{"id":1206,"date":"2021-06-30T17:02:10","date_gmt":"2021-06-30T17:02:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-physical-applications\/"},"modified":"2021-11-17T02:19:10","modified_gmt":"2021-11-17T02:19:10","slug":"problem-set-physical-applications","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-physical-applications\/","title":{"raw":"Problem Set: Physical Applications","rendered":"Problem Set: Physical Applications"},"content":{"raw":"

For the following exercises (1-6), find the work done.<\/p>\r\n\r\n

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1.\u00a0<\/strong>Find the work done when a constant force [latex]F=12[\/latex] lb moves a chair from [latex]x=0.9[\/latex] to [latex]x=1.1[\/latex] ft.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.<\/strong> How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?<\/p>\r\n[reveal-answer q=\"fs-id1167793590314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793590314\"]\r\n

150 ft-lb<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.<\/strong> Find the work done when you push a box along the floor 2 m, when you apply a constant force of [latex]F=100\\text{N}.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793705293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793705293\"]\r\n

[latex]200\\text{J}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>Compute the work done for a force [latex]F=\\frac{12}{{x}^{2}}[\/latex] N from [latex]x=1[\/latex] to [latex]x=2[\/latex] m.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6. <\/strong>What is the work done moving a particle from [latex]x=0[\/latex] to [latex]x=1[\/latex] m if the force acting on it is [latex]F=3{x}^{2}[\/latex] N?<\/p>\r\n[reveal-answer q=\"fs-id1167793274893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793274893\"]\r\n

1 J<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (7-11), find the mass of the one-dimensional object.<\/p>\r\n\r\n

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7.\u00a0<\/strong>A wire that is 2 ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)={x}^{2}+2x[\/latex] lb\/ft<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>A car antenna that is 3 ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)=3x+2[\/latex] lb\/ft<\/p>\r\n[reveal-answer q=\"fs-id1167793724747\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793724747\"]\r\n

[latex]\\frac{39}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>A metal rod that is 8 in. long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)={e}^{1\\text{\/}2x}[\/latex] lb\/in.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>A pencil that is 4 in. long (starting at [latex]x=2)[\/latex] and has a density function of [latex]\\rho (x)=5\\text{\/}x[\/latex] oz\/in.<\/p>\r\n[reveal-answer q=\"fs-id1167793630148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793630148\"]\r\n

[latex]\\text{ln}(243)[\/latex]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>A ruler that is 12 in. long (starting at [latex]x=5)[\/latex] and has a density function of [latex]\\rho (x)=\\text{ln}(x)+(1\\text{\/}2){x}^{2}[\/latex] oz\/in.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (12-16), find the mass of the two-dimensional object that is centered at the origin.<\/p>\r\n\r\n

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12.\u00a0<\/strong>An oversized hockey puck of radius 2 in. with density function [latex]\\rho (x)={x}^{3}-2x+5[\/latex]<\/p>\r\n[reveal-answer q=\"30087466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"30087466\"]\r\n\r\n[latex]\\frac{332\\pi }{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>A frisbee of radius 6 in. with density function [latex]\\rho (x)={e}^{\\text{\u2212}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>A plate of radius 10 in. with density function [latex]\\rho (x)=1+ \\cos (\\pi x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793941017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793941017\"]\r\n

[latex]100\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.<\/strong> A jar lid of radius 3 in. with density function [latex]\\rho (x)=\\text{ln}(x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>A disk of radius 5 cm with density function [latex]\\rho (x)=\\sqrt{3x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793244624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793244624\"]\r\n

[latex]20\\pi \\sqrt{15}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>A 12-in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?<\/p>\r\n[reveal-answer q=\"fs-id1167793255054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793255054\"]\r\n

6 J<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>A 1-m spring requires 10 J to stretch the spring to 1.1 m. How much work would it take to stretch the spring from 1 m to 1.2 m?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?<\/p>\r\n[reveal-answer q=\"fs-id1167793545960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793545960\"]\r\n

5 cm<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>A force of [latex]F=20x-{x}^{3}[\/latex] N stretches a nonlinear spring by [latex]x[\/latex] meters. What work is required to stretch the spring from [latex]x=0[\/latex] to [latex]x=2[\/latex] m?<\/p>\r\n[reveal-answer q=\"fs-id1167793630566\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793630566\"]\r\n

36 J<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb\/ft.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>For the cable in the preceding exercise, how much work is done to lift the cable 50 ft?<\/p>\r\n[reveal-answer q=\"fs-id1167793496034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793496034\"]\r\n

18,750 ft-lb<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n25.\u00a0<\/strong>For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?\r\n\r\n<\/div>\r\n<\/div>\r\n
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26. [T]<\/strong> A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area [latex]A[\/latex] at height [latex]h.[\/latex] If the rock used to build the pyramid weighs approximately [latex]w=100{\\text{lb\/ft}}^{3},[\/latex] how much work did it take to lift all the rock?<\/p>\r\n[reveal-answer q=\"fs-id1167793387254\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793387254\"]\r\n

