D1.05: Exercises

Use a calculator and make the graphs by hand to work the following problems. Don’t calculate very many points—use those given. In this class we will usually use a spreadsheet to automate the calculations. The work in this lesson is mainly to help you recall how to graph formulas by hand so that you will fully understand the graphs you produce with a spreadsheet.

Part I

  1. If we invest $750 at 6% annual interest, compounded quarterly, the formula for the amount A that the investment is worth after t years is [latex]A=750\cdot{{\left(1+\frac{0.06}{4}\right)}^{4t}}[/latex].
    1. Find the amount the investment is worth after five years. Then evaluate this for two more values of t between 0 and 30 years. Use the rest of the values that were given in the example, and sketch a graph of this formula.
    2. For the previous problem, how long should we leave the investment so that it will be worth $4000? Estimate the time, correct to the nearest year.
  2. Then the formula is [latex]A=750\cdot{{\left(1+\frac{r}{n}\right)}^{n\cdot{t}}}[/latex]. This is an example of a formula with several input values. Use this formula to find the amount of the investment after 5 years if the interest rate is 8% and it is compounded monthly.
  3. Find the area of a trapezoid with bases 5 feet and 9 feet and height 7 feet. [latex]A=\frac{1}{2}h\left({{b}_{1}}+{{b}_{2}}\right)[/latex]
  4. Graph the formula for the volume of a sphere, [latex]V=\frac{4}{3}\pi{{r}^{3}}[/latex], for the values of r from 0 to 6 feet and use the graph to approximate what radius will give a volume of 400 cubic feet, correct to within 10 cubic feet. Evaluate about three values by hand/calculator and then use the rest from the example.
  5. A certain river in Georgia is polluted with coliform bacteria. This has been blamed primarily on the dairy farmers whose farms are in the watershed of the river. This formula is used to model the cost in dollars, y, to remove p% of the bacterial pollution from the river. How much of the pollution can be removed if $1.0 million is available to spend on this?  Evaluate about three values by hand/calculator and then use the rest form the example. [latex]y=\frac{380,000p}{100-p}[/latex]
  6. A certain kind of appliance is sold for $10 each and the manufacturer can sell all she produces at that price. To produce these appliances requires a fixed cost of $14,220 per month (equipment depreciation, salaries, utilities, etc.) and the variable cost per appliance is $2.10.
    1. Write a formula for the cost of producing x appliances.
    2. Write another formula for the revenue produced by selling x appliances.
    3. Graph both formulas for [latex]0\le{x}\le2500[/latex].
    4. For what value of x is the cost equal to the revenue? (That is called the “break-even” point.

Part II

  1. The concentration of a particular drug in a patient’s blood is given by the formula [latex]C=\frac{7}{0.3{{t}^{2}}+1.1}[/latex] where t is the number of hours after the patient takes the medicine and C is the concentration in milliliters per liter of blood. Of course, [latex]t\ge0[/latex].  (Hint:       When you are using your calculator to evaluate this, be sure to put parentheses around the entire denominator of the fraction.)
t C

a.     Use your calculator to confirm at least one of the values of C in the table of values and then fill in the missing value.

b.     Which is the input variable? (When you graph this, put that along the horizontal axis.)

c.     Using graph paper, graph this formula by hand.
d.     It is recommended that the patient not take another dose until the concentration is less than 2.0 ml/l. According to your graph of the formula, approximately how long will that take?   (Estimate to the nearest hour.)

0 6.363636
1 5.000000
2 3.043478
3
4 1.186441
5 0.813953
  1. The landing speed S , in feet per second, of a particular type of small plane can be modeled by the formula [latex]S=\sqrt{1.496w}[/latex] where w is the weight in pounds of the plane.
w S

a.     Use your calculator to confirm at least one of the values of C in the table of values and then fill in the missing value..

b.     Which is the input variable? (When you graph this, put that along the horizontal axis.)

c.     Using graph paper, graph this formula by hand.

d.     What weight would correspond to a landing speed of 100 ft/sec? Use your graph to estimate this to the nearest thousand pounds.

4000 77.35632
6000 94.74175
8000
10000 122.3111
  1. The logistic formula can be used to describe population growth in a situation with limited resources. Consider this logistic formula [latex]P=\frac{1000}{1+10\cdot({{2}^{-t}})}[/latex], where P is the population at time t. (Hint:       When you are using your calculator to evaluate this, be sure to put parentheses around the entire denominator of the fraction.)
t P(t)

a.     Use your calculator to confirm at least one of the values of P in the table of values and then fill in the missing value.

b.     Which is the input variable? (When you graph this, put that along the horizontal axis.)

c.     Using graph paper, graph this formula by hand.

d.     What time would we expect to have a value for P of 900? Use your graph to estimate this.

0 90.9091
1 166.6667
2 285.7143
3 444.4444
4
5 761.9048
6 864.8649
7
8 962.4060
9 980.8429
  1. A certain kind of thermometer is sold for $5 each and the manufacturer can sell all she produces at that price. To produce these thermometers requires a fixed cost of $3500 per month (equipment depreciation, salaries, utilities, etc.) and the variable cost per thermometer is $1.50.
    1. Write a formula for the cost of producing x thermometers.
    2. Write a formula for the revenue produced by selling x thermometers.
    3. Graph both formulas for [latex]0\le{x}\le2500[/latex] in increments of 500.
    4. For what value of x is the cost equal to the revenue? (That point on the graph is called the “break-even” point.)
    5. What is the total cost of making that many thermometers?
    6. What is the total revenue earned from that many thermometers?