Max & min - how many shoes should be made?

[ayrestaurant]

Homework Statement

A show company wants to determine how many shoes to manufacture in order to maximize its profit. The cost for making x shoes is given by C(x) = 66x + 3690. The price of a single pair of shoes is given by p(x)=(3690/x)+94-(7/17)x, where x is the total number of shoes made. Assuming that all the shoes manufactured are sold, how many shoes should be made? (Hint: P(x) = R(x) - C(x), where R(x) (revenue) is equal to the price of a single pair of shoes multiplied by the number of shoes sold.

a. 27
b. 7
c. 48
d. 34

Homework Equations

help...

The Attempt at a Solution

differentiate and equal to cero.. but I need help with equation..


help.. thanx!

 

[mighty2000]

Hi there,

To determine the number of shoes that should be made in order to maximize profit, we will use the equation P(x) = R(x) - C(x), where P(x) is the profit, R(x) is the revenue, and C(x) is the cost.

We can rewrite the given equations for R(x) and C(x) as:

R(x) = (3690/x + 94 - (7/17)x)x = 3690 + 94x - (7/17)x^2

C(x) = 66x + 3690

Now, we can substitute these equations into P(x) = R(x) - C(x) to get:

P(x) = 3690 + 94x - (7/17)x^2 - (66x + 3690)

Simplifying this, we get:

P(x) = 94x - (7/17)x^2 - 66x

P(x) = -(7/17)x^2 + 28x

Now, to maximize profit, we need to find the value of x that makes P(x) the largest. To do this, we can take the derivative of P(x) and set it equal to 0:

P'(x) = -(14/17)x + 28

Setting this equal to 0, we get:

0 = -(14/17)x + 28

(14/17)x = 28

x = (17/14) * 28

x = 34

Therefore, the company should make 34 shoes in order to maximize profit. I hope this helps!