[latex]\\frac{32}{3}\u00d7{10}^{9}\\text{ft-lb}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [T]<\/strong> For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2 ft\/hr, how long did it take to build the pyramid?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [T]<\/strong> The force of gravity on a mass [latex]m[\/latex] is [latex]F=\\text{\u2212}((GMm)\\text{\/}{x}^{2})[\/latex] newtons. For a rocket of mass [latex]m=1000\\text{kg},[\/latex] compute the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=6500[\/latex] km. (Note<\/em>: [latex]G=6\u00d7{10}^{-17}{\\text{N m}}^{2}\\text{\/}{\\text{kg}}^{2}[\/latex] and [latex]M=6\u00d7{10}^{24}\\text{kg}\\text{.})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793551051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793551051\"]\r\n

[latex]8.65\u00d7{10}^{5}\\text{J}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n30. [T]<\/strong> For the rocket in the preceding exercise, find the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=\\infty .[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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31. [T]<\/strong> A rectangular dam is 40 ft high and 60 ft wide. Compute the total force [latex]F[\/latex] on the dam when<\/p>\r\n\r\n

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  1. the surface of the water is at the top of the dam and<\/li>\r\n \t
  2. the surface of the water is halfway down the dam.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1167793957832\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793957832\"]\r\n

    a. 3,000,000 lb, b. 749,000 lb<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    32. [T]<\/strong> Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb\/ft3<\/sup>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    33. [T]<\/strong> Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.<\/p>\r\n[reveal-answer q=\"fs-id1167793957919\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793957919\"]\r\n

    [latex]23.25\\pi [\/latex] million ft-lb<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    34. [T]<\/strong> How much work is required to pump out a swimming pool if the area of the base is 800 ft2<\/sup>, the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb\/ft3<\/sup>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    35.\u00a0<\/strong>A cylinder of depth [latex]H[\/latex] and cross-sectional area [latex]A[\/latex] stands full of water at density [latex]\\rho .[\/latex] Compute the work to pump all the water to the top.<\/p>\r\n[reveal-answer q=\"fs-id1167794043046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794043046\"]\r\n

    [latex]\\frac{A\\rho {H}^{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    36.\u00a0<\/strong>For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    \r\n\r\n37.\u00a0<\/strong>A cone-shaped tank has a cross-sectional area that increases with its depth: [latex]A=(\\pi {r}^{2}{h}^{2})\\text{\/}{H}^{3}.[\/latex] Show that the work to empty it is half the work for a cylinder with the same height and base.\r\n\r\n[reveal-answer q=\"fs-id1167793975996\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793975996\"]\r\n

    Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

    For the following exercises (1-6), find the work done.<\/p>\n

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    1.\u00a0<\/strong>Find the work done when a constant force [latex]F=12[\/latex] lb moves a chair from [latex]x=0.9[\/latex] to [latex]x=1.1[\/latex] ft.<\/p>\n<\/div>\n<\/div>\n

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    2.<\/strong> How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?<\/p>\n

    Show Solution<\/span><\/p>\n
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    150 ft-lb<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    3.\u00a0<\/strong>What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)<\/p>\n<\/div>\n<\/div>\n

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    4.<\/strong> Find the work done when you push a box along the floor 2 m, when you apply a constant force of [latex]F=100\\text{N}.[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]200\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    5.\u00a0<\/strong>Compute the work done for a force [latex]F=\\frac{12}{{x}^{2}}[\/latex] N from [latex]x=1[\/latex] to [latex]x=2[\/latex] m.<\/p>\n<\/div>\n<\/div>\n

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    6. <\/strong>What is the work done moving a particle from [latex]x=0[\/latex] to [latex]x=1[\/latex] m if the force acting on it is [latex]F=3{x}^{2}[\/latex] N?<\/p>\n

    Show Solution<\/span><\/p>\n
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    1 J<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    For the following exercises (7-11), find the mass of the one-dimensional object.<\/p>\n

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    7.\u00a0<\/strong>A wire that is 2 ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)={x}^{2}+2x[\/latex] lb\/ft<\/p>\n<\/div>\n<\/div>\n

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    8.\u00a0<\/strong>A car antenna that is 3 ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)=3x+2[\/latex] lb\/ft<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{39}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    9.\u00a0<\/strong>A metal rod that is 8 in. long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)={e}^{1\\text{\/}2x}[\/latex] lb\/in.<\/p>\n<\/div>\n<\/div>\n

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    10.\u00a0<\/strong>A pencil that is 4 in. long (starting at [latex]x=2)[\/latex] and has a density function of [latex]\\rho (x)=5\\text{\/}x[\/latex] oz\/in.<\/p>\n

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

    [latex]\\text{ln}(243)[\/latex]<\/p>\n

     <\/p>\n<\/div>\n<\/div>\n

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    11.\u00a0<\/strong>A ruler that is 12 in. long (starting at [latex]x=5)[\/latex] and has a density function of [latex]\\rho (x)=\\text{ln}(x)+(1\\text{\/}2){x}^{2}[\/latex] oz\/in.<\/p>\n<\/div>\n<\/div>\n

    For the following exercises (12-16), find the mass of the two-dimensional object that is centered at the origin.<\/p>\n

