Question
Fatima measures this square-based pyramid. She finds that its volume is 52 cubic feet. What is the height of Fatima’s pyramid? (1 point) Responses 10 ft. 10 ft. 12 ft. 12 ft. 4 ft. 4 ft. 225 ft.
Answers
GPT-4o mini
To find the height of a square-based pyramid given its volume, we can use the formula for the volume of a pyramid:
\[
V = \frac{1}{3} \times B \times h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height.
Since the base is a square, if we let \( s \) be the length of a side of the square base, then the area of the base is:
\[
B = s^2
\]
Substituting this into the volume formula, we get:
\[
V = \frac{1}{3} \times s^2 \times h
\]
We are given that the volume \( V = 52 \) cubic feet. Therefore, we can write:
\[
52 = \frac{1}{3} \times s^2 \times h
\]
We need more information to find the exact values for \( s \) and \( h \) since we have two variables. However, let's examine the response options given:
1. 10 ft
2. 12 ft
3. 4 ft
4. 225 ft
We can arbitrarily test each height option one by one to see if it can yield a square base area that results in a volume of 52 cubic feet.
**1. Testing \( h = 10 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 10
\]
\[
52 = \frac{10}{3} s^2
\]
\[
s^2 = \frac{52 \cdot 3}{10} = \frac{156}{10} = 15.6
\]
Since \( s^2 \) is a valid area, this option could potentially work.
---
**2. Testing \( h = 12 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 12
\]
\[
52 = 4 s^2
\]
\[
s^2 = \frac{52}{4} = 13
\]
This is also a valid area, so this option could also work.
---
**3. Testing \( h = 4 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 4
\]
\[
52 = \frac{4}{3} s^2
\]
\[
s^2 = \frac{52 \cdot 3}{4} = \frac{156}{4} = 39
\]
This area is also valid.
---
**4. Testing \( h = 225 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 225
\]
\[
52 = 75 s^2
\]
\[
s^2 = \frac{52}{75} \approx 0.6933
\]
This area is valid as well but yields a much larger height compared to the other options.
---
Since all tested heights (10 ft, 12 ft, and 4 ft) produced valid areas, but we typically seek the height that seems reasonable considering the volume, the most plausible answer is not 225 ft. The most realistic height seems to be **10 ft** based on the area values we processed.
Therefore, one of the answers could be \( \text{10 ft} \). Still, \( 4 ft \) or \( 12 ft \) can also yield feasible pyramid solutions. Ultimately, for typical pyramid dimensions and practicality for the volume given, the most reasonable answer is likely \( \text{10 ft} \).
\[
V = \frac{1}{3} \times B \times h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height.
Since the base is a square, if we let \( s \) be the length of a side of the square base, then the area of the base is:
\[
B = s^2
\]
Substituting this into the volume formula, we get:
\[
V = \frac{1}{3} \times s^2 \times h
\]
We are given that the volume \( V = 52 \) cubic feet. Therefore, we can write:
\[
52 = \frac{1}{3} \times s^2 \times h
\]
We need more information to find the exact values for \( s \) and \( h \) since we have two variables. However, let's examine the response options given:
1. 10 ft
2. 12 ft
3. 4 ft
4. 225 ft
We can arbitrarily test each height option one by one to see if it can yield a square base area that results in a volume of 52 cubic feet.
**1. Testing \( h = 10 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 10
\]
\[
52 = \frac{10}{3} s^2
\]
\[
s^2 = \frac{52 \cdot 3}{10} = \frac{156}{10} = 15.6
\]
Since \( s^2 \) is a valid area, this option could potentially work.
---
**2. Testing \( h = 12 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 12
\]
\[
52 = 4 s^2
\]
\[
s^2 = \frac{52}{4} = 13
\]
This is also a valid area, so this option could also work.
---
**3. Testing \( h = 4 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 4
\]
\[
52 = \frac{4}{3} s^2
\]
\[
s^2 = \frac{52 \cdot 3}{4} = \frac{156}{4} = 39
\]
This area is also valid.
---
**4. Testing \( h = 225 \) ft:**
\[
52 = \frac{1}{3} \times s^2 \times 225
\]
\[
52 = 75 s^2
\]
\[
s^2 = \frac{52}{75} \approx 0.6933
\]
This area is valid as well but yields a much larger height compared to the other options.
---
Since all tested heights (10 ft, 12 ft, and 4 ft) produced valid areas, but we typically seek the height that seems reasonable considering the volume, the most plausible answer is not 225 ft. The most realistic height seems to be **10 ft** based on the area values we processed.
Therefore, one of the answers could be \( \text{10 ft} \). Still, \( 4 ft \) or \( 12 ft \) can also yield feasible pyramid solutions. Ultimately, for typical pyramid dimensions and practicality for the volume given, the most reasonable answer is likely \( \text{10 ft} \).