Calculus Problem

Question

A company making espresso machines has the demand price that can be modeled by p=500-0.5q^2 and the total cost modeled by C=50q+40, where q is the number of espresso machines sold.

A) What quantity will maximize profit?

B) What is the maximum profit?

Answer 1

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There is an error or typo somewhere in one of your modeling functions.

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Normally we can solve this in 2 different ways using Calculus or Algebra 2.

First we recognize that profit = demand price - cost, so we can say that profit P is a function of quantity q with

P(q) = 500 - 0.5q^2 -50q - 40

Method 1 using calculus:

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Find P'(q) and set it equal to zero. However, given your modeling functions we will end up with a negative quantity q = -50.

Next we would plug in our quantity q that we just found into our profit function P(q) to calculate the maximum profit.

Method 2 using Algebra 2:

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We recognize that P(q) is a quadratic function whose graph is a parabola. We know it opens downward, so its vertex must be the point of maximum profit.

We can find the vertex easily using q = -b/2a, but again this gives us a negative quantity q = -50.

Next we would plug in our quantity q that we just found into our profit function P(q) to calculate the maximum profit.