Question
The area of square-based pyramid is 50 square yards. The volume of the pyramid is 250 cubic yards. What is the height of the pyramid?
The height of the pyramid is __ yards.
The height of the pyramid is __ yards.
Answers
GPT-4o mini
To find the height of the square-based pyramid, we need to use the formulas for the area and the volume of a pyramid.
1. The area \( A \) of the base of a square-based pyramid is given as 50 square yards. For a square base with side length \( s \), the area can be expressed as:
\[
A = s^2
\]
Therefore, we have:
\[
s^2 = 50 \quad \Rightarrow \quad s = \sqrt{50} = 5\sqrt{2} \text{ yards}
\]
2. The volume \( V \) of the pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times A \times h
\]
Substituting in the known values:
\[
250 = \frac{1}{3} \times 50 \times h
\]
3. Simplifying this equation:
\[
250 = \frac{50h}{3} \quad \Rightarrow \quad 250 \times 3 = 50h \quad \Rightarrow \quad 750 = 50h
\]
4. Solving for \( h \):
\[
h = \frac{750}{50} = 15 \text{ yards}
\]
Thus, the height of the pyramid is \( \boxed{15} \) yards.
1. The area \( A \) of the base of a square-based pyramid is given as 50 square yards. For a square base with side length \( s \), the area can be expressed as:
\[
A = s^2
\]
Therefore, we have:
\[
s^2 = 50 \quad \Rightarrow \quad s = \sqrt{50} = 5\sqrt{2} \text{ yards}
\]
2. The volume \( V \) of the pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times A \times h
\]
Substituting in the known values:
\[
250 = \frac{1}{3} \times 50 \times h
\]
3. Simplifying this equation:
\[
250 = \frac{50h}{3} \quad \Rightarrow \quad 250 \times 3 = 50h \quad \Rightarrow \quad 750 = 50h
\]
4. Solving for \( h \):
\[
h = \frac{750}{50} = 15 \text{ yards}
\]
Thus, the height of the pyramid is \( \boxed{15} \) yards.