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    12.\u00a0<\/strong>An oversized hockey puck of radius 2 in. with density function [latex]\\rho (x)={x}^{3}-2x+5[\/latex]<\/p>\n

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

    [latex]\\frac{332\\pi }{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    13.\u00a0<\/strong>A frisbee of radius 6 in. with density function [latex]\\rho (x)={e}^{\\text{\u2212}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    14.\u00a0<\/strong>A plate of radius 10 in. with density function [latex]\\rho (x)=1+ \\cos (\\pi x)[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]100\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    15.<\/strong> A jar lid of radius 3 in. with density function [latex]\\rho (x)=\\text{ln}(x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    16.\u00a0<\/strong>A disk of radius 5 cm with density function [latex]\\rho (x)=\\sqrt{3x}[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]20\\pi \\sqrt{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    17.\u00a0<\/strong>A 12-in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?<\/p>\n<\/div>\n<\/div>\n

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    18.\u00a0<\/strong>A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?<\/p>\n

    Show Solution<\/span><\/p>\n
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    6 J<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    19.\u00a0<\/strong>A 1-m spring requires 10 J to stretch the spring to 1.1 m. How much work would it take to stretch the spring from 1 m to 1.2 m?<\/p>\n<\/div>\n<\/div>\n

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    20.\u00a0<\/strong>A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the spring?<\/p>\n

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

    5 cm<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    21.\u00a0<\/strong>A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?<\/p>\n<\/div>\n<\/div>\n

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    22.\u00a0<\/strong>A force of [latex]F=20x-{x}^{3}[\/latex] N stretches a nonlinear spring by [latex]x[\/latex] meters. What work is required to stretch the spring from [latex]x=0[\/latex] to [latex]x=2[\/latex] m?<\/p>\n

    Show Solution<\/span><\/p>\n
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    36 J<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    23.\u00a0<\/strong>Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb\/ft.<\/p>\n<\/div>\n<\/div>\n

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    24.\u00a0<\/strong>For the cable in the preceding exercise, how much work is done to lift the cable 50 ft?<\/p>\n

    Show Solution<\/span><\/p>\n
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    18,750 ft-lb<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    25.\u00a0<\/strong>For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?<\/p>\n<\/div>\n<\/div>\n

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    26. [T]<\/strong> A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area [latex]A[\/latex] at height [latex]h.[\/latex] If the rock used to build the pyramid weighs approximately [latex]w=100{\\text{lb\/ft}}^{3},[\/latex] how much work did it take to lift all the rock?<\/p>\n

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

    [latex]\\frac{32}{3}\u00d7{10}^{9}\\text{ft-lb}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    27. [T]<\/strong> For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2 ft\/hr, how long did it take to build the pyramid?<\/p>\n<\/div>\n<\/div>\n

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    28. [T]<\/strong> The force of gravity on a mass [latex]m[\/latex] is [latex]F=\\text{\u2212}((GMm)\\text{\/}{x}^{2})[\/latex] newtons. For a rocket of mass [latex]m=1000\\text{kg},[\/latex] compute the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=6500[\/latex] km. (Note<\/em>: [latex]G=6\u00d7{10}^{-17}{\\text{N m}}^{2}\\text{\/}{\\text{kg}}^{2}[\/latex] and [latex]M=6\u00d7{10}^{24}\\text{kg}\\text{.})[\/latex]<\/p>\n

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    [latex]8.65\u00d7{10}^{5}\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    30. [T]<\/strong> For the rocket in the preceding exercise, find the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    31. [T]<\/strong> A rectangular dam is 40 ft high and 60 ft wide. Compute the total force [latex]F[\/latex] on the dam when<\/p>\n

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    1. the surface of the water is at the top of the dam and<\/li>\n
    2. the surface of the water is halfway down the dam.<\/li>\n<\/ol>\n
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      a. 3,000,000 lb, b. 749,000 lb<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      32. [T]<\/strong> Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb\/ft3<\/sup>.<\/p>\n<\/div>\n<\/div>\n

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      33. [T]<\/strong> Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.<\/p>\n

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      [latex]23.25\\pi[\/latex] million ft-lb<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      34. [T]<\/strong> How much work is required to pump out a swimming pool if the area of the base is 800 ft2<\/sup>, the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb\/ft3<\/sup>.<\/p>\n<\/div>\n<\/div>\n

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      35.\u00a0<\/strong>A cylinder of depth [latex]H[\/latex] and cross-sectional area [latex]A[\/latex] stands full of water at density [latex]\\rho .[\/latex] Compute the work to pump all the water to the top.<\/p>\n

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      [latex]\\frac{A\\rho {H}^{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      36.\u00a0<\/strong>For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.<\/p>\n<\/div>\n<\/div>\n

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      37.\u00a0<\/strong>A cone-shaped tank has a cross-sectional area that increases with its depth: [latex]A=(\\pi {r}^{2}{h}^{2})\\text{\/}{H}^{3}.[\/latex] Show that the work to empty it is half the work for a cylinder with the same height and base.<\/p>\n

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      Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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