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 $1,400 for setup costs and $4.00 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 $4.00? If so, identify a value that supports your answer. If not, demonstrate or explain how you know.
e. How many animals should be tested for the average cost to be $6.00 per animal? (Work must be shown algebraically, not with trial and error or guess and check attempts.)
Is there something you don't understand here? I will be happy to critique your thinking.
a. The expression that gives the total cost to test x animals is: $1,400 + $4.00 * x
To calculate the total cost, you add the setup costs of $1,400 to the cost of administering each test, which is $4.00 multiplied by the number of animals being tested, represented by x.
b. The expression that gives the average cost per animal is: ($1,400 + $4.00 * x) / x
To find the average cost per animal, you divide the total cost (which is $1,400 plus $4.00 multiplied by the number of animals) by the number of animals being tested, which is x.
c. To find the average cost per animal for 10, 100, and 1,000 animals, substitute the respective numbers into the expression for average cost per animal:
For 10 animals: ($1,400 + $4.00 * 10) / 10 = ($1,400 + $40) / 10 = $1,440 / 10 = $144.00 /animal
For 100 animals: ($1,400 + $4.00 * 100) / 100 = ($1,400 + $400) / 100 = $1,800 / 100 = $18.00 /animal
For 1,000 animals: ($1,400 + $4.00 * 1,000) / 1,000 = ($1,400 + $4,000) / 1,000 = $5,400 / 1,000 = $5.40 /animal
d. As the number of animals tested increases, the average cost per animal decreases. This is because the fixed setup cost of $1,400 is spread over a larger number of animals.
The average cost will never fall below $4.00 because each test has a cost of $4.00. Regardless of the number of animals being tested, there will always be a cost of $4.00 per animal for administering the test.
e. To find the number of animals that should be tested for the average cost to be $6.00 per animal, we can set up an equation using the average cost expression:
($1,400 + $4.00 * x) / x = $6.00
Multiply both sides of the equation by x:
$1,400 + $4.00 * x = $6.00 * x
$1,400 = $6.00 * x - $4.00 * x
$1,400 = $2.00 * x
Divide both sides of the equation by $2.00:
$1,400 / $2.00 = x
x = 700
Therefore, to achieve an average cost of $6.00 per animal, 700 animals should be tested.