Expert: Scott A Wilson - 10/12/2009

Question
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  C = 6,000/200-2x
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–d match your expectations? Why or why not?


(Source: Sea World http://www.seaworld.org/animal-info/info-books/flamingo/physical-characteristics... )

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. 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 the number of white tail deer in a 100 acre parcel of land in a nature preserve and found 12 deer. 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.  

Answer
1. The cost, in millions of dollars, to remove x % of pollution in a lake is modeled by  
C = 6,000/(200-2x) = 3000/(100-x).  In this way, I see that x is suppose to be between
0 and 100, signifying 0.00 to 1.00.

To do each of these, put in x = 75, x = 90, and x = 99.

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?
x = 100 (which means all, for it is really 100%).

e. Do the answers to sections a–d match your expectations? Why or why not?
Well, for more out, it costs more.  Doesn't that make sense?

(Source: Sea World
http://www.seaworld.org/animal-info/info-books/flamingo/physical-characteristics... )

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.
Total cost C = 2500 + 3x.

b. Write the expression that gives the average cost per animal.
x = animals, so the average cost is A(x) = C(x)/x = (2500 + 3x)/x

c. Find the average cost per animal for 10 animals. 100 animals. 1,000 animals.
Take A(x) = (2500 + 3x)/x.  Find A(10), A(100), A(1,000).

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.
The more animales, the cheaper it gets.
$3 is the lower limit of that curve.
Divide each term by x and the answer is 2500/x + 3, so as x->∞, 2500/x ->0,
and the entire equation goes go 3.

e. How many animals should be tested for the average cost to be $5.00 per animal?
Solve 5 = (2500 + 3x)/x for x.  It starts by multiplying by x, giving 5x = 2500 + 3x,
so subtract 3x from both sides and get 2x = 2500.  Divide by 2 and you've got it.

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.
P = kx where k is 1/(# of acres) and P is the population in the total acres.

b. A biologist counted the number of white tail deer in a 100 acre parcel of land in a nature preserve and found 12 deer. Find the constant of variation k.
k = 12/100, so P = kx, where x is the number of acres.

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.
P(x) = 12x/100, put in x = 2,500.  12(2500)/100 = 12*25 = 300.