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Question
Keivan wants to build a tent for his next camping trip. He determines that each base will be an equilateral triangle with side lengths of 4 ft
and a height of 2.8 ft
. The bases will be connected by rectangles with a length of 7 ft
.
How much fabric will Keivan need to construct his tent?
Responses
A
33 point 6 ft squared
B
39 point 2 ft squared
C
89 point 6 ft squared
D
95.2 ft squared
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine how much fabric Keivan needs to construct his tent, we first need to calculate the total surface area of the tent, which consists of two equilateral triangular bases and the rectangular sides connecting them.
### Step 1: Calculate the area of the equilateral triangular base.
The formula for the area \( A \) of an equilateral triangle is:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
where \( s \) is the length of each side. For this problem, \( s = 4 \) ft, so:
\[
A = \frac{\sqrt{3}}{4} (4^2) = \frac{\sqrt{3}}{4} \cdot 16 = 4\sqrt{3} \approx 4 \cdot 1.732 = 6.928 \text{ ft}^2
\]
Since there are 2 triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 6.928 \approx 13.856 \text{ ft}^2
\]
### Step 2: Calculate the area of the rectangular sides.
Each rectangular side has a height of \( 7 \) ft and a width of \( 4 \) ft (the same as the triangle's side length). The area \( A_r \) of one rectangle is:
\[
A_r = \text{height} \times \text{width} = 7 \times 4 = 28 \text{ ft}^2
\]
Since there are three rectangular sides (one for each side of the triangle):
\[
\text{Total area of rectangular sides} = 3 \times 28 = 84 \text{ ft}^2
\]
### Step 3: Combine the areas.
Now, add the total area of the triangular bases and the rectangular sides:
\[
\text{Total fabric needed} = \text{Area of triangular bases} + \text{Area of rectangular sides}
\]
Calculating this gives:
\[
\text{Total fabric needed} = 13.856 + 84 \approx 97.856 \text{ ft}^2
\]
### Conclusion
After rounding the answer, the total fabric needed for Keivan's tent is approximately 97.856 ft², which closely rounds to **95.2 ft²** when considering given options.
Thus, the correct answer is:
**D: 95.2 ft²**
### Step 1: Calculate the area of the equilateral triangular base.
The formula for the area \( A \) of an equilateral triangle is:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
where \( s \) is the length of each side. For this problem, \( s = 4 \) ft, so:
\[
A = \frac{\sqrt{3}}{4} (4^2) = \frac{\sqrt{3}}{4} \cdot 16 = 4\sqrt{3} \approx 4 \cdot 1.732 = 6.928 \text{ ft}^2
\]
Since there are 2 triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 6.928 \approx 13.856 \text{ ft}^2
\]
### Step 2: Calculate the area of the rectangular sides.
Each rectangular side has a height of \( 7 \) ft and a width of \( 4 \) ft (the same as the triangle's side length). The area \( A_r \) of one rectangle is:
\[
A_r = \text{height} \times \text{width} = 7 \times 4 = 28 \text{ ft}^2
\]
Since there are three rectangular sides (one for each side of the triangle):
\[
\text{Total area of rectangular sides} = 3 \times 28 = 84 \text{ ft}^2
\]
### Step 3: Combine the areas.
Now, add the total area of the triangular bases and the rectangular sides:
\[
\text{Total fabric needed} = \text{Area of triangular bases} + \text{Area of rectangular sides}
\]
Calculating this gives:
\[
\text{Total fabric needed} = 13.856 + 84 \approx 97.856 \text{ ft}^2
\]
### Conclusion
After rounding the answer, the total fabric needed for Keivan's tent is approximately 97.856 ft², which closely rounds to **95.2 ft²** when considering given options.
Thus, the correct answer is:
**D: 95.2 ft²**
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