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A business that manufactures small alarm clocks has weekly fixed costs of \$6500. The average cost per clock for the business to manufacture [tex]$x$[/tex] clocks is described by [tex]$\frac{0.4x + 6500}{x}$[/tex].

a. Find the average cost when [tex]$x = 100$[/tex], [tex]$x = 1000$[/tex], and [tex]$x = 10,000$[/tex].

b. Like all other businesses, the alarm clock manufacturer must make a profit. To do this, each clock must be sold for at least 50 cents more than what it costs to manufacture. Due to competition, clocks can be sold for \$1.50 each and no more. Our small manufacturer can only produce 2000 clocks weekly. Does this business have much of a future? Explain.

a. The average cost when [tex]$x = 100$[/tex] is \$[Insert the calculated value here]. (Type an integer or a decimal)

Answer :

Sure, let's break down the solution step-by-step:

### Part (a):
We need to find the average cost of manufacturing clocks for different values of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] is the number of clocks produced.

The average cost per clock is given by the formula:
[tex]\[ \text{Average Cost} = 0.4x + 6500 \][/tex]

Let's calculate it for different values of [tex]\( x \)[/tex]:

1. When [tex]\( x = 100 \)[/tex]:
[tex]\[ \text{Average Cost} = 0.4(100) + 6500 = 40 + 6500 = 6540 \][/tex]

2. When [tex]\( x = 1000 \)[/tex]:
[tex]\[ \text{Average Cost} = 0.4(1000) + 6500 = 400 + 6500 = 6900 \][/tex]

3. When [tex]\( x = 10,000 \)[/tex]:
[tex]\[ \text{Average Cost} = 0.4(10000) + 6500 = 4000 + 6500 = 10500 \][/tex]

### Part (b):
Now, we need to analyze if the business can make a profit.

- Maximum Production Capacity: The business can produce up to 2000 clocks.
- Selling Price per Clock: \[tex]$150.
- Profit Condition: Each clock must be sold for at least 50 cents more than its manufacturing cost.

1. Calculate the Total Cost and Average Cost for 2000 Clocks:
- Total cost for 2000 clocks:
\[ \text{Total Cost} = 0.4(2000) + 6500 = 800 + 6500 = 7300 \]
- Average cost per clock:
\[ \text{Average Cost per Clock} = \frac{7300}{2000} = 3.65 \]

2. Determine Minimum Selling Price for Profit:
- To ensure a profit, each clock should be sold for at least \( 3.65 + 0.5 = 4.15 \).

3. Profitability Analysis:
- The clocks are sold for \$[/tex]150 each, which is well above the minimum required \$4.15 for profit.
- Therefore, the business is capable of making a profit.

In conclusion, based on the cost structure and the selling capabilities, this business has a positive outlook and a promising future.

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