \\, b=2)[\/latex] b.<\/p>\n

                  \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  24.\u00a0<\/strong>[latex]6x-5y+15=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                  For the following exercises (25-29), for each polynomial,<\/p>\n

                    \n
                  1. find the degree<\/li>\n
                  2. find the zeros, if any<\/li>\n
                  3. find the [latex]y[\/latex]-intercept(s), if any<\/li>\n
                  4. use the leading coefficient to determine the graph\u2019s end behavior<\/li>\n
                  5. determine algebraically whether the polynomial is even, odd, or neither.<\/li>\n<\/ol>\n
                    \n
                    \n

                    25.\u00a0<\/strong>[latex]f(x)=2x^2-3x-5[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    a. 2; b. [latex]\\frac{5}{2}, \\, -1[\/latex]; c. \u22125; d. Both ends rise; e. Neither<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    26.\u00a0<\/strong>[latex]f(x)=-3x^2+6x[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    27.\u00a0<\/strong>[latex]f(x)=\\frac{1}{2}x^2-1[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    a. 2; b. [latex]\\pm \\sqrt{2}[\/latex]; c. \u22121; d. Both ends rise; e. Even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    28.\u00a0<\/strong>[latex]f(x)=x^3+3x^2-x-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    29.\u00a0<\/strong>[latex]f(x)=3x-x^3[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    a. 3; b. 0, [latex]\\pm \\sqrt{3}[\/latex]; c. 0; d. Left end rises, right end falls; e. Odd<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (30-31), use the graph of [latex]f(x)=x^2[\/latex] to graph each transformed function [latex]g[\/latex].<\/p>\n

                    \n
                    \n

                    30.\u00a0<\/strong>[latex]g(x)=x^2-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    31.\u00a0<\/strong>[latex]g(x)=(x+3)^2+1[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (32-33), use the graph of [latex]f(x)=\\sqrt{x}[\/latex] to graph each transformed function [latex]g[\/latex].<\/p>\n

                    \n
                    \n

                    32.\u00a0<\/strong>[latex]g(x)=\\sqrt{x+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    33.\u00a0<\/strong>[latex]g(x)=\u2212\\sqrt{x}-1[\/latex]<\/p>\n

                    \n
                    Show Solution<\/span><\/p>\n
                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n

                    \n<\/div>\n<\/div>\n

                    For the following exercises (34-35), use the graph of [latex]y=f(x)[\/latex] to graph each transformed function [latex]g[\/latex]<\/p>\n

                    \"An<\/span><\/p>\n

                    \n
                    \n

                    34.\u00a0<\/strong>[latex]g(x)=f(x)+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    35.\u00a0<\/strong>[latex]g(x)=f(x-1)+2[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (36-39), for each of the piecewise-defined functions, (a)<\/strong> evaluate at the given values of the independent variable and (b)<\/strong> sketch the graph.<\/p>\n

                    \n
                    \n

                    36.\u00a0<\/strong>[latex]f(x)=\\begin{cases} 4x+3, & x \\le 0 \\\\ -x+1, & x > 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-3); \\, f(0); \\, f(2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    37.\u00a0<\/strong>[latex]f(x)=\\begin{cases}x^2-3, & x < 0 \\\\ 4x-3, & x \\ge 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-4); \\, f(0); \\, f(2)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    a. [latex]13, \\, -3, \\, 5[\/latex] b.<\/p>\n

                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    38.\u00a0<\/strong>[latex]h(x)=\\begin{cases} x+1, & x \\le 5 \\\\ 4, & x > 5 \\end{cases}[\/latex];\u00a0 \u00a0[latex]h(0); \\, h(\\pi ); \\, h(5)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    39.\u00a0<\/strong>[latex]g(x)=\\begin{cases} \\left(\\dfrac{3}{x-2}\\right), & x \\ne 2 \\\\ 4, & x = 2 \\end{cases}[\/latex];\u00a0 \u00a0[latex]g(0); \\, g(-4); \\, g(2)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    a. [latex]\\frac{-3}{2}, \\, \\frac{-1}{2}, \\, 4[\/latex] b.<\/p>\n

                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (40-44), determine whether the statement is true<\/strong> or false<\/strong><\/em>. Explain why.<\/p>\n

                    \n
                    \n

                    40.\u00a0<\/strong>[latex]f(x)=\\dfrac{(4x+1)}{(7x-2)}[\/latex] is a transcendental function.<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    41.\u00a0<\/strong>[latex]g(x)=\\sqrt[3]{x}[\/latex] is an odd root function<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    True, because [latex]n=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    42.\u00a0<\/strong>A logarithmic function is an algebraic function.<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    43.\u00a0<\/strong>A function of the form [latex]f(x)=x^b[\/latex], where [latex]b[\/latex] is a real valued constant, is an exponential function.<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    False, because [latex]f(x)=x^b[\/latex] – where [latex]b[\/latex] is a real-valued constant – is a power function.\u00a0 Exponential functions are of the form [latex]f(x)=b^x[\/latex], where [latex]b[\/latex] is a real-valued constant.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    44.\u00a0<\/strong>The domain of an even root function is all real numbers.<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    45. [T]<\/strong> A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.<\/p>\n

