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.

Answer 1

To find the height of the rectangular pyramid, 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 of the pyramid.

### Step 1: Calculate the area of the base

The base of the rectangular pyramid is a rectangle with length (\( l \)) and width (\( w \)). The area (\( B \)) can be calculated using the formula:

\[
B = l \times w
\]

Given:
- Length (\( l \)) = 7 cm
- Width (\( w \)) = 9 cm

Calculating \( B \):

\[
B = 7 \, \text{cm} \times 9 \, \text{cm} = 63 \, \text{cm}^2
\]

### Step 2: Plug the values into the volume formula and solve for height

We know the volume \( V \) is 231 cubic centimeters. Now, we can use the volume formula:

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

### Step 3: Multiply both sides by 3 to eliminate the fraction

\[
3 \times 231 = 63 \times h
\]

Calculating the left side:

\[
693 = 63 \times h
\]

### Step 4: Solve for \( h \)

Now, we divide both sides by 63:

\[
h = \frac{693}{63}
\]

Calculating \( h \):

\[
h = 11 \, \text{cm}
\]

### Conclusion

The height of the pyramid is \( 11 \, \text{cm} \).