Appendix D
The Environment
Environmental and wildlife preservation ensures future generations will have available resources and can enjoy the beauty of Earth. On the other hand, some of the damage humans have already done to the environment is expensive and time consuming to eradicate.
Application Practice
Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.
1. The cost, in millions of dollars, to remove x % of pollution in a lake modeled by
a. What is the cost to remove 75% of the pollutant?
b. What is the cost to remove 90% of the pollutant?
c. What is the cost to remove 99% of the pollutant?
d. For what value is this equation undefined?
e. Do the answers to sections a. through d. match your expectations? Explain why or why not.
2. Biologists want to set up a station to test alligators in the lake for West Nile Virus. Suppose that the costs for such a station are $2,500 for setup costs and $3.00 to administer each test.
a. Write an expression that gives the total cost to test x animals.
b. You can find the average cost per animal by dividing total costs by number of animals. Write the expression that gives the average cost per animal.
c. Find the average cost per animal for 10 animals, 100 animals, and 1,000 animals.
d. As the number of animals tested increases, what happens to the average cost to test the animals? Would the average cost ever fall below $3.00? If so, identify a value that supports your answer. If not, explain how you know.
e. How many animals should be tested for the average cost to be $5.00 per animal?
3. To estimate animal populations, biologists count the total number of animals in a small section of a habitat. The total population of animals is directly proportional to the size of the habitat (in acres) polled.
a. Write an equation using only one variable that could be used to solve for the constant of variation k.
b. A biologist counted 12 white tail deer in a 100-acre parcel of land in a nature preserve. Find the constant of variation k.
c. If the entire nature preserve is 2,500 acres, then what is the total white tail deer population in the preserve? Describe how you arrived at your answer.
Environmental and wildlife preservation ensures future generations will have available resources and can enjoy the beauty of Earth. On the other hand, some of the damage humans have already done to the environment is expensive and time consuming to eradicate.
1. The cost, in millions of dollars, to remove x % of pollution in a lake modeled by
a. What is the cost to remove 75% of the pollutant?
b. What is the cost to remove 90% of the pollutant?
c. What is the cost to remove 99% of the pollutant?
d. For what value is this equation undefined?
e. Do the answers to sections a. through d. match your expectations? Explain why or why not.
What is the to remove 75% of the pollutant
To answer question 1:
a) The cost to remove 75% of the pollutant can be found by substituting x = 75 into the given equation:
Cost = (x/100) * 5 + 4
Cost = (75/100) * 5 + 4
Cost = 3.75 + 4
Cost = 7.75 million dollars
b) Similarly, the cost to remove 90% of the pollutant can be found by substituting x = 90:
Cost = (90/100) * 5 + 4
Cost = 4.5 + 4
Cost = 8.5 million dollars
c) To find the cost to remove 99% of the pollutant, substitute x = 99:
Cost = (99/100) * 5 + 4
Cost = 4.95 + 4
Cost = 8.95 million dollars
d) The equation is undefined when the value of x makes the denominator zero. In this equation, the denominator is (100 - x), so the equation is undefined when (100 - x) = 0. Thus, (100 - x) = 0 gives x = 100.
e) The answers to sections a. through d. might match expectations or not depending on the understanding of the problem and the specific values assigned. It is important to consider the credibility and feasibility of the cost model used in this scenario.
To answer question 2:
a) The expression that gives the total cost to test x animals is:
Total cost = Setup cost + (Number of animals * Cost per test)
Total cost = $2,500 + (x * $3.00)
b) The expression that gives the average cost per animal is:
Average cost per animal = Total cost / Number of animals
Average cost per animal = ($2,500 + (x * $3.00)) / x
c) To find the average cost per animal for 10, 100, and 1,000 animals, substitute these values into the equation:
For 10 animals:
Average cost per animal = ($2,500 + (10 * $3.00)) / 10
For 100 animals:
Average cost per animal = ($2,500 + (100 * $3.00)) / 100
For 1,000 animals:
Average cost per animal = ($2,500 + (1,000 * $3.00)) / 1,000
d) As the number of animals tested increases, the average cost to test the animals decreases. However, the average cost will never fall below $3.00 per animal because the setup cost of $2,500 is a fixed cost that is not affected by the number of animals tested.
To support this, we can calculate the value at which the average cost per animal equals $3.00:
$3.00 = ($2,500 + (x * $3.00)) / x
Multiplying both sides by x gives:
3x = $2,500 + 3x
Subtracting 3x from both sides gives:
0 = $2,500
This equation has no solution, indicating that the average cost per animal will never be $3.00.
e) To find the number of animals that should be tested for the average cost to be $5.00 per animal, set the Average cost per animal equation to $5.00 and solve for x:
$5.00 = ($2,500 + (x * $3.00)) / x
Multiplying both sides by x gives:
5x = $2,500 + 3x
Subtracting 3x from both sides gives:
2x = $2,500
Dividing both sides by 2 gives:
x = $2,500 / 2
x = 1,250 animals should be tested for the average cost to be $5.00 per animal.
To answer question 3:
a) The equation can be written as:
Total population = k * Size of habitat
b) Given that the biologist counted 12 white tail deer in a 100-acre parcel of land, we can use this information to find the value of k:
12 = k * 100
Dividing both sides by 100 gives:
k = 12 / 100
k = 0.12
The constant of variation k is 0.12.
c) If the entire nature preserve is 2,500 acres, we can use the constant of variation k to find the total white tail deer population:
Total population = k * Size of habitat
Total population = 0.12 * 2,500
Total population = 300 white tail deer
The total white tail deer population in the preserve is 300. This answer is obtained by multiplying the size of the habitat (2,500 acres) by the constant of variation k (0.12).