Question
Calculate the surface area of a triangular prism shaped tent if the height is 9 feet, the length is 24 feet, the width is 20 feet, and the hypotenuse of the cover is 13.45 feet
1,462.80 ft.
982.8 ft.
1,305.60 ft.
1,215.60 ft
1,462.80 ft.
982.8 ft.
1,305.60 ft.
1,215.60 ft
Answers
GPT-4o mini
To calculate the surface area of a triangular prism, we can use the following formula:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
Where:
- The Base Area of the triangular base can be found using the formula for the area of a triangle.
- The Lateral Area is the area of the three rectangular sides of the prism.
### Step 1: Calculate the Base Area
The base of the triangular side is a right triangle with a height and a base. Typically, we assume the width \(w = 20\) ft is the base of the triangle. The height \(h = 9\) ft can be taken as the height of the triangle.
The area \(A\) of the triangular base is calculated as follows:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 9 = 90 \text{ ft}^2
\]
### Step 2: Calculate the Lateral Area
The lateral area consists of three rectangles: two of them are formed by the triangle's sides and one is the base of the prism that connects two triangular bases.
1. **Two triangles (Sides)**:
- Each side of the prism (rectangles) will have a height equal to the length of the prism which is \(l = 24\) ft.
- Each of the two triangular sides will be \( (Hypotenuse \times length) \)
- Lateral area from the two triangular sides:
The lateral area \(L_A\) will include:
\[
L_A = \text{height} \times \text{length} + \text{hypotenuse} \times \text{length}
\]
Because we can consider the hypotenuse as one side of the triangle, here we have:
\[
L_A = (20 \times 24) + (13.45 \times 24) + (9 \times 24)
\]
- \(L_A = (20 \times 24) + (13.45 \times 24) + (9 \times 24)\)
- \( (480) + (322.8) + (216) = 1018.8 \)
### Step 3: Combining All Areas
Now, combining the base area and the lateral area gives us the total surface area:
\[
\text{Total Surface Area} = 2 \times A + L_A
\]
\[
\text{Total Surface Area} = 2 \times 90 + 1018.8 = 180 + 1018.8 = 1198.8
\]
Since most options given were much larger, this may indicate a misunderstanding.
The final calculated *surface area* is just over 1198.8 square feet.
Now, if we take into account if there's any adjustment we need to make based on some other aspect but since none of these match entirely the options you mentioned, possibly the calculation hints for the dimensions or the approach might need to be updated.
#### Final result may need adjustments.
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
Where:
- The Base Area of the triangular base can be found using the formula for the area of a triangle.
- The Lateral Area is the area of the three rectangular sides of the prism.
### Step 1: Calculate the Base Area
The base of the triangular side is a right triangle with a height and a base. Typically, we assume the width \(w = 20\) ft is the base of the triangle. The height \(h = 9\) ft can be taken as the height of the triangle.
The area \(A\) of the triangular base is calculated as follows:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 9 = 90 \text{ ft}^2
\]
### Step 2: Calculate the Lateral Area
The lateral area consists of three rectangles: two of them are formed by the triangle's sides and one is the base of the prism that connects two triangular bases.
1. **Two triangles (Sides)**:
- Each side of the prism (rectangles) will have a height equal to the length of the prism which is \(l = 24\) ft.
- Each of the two triangular sides will be \( (Hypotenuse \times length) \)
- Lateral area from the two triangular sides:
The lateral area \(L_A\) will include:
\[
L_A = \text{height} \times \text{length} + \text{hypotenuse} \times \text{length}
\]
Because we can consider the hypotenuse as one side of the triangle, here we have:
\[
L_A = (20 \times 24) + (13.45 \times 24) + (9 \times 24)
\]
- \(L_A = (20 \times 24) + (13.45 \times 24) + (9 \times 24)\)
- \( (480) + (322.8) + (216) = 1018.8 \)
### Step 3: Combining All Areas
Now, combining the base area and the lateral area gives us the total surface area:
\[
\text{Total Surface Area} = 2 \times A + L_A
\]
\[
\text{Total Surface Area} = 2 \times 90 + 1018.8 = 180 + 1018.8 = 1198.8
\]
Since most options given were much larger, this may indicate a misunderstanding.
The final calculated *surface area* is just over 1198.8 square feet.
Now, if we take into account if there's any adjustment we need to make based on some other aspect but since none of these match entirely the options you mentioned, possibly the calculation hints for the dimensions or the approach might need to be updated.
#### Final result may need adjustments.