Question

Alexis runs a small business and creates recordings of her friend’s skateboard stunts. She puts the recordings onto blu-ray and sells them online. She realized that the average cost of each blu-ray depends on the number of blu-rays she creates, because of the fixed cost. The cost of producing x blu-rays is given by B(x) = 1250 +1.75x
1) Alexis wants to figure out the price to charge friends for the blu-rays. She doesn’t want to make money at this point since it is a brand new business, but does want to cover her costs. Suppose Alexis created 50 blu-rays. What is the cost of producing those 50 blu-rays? How much is it for each blu-ray?
2) What is the total cost and cost per blu-ray for 100, 1000, and 10,000?

Answer 1

Answer:

Question 1)

a) What is the cost of producing those 50 blu-rays?

• B(50) = 1,337.50

b) How much is it for each blu-ray:

• B(50) / 50 = 26.75

Question 2) What is the total cost and cost per blu-ray for 100, 1000, and 10,000?

Number of blu-rays     Total cost        Cost per blu-ray

a) 100                               1,425                  14.25

b) 1000                             3,000                3.00

c) 10,000                          12,500               1.88

Explanation:

Question 1) What is the cost of producing those 50 blu-rays? How much is it for each blu-ray?

The cost of producing the blu-rays is modeled by the function

•  B(x) = 1250 +1.75x, where:

• x is the number of blu-rays produced

• 1.75 is the cost added to the total cost for producing each unit of blu-ray

• 1.75 x is the cost added to the total cost for producing x units

• 1250 is the fixed cost, which is incurred no matter how many blu-rays are produced

• B(x) is the total cost of producing x blu-rays.

Thus, to calculate the cost of producing 50 blu-rays you just substitute 50 for x in the function and compute:

• B(50) = 1250 +1.75x = 1250 + 1.75 (50)
• B(50) = 1250 + 87.5
• B(50) = 1,337.50

To calculate the cost of each of the 50 blu-rays, you divide the total cost (1,337.50) by the number of blu-rays (50).

• B(50) / 50 = 1,337.50 / 50 = 26.75

Question 2) What is the total cost and cost per blu-ray for 100, 1000, and 10,000?

The general expresssion to calculate the total cost per blu-ray unit is found by dividing the total cost, B(x) = 1250 + 1.75x, by the number of blu-rays, x:

• B(x)/x = [1250 + 1.75x] / x

To calcualte the total cost and cost per blu-ray, substitute 100, 1000, and 10,000 in the expressions:

# of blu-rays     Total cost                                    Cost per blu-ray

100                    1250 + 1.75 (100) = 1,425             1,425/100 = 14.25

1000                  1250 + 1.75 (1000) = 3,000          3000/1000 = 3.00

10,000               1250 + 1.75(10,000) = 12,500      18,750/10,000 = 1.875

As you see, the cost per blu-ray decreases as the number of blu-rays increases, and at the limit (for a huge amount of blu-rays) the cost of blu-rays goes closer to 1.75, which is the unit rate of change of the function (the slope of the graph).