Sarah Sandoval is a coffee farmer trying to decide how many tons of coffee to produce. She can sell each ton of coffee for $2,500. The cost of producing the first ton of coffee is $500, for the second ton, it's $1000. For each additional ton of coffee produced, the marginal cost increases by $500. How many tons of coffee should Sarah produce, and what is the total cost of her coffee production?

Question

Quizplus · Accepted Answer

The marginal cost increases by $500 for each additional ton, so the total cost of producing n tons will be:
$500 + $1000 + $1500 + ... + $500n
This is an arithmetic sequence with first term 500 and common difference 500. The formula for the sum of an arithmetic sequence is:
sum = (n/2)(first term + last term)
The last term can be found by adding the common difference (500) n-1 times to the first term (500), so:
last term = 500 + 500(n-1) = 500n
Plugging this into the formula for the sum, we get:
sum = (n/2)(first term + last term) = (n/2)(500 + 500n) = 250n + 250n^2
To maximize profit, Sarah should produce the quantity where marginal cost equals marginal revenue. Marginal revenue is equal to the selling price per ton, which is $2,500. The marginal cost of producing the n-th ton is $500 + $500(n-1) = $500n. Therefore, we need to solve the equation:
2,500 = 500n
n = 5
So Sarah should produce 5 tons of coffee, and the total cost will be:
sum = 250n + 250n^2 = 2,500 + 6,250 = $7,750
The best choice is B, she will produce five tons at a total cost of $7,500.

Sarah Sandoval Is a Coffee Farmer Trying to Decide How