Average Cost of Golf Balls The cost, in dollars, of producing x golf balls is given by
C(x) = 0.43x + 76,000
The average cost per golf ball is given by
a. Find the average cost per cell phone when 1000, 10,000, and 100,000 phones are produced.
b. What is the minimum average cost per cell phone? How many cell phones should be produced to minimize the aver- age cost per phone?
The Correct Answer and Explanation is :
Let’s start by defining the formulas and breaking down the problem step by step.
The total cost ( C(x) ) for producing ( x ) golf balls is given by:
[
C(x) = 0.43x + 76,000
]
Where:
- ( x ) is the number of golf balls produced.
- 0.43 is the cost per golf ball, and 76,000 is a fixed cost (likely overhead or fixed production costs).
Now, the average cost per golf ball is given by:
[
\text{Average Cost per golf ball} = \frac{C(x)}{x}
]
This can be simplified by substituting the formula for ( C(x) ):
[
\text{Average Cost per golf ball} = \frac{0.43x + 76,000}{x}
]
This simplifies to:
[
\text{Average Cost per golf ball} = 0.43 + \frac{76,000}{x}
]
a. Finding the Average Cost for Different Production Quantities
We need to find the average cost per golf ball for the following quantities: 1,000, 10,000, and 100,000 golf balls.
- When ( x = 1,000 ):
[
\text{Average Cost} = 0.43 + \frac{76,000}{1,000} = 0.43 + 76 = 76.43 \, \text{dollars}
]
- When ( x = 10,000 ):
[
\text{Average Cost} = 0.43 + \frac{76,000}{10,000} = 0.43 + 7.6 = 8.03 \, \text{dollars}
]
- When ( x = 100,000 ):
[
\text{Average Cost} = 0.43 + \frac{76,000}{100,000} = 0.43 + 0.76 = 1.19 \, \text{dollars}
]
b. Minimum Average Cost
To minimize the average cost per golf ball, we need to look at the formula for average cost:
[
\text{Average Cost per golf ball} = 0.43 + \frac{76,000}{x}
]
As ( x ) increases, the term ( \frac{76,000}{x} ) decreases, which means the average cost approaches 0.43 as ( x ) increases. Therefore, the minimum average cost occurs when ( x ) becomes very large, approaching infinity. The minimum average cost is:
[
\text{Minimum Average Cost} = 0.43
]
Conclusion
To minimize the average cost per golf ball, the production quantity should be as large as possible. The minimum average cost is 0.43 dollars per golf ball, and it occurs when production is very high, theoretically approaching an infinite number of balls. However, practically speaking, economies of scale generally reach a point where further increases in production provide diminishing returns, but based purely on the mathematical model, the cost tends toward 0.43 as production increases.
Thus, the minimum cost is achieved by producing as many golf balls as possible.