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Question 277935: 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?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! problem a:
c = 3*x + 2500
c is the total cost for setting up the station and administering the tests.
problem b:
a = (3*x+2500)/x
a is the average cost to test each animal.
that's the total cost divided by the number of animals tested.
problem c:
average cost for 10 animals is (3*10 + 2500)/10 = 2530/10 = $253.00
average cost for 100 animals is (3*100) + 2500)/100 = 2800/10 = $28.00
average cost for 1000 animals is (3*1000) + 2500)/1000 = 5500/1000 = $5.50
problem d:
average cost per animal goes down.
it will never go below $3.00 per animal.
as x gets larger, the percentage of 2500 / x gets smaller.
it can approach 0 but it will never be 0.
if it was 0, the average cost would be (3*x)/3 = $3.00
anything above that makes the average cost just a little bit higher than $3.00, but never below that.
example:
(3*(1 billion) + 2500)/1 billion equals:
3*(1 billion)/1 billion + 2500/(1 billion) which equals:
3 + (2500/(1 billion).
now 2500 / 1 billion is a very small number, but it is not equal to 0, so the average cost is going to be $3.00 + a very small number.
problem e:
the formula for average is:
a = (3*x + 2500)/x
set a = 5 and this formula becomes:
5 = (3*x + 2500)/x
multiply both sides of this equation by x to get:
5*x = 3*x + 2500
subtract 3*x from both sides of this equation to get:
2*x = 2500
divide both sides of this equation by 2 to get:
x = 1250
at 1250 tests, the average cost should be $5.00 per test.
plug this into the original equation for average to see if it holds.
a = (3*x + 2500)/x becomes:
a = (3*1250 + 2500) / 1250 which becomes:
a = 3750 + 2500) / 1250 which becomes:
a = 6250 / 1250 which becomes:
a = 5
the average cost per test of $5.00 is confirmed to be true.
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