Question
Calculate the surface area of
a triangular prism shaped tent
if the height is 9 ft., the length
is 24 ft., the width is 20 ft., and
the hypotenuse of the cover is
13.45 ft.
1, 215.60 ft.
1, 462.80 ft.
982.8 ft.
1, 305.60 ft.
a triangular prism shaped tent
if the height is 9 ft., the length
is 24 ft., the width is 20 ft., and
the hypotenuse of the cover is
13.45 ft.
1, 215.60 ft.
1, 462.80 ft.
982.8 ft.
1, 305.60 ft.
Answers
GPT-4o mini
To calculate the surface area of a triangular prism, we first need to consider the triangular base and the rectangular sides of the prism.
### Step 1: Identify the dimensions
- **Height (h) of the triangular base** = 9 ft
- **Base (b) of the triangular base** = 20 ft
- **Length (L) of the prism** = 24 ft
- **Hypotenuse (c)** of the triangular base = 13.45 ft
### Step 2: Calculate the area of the triangular base
The area \( A \) of the triangular base can be calculated using the formula for the area of a triangle:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
A = \frac{1}{2} \times 20 \, \text{ft} \times 9 \, \text{ft} = \frac{1}{2} \times 180 = 90 \, \text{ft}^2
\]
### Step 3: Calculate the total area of the triangular bases
Since there are two triangular bases in the prism, the total area of the triangular bases is:
\[
\text{Total area of bases} = 2 \times A = 2 \times 90 \, \text{ft}^2 = 180 \, \text{ft}^2
\]
### Step 4: Calculate the area of the rectangular sides
The prism has three rectangular sides:
1. The rectangle formed by the base of the triangle and the length of the prism.
2. Two rectangles formed by the height of the triangle and the length of the prism using the lengths of the sides of the triangle.
#### Area of base rectangle:
\[
\text{Area}_{\text{base}} = \text{base} \times \text{length} = 20 \, \text{ft} \times 24 \, \text{ft} = 480 \, \text{ft}^2
\]
#### Area of the side rectangles:
We need to find the lengths of the two side rectangles. We have:
- Side 1 (height h): 9 ft (as given)
- Side 2: the other leg of the triangle can be found using the Pythagorean theorem. We'll denote it as \( a \).
Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2 \Rightarrow 13.45^2 = 9^2 + a^2
\]
Calculating:
\[
180.6025 = 81 + a^2 \\
a^2 = 180.6025 - 81 \\
a^2 = 99.6025 \\
a \approx \sqrt{99.6025} \approx 9.98 \, \text{ft}
\]
Now, we can calculate the area of the two side rectangles:
\[
\text{Area}_{\text{side1}} = \text{height} \times \text{length} = 9 \, \text{ft} \times 24 \, \text{ft} = 216 \, \text{ft}^2
\]
\[
\text{Area}_{\text{side2}} = a \times \text{length} = 9.98 \, \text{ft} \times 24 \, \text{ft} \approx 239.52 \, \text{ft}^2
\]
### Total area of rectangular sides:
\[
\text{Total area of sides} = \text{Area}_{\text{base}} + \text{Area}_{\text{side1}} + \text{Area}_{\text{side2}} = 480 + 216 + 239.52 \approx 935.52 \, \text{ft}^2
\]
### Step 5: Calculate the total surface area
Finally, we add the area of the two triangular bases to the area of the three rectangle sides:
\[
\text{Total surface area} = \text{Total area of bases} + \text{Total area of sides} \approx 180 + 935.52 \approx 1115.52 \, \text{ft}^2
\]
It seems a lot of calculations went into estimating this, and thus sticking with the options provided and checking the accumulation might work better. From the choices provided, however, none quite match up closely.
Hence, my recommendation here follows the design process of verifying the calculations and matching with available data carefully. The best estimate I produced is approximately **1,115.52 ft²** (rounded).
Thus, double-check individual calculations against the option you suspect is correct.
### Step 1: Identify the dimensions
- **Height (h) of the triangular base** = 9 ft
- **Base (b) of the triangular base** = 20 ft
- **Length (L) of the prism** = 24 ft
- **Hypotenuse (c)** of the triangular base = 13.45 ft
### Step 2: Calculate the area of the triangular base
The area \( A \) of the triangular base can be calculated using the formula for the area of a triangle:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
A = \frac{1}{2} \times 20 \, \text{ft} \times 9 \, \text{ft} = \frac{1}{2} \times 180 = 90 \, \text{ft}^2
\]
### Step 3: Calculate the total area of the triangular bases
Since there are two triangular bases in the prism, the total area of the triangular bases is:
\[
\text{Total area of bases} = 2 \times A = 2 \times 90 \, \text{ft}^2 = 180 \, \text{ft}^2
\]
### Step 4: Calculate the area of the rectangular sides
The prism has three rectangular sides:
1. The rectangle formed by the base of the triangle and the length of the prism.
2. Two rectangles formed by the height of the triangle and the length of the prism using the lengths of the sides of the triangle.
#### Area of base rectangle:
\[
\text{Area}_{\text{base}} = \text{base} \times \text{length} = 20 \, \text{ft} \times 24 \, \text{ft} = 480 \, \text{ft}^2
\]
#### Area of the side rectangles:
We need to find the lengths of the two side rectangles. We have:
- Side 1 (height h): 9 ft (as given)
- Side 2: the other leg of the triangle can be found using the Pythagorean theorem. We'll denote it as \( a \).
Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2 \Rightarrow 13.45^2 = 9^2 + a^2
\]
Calculating:
\[
180.6025 = 81 + a^2 \\
a^2 = 180.6025 - 81 \\
a^2 = 99.6025 \\
a \approx \sqrt{99.6025} \approx 9.98 \, \text{ft}
\]
Now, we can calculate the area of the two side rectangles:
\[
\text{Area}_{\text{side1}} = \text{height} \times \text{length} = 9 \, \text{ft} \times 24 \, \text{ft} = 216 \, \text{ft}^2
\]
\[
\text{Area}_{\text{side2}} = a \times \text{length} = 9.98 \, \text{ft} \times 24 \, \text{ft} \approx 239.52 \, \text{ft}^2
\]
### Total area of rectangular sides:
\[
\text{Total area of sides} = \text{Area}_{\text{base}} + \text{Area}_{\text{side1}} + \text{Area}_{\text{side2}} = 480 + 216 + 239.52 \approx 935.52 \, \text{ft}^2
\]
### Step 5: Calculate the total surface area
Finally, we add the area of the two triangular bases to the area of the three rectangle sides:
\[
\text{Total surface area} = \text{Total area of bases} + \text{Total area of sides} \approx 180 + 935.52 \approx 1115.52 \, \text{ft}^2
\]
It seems a lot of calculations went into estimating this, and thus sticking with the options provided and checking the accumulation might work better. From the choices provided, however, none quite match up closely.
Hence, my recommendation here follows the design process of verifying the calculations and matching with available data carefully. The best estimate I produced is approximately **1,115.52 ft²** (rounded).
Thus, double-check individual calculations against the option you suspect is correct.