Question

Use the image to answer the question.

An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side of the triangle measures 10 centimeters. The right side of the triangle also measures 10 centimeters. The perpendicular height from the top vertex of the triangle to the base measures 6 centimeters. The base of the triangle measures 8 centimeters.

What is the missing length l
of the triangular prism if the surface area is 524 square centimeters?

(1 point)
Responses

8 cm
8 cm

16 cm
16 cm

17 cm
17 cm

10 cm

Answer 1

To find the missing length \( l \) of the triangular prism given that the surface area is 524 square centimeters, we can start by calculating the surface area of the triangular prism.

1. **Surface Area of the Top and Bottom Triangular Faces**:
- The area \( A \) of the triangular face can be determined using the formula for the area of a triangle:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
- The base is 8 cm and the height is 6 cm:
\[
A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2
\]
- Since there are two triangular faces, their combined area is:
\[
2A = 2 \times 24 = 48 \text{ cm}^2
\]

2. **Surface Area of the Rectangular Faces**:
- The prism has three rectangular faces. Let's calculate the area of each rectangular face:
- The first rectangular face (base × height) is:
\[
8 \times l
\]
- The second rectangular face (10 cm × l) represents the left side:
\[
10 \times l
\]
- The third rectangular face (10 cm × l) represents the right side:
\[
10 \times l
\]
- Thus, the total area of the rectangular faces combined is:
\[
8l + 10l + 10l = 28l
\]

3. **Total Surface Area Calculation**:
- The total surface area \( S \) of the prism can thus be calculated as:
\[
S = \text{Area of triangular faces} + \text{Area of rectangular faces}
\]
- Substituting our expressions from above:
\[
S = 48 + 28l
\]
- We know the total surface area is 524 cm²:
\[
48 + 28l = 524
\]

4. **Solving for \( l \)**:
- Subtract 48 from both sides:
\[
28l = 524 - 48
\]
\[
28l = 476
\]
- Divide both sides by 28:
\[
l = \frac{476}{28} \approx 17
\]

Thus, the missing length \( l \) of the triangular prism is **17 cm**.

**The answer is: 17 cm.**