Answer to Question #101099 in Calculus for Jeff Kleven

Question

Hi, my daughter is trying to solve a grade 12 calculus (I think) problem. Can you help us with how to think through this?
As as side business, Santa creates and sells red wagons. Santa can sell 30 red wagons at $20 each per week. If the price is increased, for each dollar increase there will be a loss of one sale per week. The cost to create a wagon is $10 each. What price will give Santa maximum profit?

Thanks!

Answer 1

Let x be the increase in price.

So,

price P=20+x,

number of wagons sold N=30-x,

revenue R=PN=(20+x)(30-x),

cost C=10N=10(30-x),

profit $P=R-C=(20+x)(30-x)-10(30-x)=(10+x)(30-x)=-x^2+20x+300.$

$\frac{dP}{dx}=0\to\;-2x+20=0\to\;x=10.$

Increase in price by $10 will give Santa maximum profit.

Thus, the price which will give Santa maximum profit is 20+10=$30.

Maximum profit is 20*20=$400.