A company with manufacturing plants located in State A and State B has​ labor-hour and wage requirements for the manufacture of three types of inflatable boats as given in the two matrices below. Complete parts​ (A) through​ (D).
Hourly Wages
​Labor-Hours Per Boat
State A
State B
Cutting Dept
Assembly Dept
Packaging Dept
17.56
17.69
 Cutting Dept
0.5
0.4
0.2
 ​1-person boat
Nequals
14.22
14.52
 Assembly Dept
Mequals
1.4
0.9
0.6
 ​2-person boat
9.57
11.42
 Packaging Dept
1.7
1.6
0.7
 ​4-person boat
Question content area bottom
Part 1
​(A) Find the labor costs for a​ one-person boat manufactured at the State A plant.


We are tasked with finding the labor costs for a one-person boat manufactured at the State A plant. 

Here are the matrices:

- The **Hourly Wages matrix** for State A (in USD) is:
  - Cutting Department: \$17.56
  - Assembly Department: \$14.22
  - Packaging Department: \$9.57

- The **Labor-Hours Per Boat matrix** for a one-person boat (for each department) is:
  - Cutting Department: 0.5 hours
  - Assembly Department: 1.4 hours
  - Packaging Department: 1.7 hours

To calculate the total labor cost for a one-person boat, we need to multiply the number of hours required in each department by the corresponding hourly wage, then sum the costs.

### Calculation:
1. **Cutting Department:**
   - Hours: $0.5$
   - Hourly Wage: \$17.56
   \[
   \text{Cost from Cutting Department} = 0.5 \times 17.56 = 8.78
   \]

2. **Assembly Department:**
   - Hours: $1.4$
   - Hourly Wage: \$14.22
   \[
   \text{Cost from Assembly Department} = 1.4 \times 14.22 = 19.908
   \]

3. **Packaging Department:**
   - Hours: $1.7$
   - Hourly Wage: \$9.57
   \[
   \text{Cost from Packaging Department} = 1.7 \times 9.57 = 16.269
   \]

### Total Labor Cost:
Now, summing up the costs from all departments:
\[
\text{Total Labor Cost} = 8.78 + 19.908 + 16.269 = 44.957
\]

Thus, the labor cost for a one-person boat manufactured at the State A plant is approximately **\$44.96**.

Would you like a more detailed explanation or have any questions?

Here are 5 relative questions:
1. What would be the labor cost for a two-person boat at the State A plant?
2. How does the labor cost for a one-person boat differ between State A and State B?
3. What is the total labor cost for a four-person boat manufactured at the State A plant?
4. How would changing hourly wages affect the total cost for a one-person boat?
5. How can you compute the labor costs for multiple boats using matrix multiplication?

**Tip:** Understanding how to break down the problem into smaller parts can make it easier to solve complex matrix-based questions.

A company with manufacturing plants located in State A and State B has labor-hour and wage requirements for the manufacture of three types of inflatable boats as given in the two matrices below. Complete parts (A) through (D). Part 1: (A) Find the labor costs for a one-person boat manufactured at the State A plant.

Calculate the labor cost for manufacturing a one-person boat at the State A plant. This problem involves matrix multiplication of labor-hour and wage data across three departments (Cutting, Assembly, Packaging) to find the total cost. Ideal for college-level students studying applied mathematics or economics.

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