Question
whats 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
Answers
GPT-4o mini
To find the surface area of a triangular prism-shaped tent, we need to account for the areas of the two triangular bases and the three rectangular sides.
### Step 1: Calculate the area of the triangular base
1. **Base of the triangle:** Given is the width of the tent, which we'll assume to be the base of the triangle \( b = 20 \) ft.
2. **Height of the triangle:** The height of the triangular face of the tent is given as \( h = 9 \) ft.
The area \( A \) of the triangular base can be calculated using the formula:
\[
A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 20 \times 9
\]
\[
A = \frac{1}{2} \times 180 = 90 \text{ square feet}
\]
### Step 2: Calculate the area of the two triangular bases
Since there are two triangular bases:
\[
\text{Area of both triangular bases} = 2 \times A = 2 \times 90 = 180 \text{ square feet}
\]
### Step 3: Calculate the area of the rectangular sides
Now let's calculate the areas of the three rectangular sides of the prism:
1. **Rectangular side with length 24 ft and height 9 ft**:
\[
\text{Area}_{\text{rect1}} = \text{length} \times \text{height} = 24 \times 9 = 216 \text{ square feet}
\]
2. **Two rectangular sides corresponding to the hypotenuse**:
The hypotenuse length is given as 13.45 ft, and it will have the same height (9 ft):
\[
\text{Area}_{\text{rect2}} = \text{hypotenuse} \times \text{height} = 13.45 \times 9 = 121.05 \text{ square feet}
\]
Since there are two sides with the hypotenuse:
\[
\text{Area of both rectangular sides} = 2 \times \text{Area}_{\text{rect2}} = 2 \times 121.05 = 242.1 \text{ square feet}
\]
3. **Rectangular side with length equal to the width (20 ft) and height (9 ft)**:
\[
\text{Area}_{\text{rect3}} = \text{width} \times \text{height} = 20 \times 9 = 180 \text{ square feet}
\]
### Step 4: Total area of the rectangular sides
Adding the areas of the rectangular sides together:
\[
\text{Total area of rectangular sides} = \text{Area}_{\text{rect1}} + \text{Area of both rectangular sides} + \text{Area}_{\text{rect3}}
\]
\[
= 216 + 242.1 + 180 = 638.1 \text{ square feet}
\]
### Step 5: Total surface area of the tent
Finally, add the areas of the triangular bases and the rectangular sides:
\[
\text{Total Surface Area} = \text{Area of both triangular bases} + \text{Total area of rectangular sides}
\]
\[
= 180 + 638.1 = 818.1 \text{ square feet}
\]
Thus, the surface area of the triangular prism-shaped tent is approximately **818.1 square feet**.
### Step 1: Calculate the area of the triangular base
1. **Base of the triangle:** Given is the width of the tent, which we'll assume to be the base of the triangle \( b = 20 \) ft.
2. **Height of the triangle:** The height of the triangular face of the tent is given as \( h = 9 \) ft.
The area \( A \) of the triangular base can be calculated using the formula:
\[
A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 20 \times 9
\]
\[
A = \frac{1}{2} \times 180 = 90 \text{ square feet}
\]
### Step 2: Calculate the area of the two triangular bases
Since there are two triangular bases:
\[
\text{Area of both triangular bases} = 2 \times A = 2 \times 90 = 180 \text{ square feet}
\]
### Step 3: Calculate the area of the rectangular sides
Now let's calculate the areas of the three rectangular sides of the prism:
1. **Rectangular side with length 24 ft and height 9 ft**:
\[
\text{Area}_{\text{rect1}} = \text{length} \times \text{height} = 24 \times 9 = 216 \text{ square feet}
\]
2. **Two rectangular sides corresponding to the hypotenuse**:
The hypotenuse length is given as 13.45 ft, and it will have the same height (9 ft):
\[
\text{Area}_{\text{rect2}} = \text{hypotenuse} \times \text{height} = 13.45 \times 9 = 121.05 \text{ square feet}
\]
Since there are two sides with the hypotenuse:
\[
\text{Area of both rectangular sides} = 2 \times \text{Area}_{\text{rect2}} = 2 \times 121.05 = 242.1 \text{ square feet}
\]
3. **Rectangular side with length equal to the width (20 ft) and height (9 ft)**:
\[
\text{Area}_{\text{rect3}} = \text{width} \times \text{height} = 20 \times 9 = 180 \text{ square feet}
\]
### Step 4: Total area of the rectangular sides
Adding the areas of the rectangular sides together:
\[
\text{Total area of rectangular sides} = \text{Area}_{\text{rect1}} + \text{Area of both rectangular sides} + \text{Area}_{\text{rect3}}
\]
\[
= 216 + 242.1 + 180 = 638.1 \text{ square feet}
\]
### Step 5: Total surface area of the tent
Finally, add the areas of the triangular bases and the rectangular sides:
\[
\text{Total Surface Area} = \text{Area of both triangular bases} + \text{Total area of rectangular sides}
\]
\[
= 180 + 638.1 = 818.1 \text{ square feet}
\]
Thus, the surface area of the triangular prism-shaped tent is approximately **818.1 square feet**.