Question

A major shoe company is determines that to minimize costs, C for producing shoes is modeled by C = .1s^2 - 2s +25. C is the total cost in billions of dollars and s is the number of units made in millions.

a) Find how many shoes (in millions)the company is able to produce while minimizing the cost.

b) Then determine the lowest cost (in billions) the company is able to achieve.

Answer 1

(A) 10 Million Shoes (B) 15 Billion Dollars

Explanation:

To find the number of shoes (in millions) that the company is able to produce while minimizing the cost, we need to find the minimum point of the cost function. This can be done by finding the derivative of the cost function with respect to the number of units made (s) and setting it equal to zero.

Given:

Cost function: C = 0.1s^2 - 2s + 25

a) Find how many shoes (in millions) the company is able to produce while minimizing the cost:

Step 1: Find the derivative of the cost function with respect to s.

dC/ds = 0.2s - 2

Step 2: Set the derivative equal to zero and solve for s.

0.2s - 2 = 0

0.2s = 2

s = 2 / 0.2

s = 10

Therefore, the company is able to produce 10 million shoes to minimize the cost.

b) Determine the lowest cost (in billions) the company is able to achieve:

To find the lowest cost, we substitute the value of s = 10 into the cost function C = 0.1s^2 - 2s + 25.

C = 0.1(10)^2 - 2(10) + 25

C = 0.1(100) - 20 + 25

C = 10 - 20 + 25

C = 15 billion dollars

Therefore, the lowest cost the company is able to achieve is 15 billion dollars