Question
The roof of a house is in the form of a square pyramid. If its volume is 225 cubic feet and the base edge is 6 feet, how high is the roof?
Answers
GPT-4o mini
To find the height of the square pyramid roof, we can use the formula for the volume of a pyramid, which is given by:
\[
V = \frac{1}{3} B h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height of the pyramid.
For a square pyramid, the area of the base \( B \) can be calculated as:
\[
B = \text{(base edge)}^2
\]
Given that the base edge is 6 feet, we can calculate \( B \):
\[
B = 6^2 = 36 \text{ square feet}
\]
We know the volume \( V \) of the pyramid is 225 cubic feet. We can now substitute the values of \( V \) and \( B \) into the volume formula to find \( h \):
\[
225 = \frac{1}{3} (36) h
\]
To eliminate the fraction, we can multiply both sides by 3:
\[
675 = 36h
\]
Now, we can solve for \( h \) by dividing both sides by 36:
\[
h = \frac{675}{36}
\]
Calculating this gives:
\[
h = 18.75
\]
Thus, the height of the roof is:
\[
\boxed{18.75} \text{ feet}
\]
\[
V = \frac{1}{3} B h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height of the pyramid.
For a square pyramid, the area of the base \( B \) can be calculated as:
\[
B = \text{(base edge)}^2
\]
Given that the base edge is 6 feet, we can calculate \( B \):
\[
B = 6^2 = 36 \text{ square feet}
\]
We know the volume \( V \) of the pyramid is 225 cubic feet. We can now substitute the values of \( V \) and \( B \) into the volume formula to find \( h \):
\[
225 = \frac{1}{3} (36) h
\]
To eliminate the fraction, we can multiply both sides by 3:
\[
675 = 36h
\]
Now, we can solve for \( h \) by dividing both sides by 36:
\[
h = \frac{675}{36}
\]
Calculating this gives:
\[
h = 18.75
\]
Thus, the height of the roof is:
\[
\boxed{18.75} \text{ feet}
\]