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Part 1
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
17.54
18.11
 Cutting Dept
0.6
0.2
0.8
 ​1-person boat
Nequals
12.91
17.92
 Assembly Dept
Mequals
1.3
0.8
1.3
 ​2-person boat
8.81
10.02
 Packaging Dept
1.4
2.1
1.6
 ​4-person boat
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Part 1
​(A) Find the labor costs for a​ one-person boat manufactured at the State A plant.
Labor costsequals​$
  
enter your response here ​(Round to the nearest cent as​ needed.)

To find the labor cost for a one-person boat manufactured at the State A plant, we need to calculate the total labor cost by multiplying the labor hours required in each department by the corresponding hourly wage for that department.

### Step-by-Step Solution:

1. **Identify the Data:**
   - **Hourly wages in State A:**
     - Cutting Dept: $17.54
     - Assembly Dept: $12.91
     - Packaging Dept: $8.81
   - **Labor hours required per one-person boat:**
     - Cutting Dept: 0.6 hours
     - Assembly Dept: 1.3 hours
     - Packaging Dept: 1.4 hours

2. **Calculate the Cost for Each Department:**
   - Cutting Dept: $0.6 \, \text{hours} \times 17.54 \, \text{\$/hour} = 10.524 \, \text{\$}$
   - Assembly Dept: $1.3 \, \text{hours} \times 12.91 \, \text{\$/hour} = 16.783 \, \text{\$}$
   - Packaging Dept: $1.4 \, \text{hours} \times 8.81 \, \text{\$/hour} = 12.334 \, \text{\$}$

3. **Sum the Costs to Get the Total Labor Cost:**
   \[
   \text{Total Labor Cost} = 10.524 + 16.783 + 12.334
   \]

Let's calculate the total cost.The labor cost for a one-person boat manufactured at the State A plant is **$39.64**.

Would you like further details or have any questions about this calculation?

### Related Questions:
1. How do you calculate the labor costs for a two-person boat in State A?
2. What is the total labor cost for manufacturing a four-person boat in State B?
3. How would the labor costs change if the wages increased by 10% in State A?
4. Can you calculate the labor costs for manufacturing all types of boats in State B?
5. What is the impact on total labor costs if labor hours are reduced by 20% for the packaging department?

**Tip:** Understanding how to multiply matrices can help in solving more complex problems involving labor costs across multiple plants.

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). (A) Find the labor costs for a one-person boat manufactured at the State A plant. Labor costs = $ (Round to the nearest cent as needed.)

Calculate the labor cost for manufacturing a one-person boat at the State A plant. This problem involves matrix multiplication to determine the total cost based on hourly wages and labor hours required for cutting, assembly, and packaging departments. Suitable for students in Grades 9-12.

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