                      \n
                    1. Find a function [latex]y=V(t)[\/latex] that determines the value [latex]V[\/latex] of the equipment at the end of [latex]t[\/latex] years.<\/li>\n
                    2. Find and interpret the meaning of the [latex]x[\/latex]– and [latex]y[\/latex]-intercepts for this situation.<\/li>\n
                    3. What is the value of the equipment at the end of 5 years?<\/li>\n
                    4. When will the value of the equipment be $3000?<\/li>\n<\/ol>\n
                      Show Solution<\/span><\/p>\n
                      \n

                      a. [latex]V(t)=-2733t+20500[\/latex] b. [latex](0, 20500)[\/latex] means that the initial purchase price of the equipment is $20,500; [latex](7.5,0)[\/latex] means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                      \n
                      \n

                      46. [T]<\/strong> Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 [latex](t=0)[\/latex], total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.<\/p>\n

                        \n
                      1. Find a linear function [latex]S[\/latex] that estimates the total online holiday sales in the year [latex]t[\/latex].<\/li>\n
                      2. Interpret the slope of the graph of [latex]S[\/latex].<\/li>\n
                      3. Use part (a) to predict the year when online shopping during Christmas will reach $60 billion.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                        \n
                        \n

                        47. [T]<\/strong> A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost [latex]C[\/latex] as a function of number of cupcakes made.<\/p>\n

                          \n
                        1. Find a linear function that relates cost [latex]C[\/latex] to [latex]x[\/latex], the number of cupcakes made.<\/li>\n
                        2. Find the cost to bake 160 cupcakes.<\/li>\n
                        3. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint<\/em>: Use the INTERSECTION function on a calculator to find this number.)<\/em><\/li>\n<\/ol>\n
                          Show Solution<\/span><\/p>\n
                          \n

                          a. [latex]C=0.75x+125[\/latex] b. $245 c. 167 cupcakes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                          \n
                          \n

                          48. [T]<\/strong> A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.<\/p>\n

                            \n
                          1. Find a linear function that models the price [latex]P[\/latex] of the house versus the number of years [latex]t[\/latex] since the original purchase.<\/li>\n
                          2. Interpret the slope of the graph of [latex]P[\/latex].<\/li>\n
                          3. Find the price of the house 15 years from when it was originally purchased.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                            \n
                            \n

                            49. [T]<\/strong> A car was purchased for $26,000. The value of the car depreciates by $1500 per year.<\/p>\n

                              \n
                            1. Find a linear function that models the value [latex]V[\/latex] of the car after [latex]t[\/latex] years.<\/li>\n
                            2. Find and interpret [latex]V(4)[\/latex].<\/li>\n<\/ol>\n
                              Show Solution<\/span><\/p>\n
                              \n

                              a. [latex]V(t)=-1500t+26,000[\/latex] b. In 4 years, the value of the car is $20,000.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                              \n
                              \n

                              50. [T]<\/strong> A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.<\/p>\n<\/div>\n<\/div>\n

                              \n
                              \n

                              51. [T]<\/strong> The total cost [latex]C[\/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[\/latex], where [latex]x[\/latex] is the number of items produced. Determine the cost to produce 175 items.<\/p>\n

                              Show Solution<\/span><\/p>\n
                              \n

                              $30,337.50<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                              \n
                              \n

                              52. [T]<\/strong> A professor asks her class to report the amount of time [latex]t[\/latex] they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.<\/p>\n

                                \n
                              1. Find the linear function [latex]y=N(t)[\/latex] that models this situation, where [latex]N[\/latex] is the number of pages typed and [latex]t[\/latex] is the time in minutes.<\/li>\n
                              2. Use part (a) to determine how many pages can be typed in 2 hours.<\/li>\n
                              3. Use part (a) to determine how long it takes to type a 20-page assignment.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                53. [T]<\/strong> The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[\/latex], where [latex]t[\/latex] is time in years and [latex]t=0[\/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.<\/p>\n

                                Show Solution<\/span><\/p>\n
                                \n

                                96% of the total capacity<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                54. [T]<\/strong> The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.<\/p>\n

                                  \n
                                1. Find the linear function [latex]y=N(x)[\/latex], where [latex]N[\/latex] is the number of students that actually enroll and [latex]x[\/latex] is the number of all students offered admission to the class of 2019.<\/li>\n
                                2. If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n\n\t\t\t
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                                  Candela Citations<\/h3>\n\t\t\t\t\t
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                                  CC licensed content, Shared previously<\/div>