To calculate the cost of the last unit for URA’s production of 20 additional cruise missile guidance controls, we need to apply the learning curve theory. The theory assumes that as production doubles, the time (or cost) per unit decreases by a constant percentage, known as the "learning rate." The goal is to determine the cost of the 22nd unit (since two units have already been produced) based on labor and material/equipment costs, factoring in the learning rate.

Step-by-Step Calculation Process:

1. Determine the Learning Rate: Based on the data given:

   • The first unit took 5,000 hours.
   • The second unit took 4,750 hours.

   The percentage reduction in labor between the first and second unit gives us a learning rate.

   The learning curve equation can be written as:

   $T_n = T_1 \times n^{\log_2(b)}$

   Where:

   • $T_n$ = Time for the $n$th unit
   • $T_1$ = Time for the first unit (5,000 hours)
   • $n$ = Unit number
   • $b$ = Learning rate

   We can calculate the learning rate by using the data for the second unit:

   $4,750 = 5,000 \times 2^{\log_2(b)}$

   Taking the ratio:

   $\frac{4,750}{5,000} = b$

   $b = 0.95$

   Therefore, the learning rate is 95%, which means for every doubling of production, the labor cost reduces by 5%.

2. Determine the Hours for the 22nd Unit: Using the learning rate of 95%, we can calculate the time it takes to build the 22nd unit using the learning curve formula again:

   $T_{22} = T_1 \times 22^{\log_2(0.95)}$

   Now let's calculate this equation to find the labor hours for the 22nd unit.

The time required to produce the 22nd unit is approximately 3,977.68 hours.

Step 3: Calculate Labor Cost for the 22nd Unit

Labor cost is charged at $20 per hour, so the labor cost for the 22nd unit is:

$\text{Labor Cost}_{22} = 3,977.68 \times 20 = 79,553.62 \text{ dollars}$

Step 4: Estimate Material and Equipment Costs

From the given data, the material cost also shows a decreasing trend:

• First unit: $40,000
• Second unit: $34,000

The reduction in material cost between the first two units is $\frac{34,000}{40,000} = 0.85$, or 85% of the original cost. This suggests a similar learning curve for material costs.

To estimate the material cost for the 22nd unit, we can apply this trend as well.

$\text{Material Cost}_{22} = 40,000 \times 0.85^{21}$

Let's calculate that now.The estimated material and equipment cost for the 22nd unit is approximately $1,317.82.

Step 5: Total Cost for the 22nd Unit

Now, we can sum up the labor and material costs to find the total cost of producing the 22nd unit:

$\text{Total Cost}_{22} = \text{Labor Cost}_{22} + \text{Material Cost}_{22} = 79,553.62 + 1,317.82 = 80,871.44 \text{ dollars}$

Thus, the total cost to build the 22nd unit is approximately $80,871.44.

Would you like more details or have any questions? Here are some related questions to further explore this topic:

1. How would the total cost change if the learning rate was different?
2. What is the cumulative labor cost for producing all 22 units?
3. How does the learning curve affect production planning and pricing strategies?
4. Can the material cost reduction follow a different learning rate than labor?
5. What strategies can companies use to accelerate the learning curve?

Tip: Learning curves are widely applicable not just to labor but also to materials and processes, helping businesses predict costs and improve efficiency over time.