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. Once answered then simplify

Answer 1

I said then simplify

Answer 2

I meant simplify the explanation

Answer 3

To find the height of the rectangular pyramid, we will use the formula for the volume of a pyramid, which is given by:

\[
V = \frac{1}{3} \times B \times h
\]

where:
- \( V \) is the volume of the pyramid,
- \( B \) is the area of the base, and
- \( h \) is the height of the pyramid.

For a rectangular base, the area \( B \) can be calculated as:

\[
B = \text{length} \times \text{width}
\]

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

Calculating the area \( B \):

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

Now, substituting \( B \) and \( V = 231 \, \text{cm}^3 \) into the volume formula:

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

To isolate \( h \), first, multiply both sides by 3:

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

\[
693 = 63h
\]

Next, divide both sides by 63 to solve for \( h \):

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

Now performing the division:

\[
h = 11
\]

Thus, the height \( h \) of the pyramid is:

\[
\boxed{11 \, \text{cm}}
\]

In conclusion, the height of the pyramid simplifies directly to 11 cm based on our calculations.

Answer 4

I apologize for the oversight regarding the instruction to simplify the solution. In this context, the solution is straightforward because we've arrived at a specific integer value for the height. However, let’s reflect on how we could summarize or present it more clearly:

1. **Calculate the area of the base**:
\[
B = 7 \, \text{cm} \times 9 \, \text{cm} = 63 \, \text{cm}^2
\]

2. **Use the volume formula**:
\[
V = \frac{1}{3} \times B \times h
\]
Plugging in the values:
\[
231 = \frac{1}{3} \times 63 \times h
\]

3. **Multiply by 3 to eliminate the fraction**:
\[
3 \times 231 = 63h
\]
\[
693 = 63h
\]

4. **Divide by 63 to find \( h \)**:
\[
h = \frac{693}{63}
\]

5. **Simplifying \( \frac{693}{63} \)**:
- Finding the greatest common divisor (GCD) of 693 and 63:
- Both 693 and 63 are divisible by 9.
\[
\frac{693 \div 9}{63 \div 9} = \frac{77}{7} = 11
\]

So, the simplified height of the pyramid is:

\[
\boxed{11 \, \text{cm}}
\]

The height remains \( 11 \text{ cm} \), which is already in its simplest form as a whole number. Thank you for your patience, and I hope this meets the requirement to "simplify"!

Answer 5

I apologize for misunderstanding your request! Here’s a simplified version of the explanation:

1. **Calculate the area of the base**:
\[
B = \text{length} \times \text{width} = 7 \, \text{cm} \times 9 \, \text{cm} = 63 \, \text{cm}^2
\]

2. **Use the volume formula**:
\[
V = \frac{1}{3} \times B \times h
\]
Substitute \( V = 231 \, \text{cm}^3 \) and \( B = 63 \, \text{cm}^2 \):
\[
231 = \frac{1}{3} \times 63 \times h
\]

3. **Multiply by 3** to eliminate the fraction:
\[
693 = 63h
\]

4. **Divide by 63** to find \( h \):
\[
h = \frac{693}{63} = 11 \, \text{cm}
\]

Thus, the height of the pyramid is:
\[
\boxed{11 \, \text{cm}}
\]

This version delivers the essential steps in a concise manner. Thank you for your guidance!