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Use a two-dimensional model and the dimensions provided to calculate the perimeter and area of the sail. Round the area to the nearest tenth, if necessary.

To calculate the perimeter and area of the triangular sail based on the given dimensions, we can first convert all the measurements to the same unit (feet) and then apply the formulas for perimeter and area of a triangle.

### 1. **Conversion of measurements:**

The dimensions are provided in feet and inches, so we need to convert everything to feet:

- \( 14 \text{ ft } 7 \text{ in} \)
  - \( 7 \text{ in} = \frac{7}{12} \text{ ft} \approx 0.583 \text{ ft} \)
  - Total: \( 14 + 0.583 = 14.583 \text{ ft} \)

- \( 13 \text{ ft } 4 \text{ in} \)
  - \( 4 \text{ in} = \frac{4}{12} \text{ ft} \approx 0.333 \text{ ft} \)
  - Total: \( 13 + 0.333 = 13.333 \text{ ft} \)

- \( 11 \text{ ft } 2 \text{ in} \)
  - \( 2 \text{ in} = \frac{2}{12} \text{ ft} \approx 0.167 \text{ ft} \)
  - Total: \( 11 + 0.167 = 11.167 \text{ ft} \)

Now we have the side lengths of the triangle in feet as:
- \( a = 14.583 \text{ ft} \)
- \( b = 13.333 \text{ ft} \)
- \( c = 11.167 \text{ ft} \)

### 2. **Perimeter Calculation:**

The perimeter \( P \) of a triangle is the sum of all its sides:

\[
P = a + b + c = 14.583 + 13.333 + 11.167 = 39.083 \text{ ft}
\]

### 3. **Area Calculation (Heron's Formula):**

We can use **Heron's formula** to calculate the area of a triangle when all side lengths are known:

1. First, calculate the semi-perimeter \( s \):

\[
s = \frac{a + b + c}{2} = \frac{14.583 + 13.333 + 11.167}{2} = 19.541 \text{ ft}
\]

2. Now, apply Heron's formula for area \( A \):

\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Substitute the values:

\[
A = \sqrt{19.541(19.541 - 14.583)(19.541 - 13.333)(19.541 - 11.167)}
\]
\[
A = \sqrt{19.541 \times 4.958 \times 6.208 \times 8.374}
\]

Let me calculate this for you.The area of the sail is approximately \( 70.98 \, \text{square feet} \).

### Summary:
- **Perimeter**: \( 39.08 \, \text{feet} \)
- **Area**: \( 70.98 \, \text{square feet} \)

Would you like more details, or do you have any questions?

Here are some related questions to consider:
1. How does Heron's formula help in calculating the area of any triangle?
2. Can you use trigonometry to calculate the area if an angle is known?
3. What would happen to the perimeter and area if the dimensions were doubled?
4. How do you convert between square feet and other area measurements like square meters?
5. How can knowing the area help in practical sailing situations?

**Tip**: Always make sure to convert all units to the same system (e.g., feet, inches, or meters) before performing calculations!

Learn how to calculate the perimeter and area of a triangular sail using given dimensions and Heron's formula. This problem covers geometric concepts and unit conversion, ideal for Grades 9-12. Includes step-by-step explanations of key formulas.

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