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?
a. C = 2500 + 3X.
b. Avg = ( 2500 + 3X ) / X
c. Avg = ( 2500 + 3*10 ) / 10 = 253
Avg = ( 2500 + 3*100 ) / 100 = 28
Avg = ( 2500 + 3*1000 ) /1000 = 5.5
d. Decreases. No. Rearranginging
equation: Avg = 2500 / x + 3X/X
As X increases, the first term
approaches 0; but 2nd term re-
mains constant at 3. Therefore
the min. is 3.
e. Avg = ( 2500 + 3X ) / X = 5.00
Solve for X by multiplying both
sides by X: 2500 + 3X = 5X,
2X = 2500, X = 1250.
a. The total cost to test x animals can be represented by the expression: Total Cost = Setup Cost + (Number of Animals * Cost per Test)
In this case, the setup cost is $2,500 and the cost per test is $3.00. So the expression becomes: Total Cost = $2,500 + ($3.00 * x)
b. The expression that gives the average cost per animal can be derived by dividing the total cost by the number of animals tested: Average Cost per Animal = Total Cost / Number of Animals
c. To find the average cost per animal for 10 animals, 100 animals, and 1,000 animals, substitute the respective values into the average cost expression:
For 10 animals:
Average Cost per Animal = ($2,500 + ($3.00 * 10)) / 10
Average Cost per Animal = ($2,500 + $30) / 10
Average Cost per Animal = $2,530 / 10
Average Cost per Animal = $253
For 100 animals:
Average Cost per Animal = ($2,500 + ($3.00 * 100)) / 100
Average Cost per Animal = ($2,500 + $300) / 100
Average Cost per Animal = $2,800 / 100
Average Cost per Animal = $28
For 1,000 animals:
Average Cost per Animal = ($2,500 + ($3.00 * 1,000)) / 1,000
Average Cost per Animal = ($2,500 + $3,000) / 1,000
Average Cost per Animal = $5,500 / 1,000
Average Cost per Animal = $5.50
d. As the number of animals tested increases, the average cost to test the animals decreases. This is because the setup cost ($2,500) becomes a smaller proportion of the total cost when the number of animals increases.
The average cost per animal will never fall below $3.00 because the cost per test is fixed at $3.00. Regardless of the number of animals tested, there will always be a minimum cost of $3.00 per animal.
e. To determine the number of animals needed for the average cost to be $5.00 per animal, we can set up the following equation:
$5.00 = ($2,500 + ($3.00 * x)) / x
Multiply both sides of the equation by x to eliminate the denominator:
$5.00 * x = $2,500 + ($3.00 * x)
Simplify the equation:
5x = 2500 + 3x
Subtract 3x from both sides:
2x = 2500
Divide both sides by 2:
x = 1250
Therefore, 1250 animals should be tested for the average cost to be $5.00 per animal.