Question

A rectangular pyramid has a length of 7 cm and a width of 9 cm. Its volume is 231 cubic centimeters. Use the formula for the volume of a pyramid to calculate the height of this pyramid. Show all your work.

40 words limit

Answer 1

Where did 231 and 693 come from?

Answer 2

The volume \( V \) of a pyramid is given by the formula:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

For a rectangular pyramid:

\[
\text{Base Area} = \text{Length} \times \text{Width} = 7 \, \text{cm} \times 9 \, \text{cm} = 63 \, \text{cm}^2
\]

Substituting into the volume formula:

\[
231 = \frac{1}{3} \times 63 \times h
\]

Multiplying both sides by 3:

\[
693 = 63h
\]

Solving for \( h \):

\[
h = \frac{693}{63} = 11 \, \text{cm}
\]

Thus, the height of the pyramid is **11 cm**.

Answer 3

Apologies for any confusion. Here’s a clearer breakdown:

1. **Given Information**:
- Length of the pyramid, \( l = 7 \, \text{cm} \)
- Width of the pyramid, \( w = 9 \, \text{cm} \)
- Volume of the pyramid, \( V = 231 \, \text{cm}^3 \)

2. **Base Area Calculation**:
\[
\text{Base Area} = l \times w = 7 \, \text{cm} \times 9 \, \text{cm} = 63 \, \text{cm}^2
\]

3. **Volume Formula**:
The volume \( V \) of a pyramid is calculated using:
\[
V = \frac{1}{3} \times \text{Base Area} \times h
\]

4. **Substituting in Values**:
\[
231 = \frac{1}{3} \times 63 \times h
\]

5. **Multiplying Both Sides by 3**:
This step eliminates the fraction:
\[
3 \times 231 = 63h \implies 693 = 63h
\]

6. **Solving for Height \( h \)**:
To find \( h \):
\[
h = \frac{693}{63} = 11 \, \text{cm}
\]

Thus, the height of the pyramid is \( **11 \, \text{cm}**.