Shrewsbury Technologies, which manufactures high-technology instruments for spacecraft, is considering the sale of a navigational unit to a private company that wishes to launch its own communications satellite. The company plans to purchase 8 units, although it would also consider buying 16 units. Shrewsbury has started a chart relating labor time required to units produced:

Unit Produced (*X*)Time Required to Produce the *X*th Unit1

22,000 hours

2

15,400 hours

4

10,780 hours

8

?

16

?
exact percentage of 16th unit to 1st unit cost

To determine the exact percentage of the labor time required for the 16th unit compared to the 1st unit, we can use the concept of the learning curve. The learning curve formula often used in such cases is:

\[
Y = A \times X^b
\]

Where:
- \(Y\) is the time required to produce the \(X\)th unit.
- \(A\) is the time required to produce the 1st unit.
- \(X\) is the unit number.
- \(b\) is the learning curve exponent, calculated as \(b = \log(L) / \log(2)\), where \(L\) is the learning rate.

Given dataTo determine the exact percentage of the labor time required for the 16th unit compared to the 1st unit, we can use the concept of the learning curve. The learning curve formula often used in such cases is:

\[
Y = A \times X^b
\]

Where:
- \(Y\) is the time required to produce the \(X\)th unit.
- \(A\) is the time required to produce the 1st unit.
- \(X\) is the unit number.
- \(b\) is the learning curve exponent, calculated as \(b = \log(L) / \log(2)\), where \(L\) is the learning rate.

Given 
- Time for the 1st unit: \(22,000\) hours
- Time for the 2nd unit: \(15,400\) hours
- Time for the 4th unit: \(10,780\) hours
- Time for the 8th unit: unknown
- Time for the 16th unit: unknown

### Step 1: Determine the learning rate

The learning rate \(L\) is found by comparing the time for the 2nd unit to the 1st unit:

\[
L = \frac{\text{Time for 2nd unit}}{\text{Time for 1st unit}} = \frac{15,400}{22,000} \approx 0.7
\]

### Step 2: Calculate the learning curve exponent \(b\)

\[
b = \frac{\log(L)}{\log(2)} = \frac{\log(0.7)}{\log(2)} \approx -0.5145
\]

### Step 3: Calculate the time for the 16th unit

Now, using the learning curve formula:

\[
Y_{16} = 22,000 \times 16^{-0.5145}
\]

Let's calculate \(Y_{16}\).

\[
Y_{16} \approx 22,000 \times 0.397 = 8,734 \text{ hours}
\]

### Step 4: Determine the percentage of the 16th unit compared to the 1st unit

\[
\text{Percentage} = \frac{Y_{16}}{Y_{1}} \times 100 = \frac{8,734}{22,000} \times 100 \approx 39.7\%
\]

So, the time required to produce the 16th unit is approximately **39.7%** of the time required to produce the 1st unit.

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How does the learning curve affect production costs?
2. What is the learning rate and how is it determined?
3. How would the production time change for 32 units?
4. Can the learning curve exponent be a positive value?
5. How can companies use learning curves in pricing strategies?
6. What are common learning rates in different industries?
7. How does a decreasing learning rate affect long-term production?
8. What if the learning rate was 90% instead of 70%?

**Tip:** Understanding the learning curve is essential for forecasting production times and costs in manufacturing processes.

Explore how to calculate labor time for units using the learning curve formula. Understand the relationship between unit number and labor time, with practical examples and calculations provided. Suitable for advanced mathematics enthusiasts.

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