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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

5 answers

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.

I said then simplify

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"!

I meant simplify the explanation

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!