The marginal cost to make a sound system is $50 and the fixed cost is $3600.

Question

The revenue function for the same sound system is r(x)=300x-x^2. What is the average rate of change for profit as the number of sound systems increase from 50 to 200?

Answer 1

We need the profit function:

Profit = revenue - costs

In this problem, revenue is give by r(x), where x is the quantity of stereo systems produced.

Costs are a bit trickier; if the marginal cost is 50, then the cost is 50x + F, where F is a constant. This is because the marginal cost is the derivative of the cost function-- if you take the derivative of 50x + F you get 50. The question tells us that fixed costs are 3600 so F = 3600. Note that you can also use the formula

C = Variable Costs + Fixed Costs

where the variable costs in 50x.

Finally, Profit(x) = 300x - x² - 50x - 3600 = 250x - x² - 3600.

The average rate of change of profit from 50 to 200 is given by

[ Profit(200) - Profit(50) ] / [ 200 - 50]

and calculating that will give you the answer!