Question
What is the missing length l
of the triangular prism if the surface area is 524 square centimeters? The triangles have a base of 8 and a height of 6. The rectangles are 10
(1 point)
Responses
16 cm
16 cm
8 cm
8 cm
17 cm
17 cm
10 cm
10 cm
of the triangular prism if the surface area is 524 square centimeters? The triangles have a base of 8 and a height of 6. The rectangles are 10
(1 point)
Responses
16 cm
16 cm
8 cm
8 cm
17 cm
17 cm
10 cm
10 cm
Answers
Answered by
GPT-4o mini
To find the missing length \( l \) of the triangular prism, we first need to calculate the surface area of the prism.
1. **Calculate the area of the triangular base:**
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base is 8 cm and the height is 6 cm. So,
\[
A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2
\]
2. **Calculate the total area of the two triangular bases:**
Since there are two triangular bases in a prism, their combined area is:
\[
\text{Total area of triangles} = 2 \times 24 = 48 \text{ cm}^2
\]
3. **Calculate the area of the three rectangular sides:**
- The first rectangular side (length \( l \), height from triangle) has an area:
\[
\text{Area} = l \times 8
\]
- The second rectangular side also has an area:
\[
\text{Area} = l \times 6
\]
- The third rectangular side has a base of the prism, which is the length given as 10 cm:
\[
\text{Area} = 10 \times l
\]
4. **Combine the areas to find the total surface area:**
The total surface area \( S \) of the prism is the sum of the areas of the two triangular bases and the three rectangular faces:
\[
S = \text{Area of triangles} + \text{Area of rectangles}
\]
Substituting in the areas calculated:
\[
S = 48 + (l \times 8) + (l \times 6) + (10 \times l)
\]
This simplifies to:
\[
S = 48 + 24l
\]
5. **Set the total surface area equal to the given surface area:**
We are given that the total surface area is 524 cm\(^2\):
\[
24l + 48 = 524
\]
6. **Solve for \( l \):**
First, subtract 48 from both sides of the equation:
\[
24l = 524 - 48
\]
\[
24l = 476
\]
Now, divide both sides by 24:
\[
l = \frac{476}{24}
\]
\[
l = 19.8333 \text{ cm}
\]
None of the given responses (16, 8, 17, or 10 cm) is correct for this calculation. Therefore, the answer is not among the options provided if the calculations are correct.
1. **Calculate the area of the triangular base:**
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base is 8 cm and the height is 6 cm. So,
\[
A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2
\]
2. **Calculate the total area of the two triangular bases:**
Since there are two triangular bases in a prism, their combined area is:
\[
\text{Total area of triangles} = 2 \times 24 = 48 \text{ cm}^2
\]
3. **Calculate the area of the three rectangular sides:**
- The first rectangular side (length \( l \), height from triangle) has an area:
\[
\text{Area} = l \times 8
\]
- The second rectangular side also has an area:
\[
\text{Area} = l \times 6
\]
- The third rectangular side has a base of the prism, which is the length given as 10 cm:
\[
\text{Area} = 10 \times l
\]
4. **Combine the areas to find the total surface area:**
The total surface area \( S \) of the prism is the sum of the areas of the two triangular bases and the three rectangular faces:
\[
S = \text{Area of triangles} + \text{Area of rectangles}
\]
Substituting in the areas calculated:
\[
S = 48 + (l \times 8) + (l \times 6) + (10 \times l)
\]
This simplifies to:
\[
S = 48 + 24l
\]
5. **Set the total surface area equal to the given surface area:**
We are given that the total surface area is 524 cm\(^2\):
\[
24l + 48 = 524
\]
6. **Solve for \( l \):**
First, subtract 48 from both sides of the equation:
\[
24l = 524 - 48
\]
\[
24l = 476
\]
Now, divide both sides by 24:
\[
l = \frac{476}{24}
\]
\[
l = 19.8333 \text{ cm}
\]
None of the given responses (16, 8, 17, or 10 cm) is correct for this calculation. Therefore, the answer is not among the options provided if the calculations are correct.