Question

The volume of a pyramid or a cone is \(\frac{1}{3} B h\), where \(B\) is the area of the base and \(h\) is the height. What is the height of a cone with the same volume as a pyramid with a square base 6 feet on a side and with a height of 20 feet? A. 225 feet B. 166.67 feet C. 196.3 feet D. 25.5 feet

Short answer

Rechecking the problem choices, there must be an error in problem assumptions.

Step-by-step solution

Calculate the volume of the pyramid

First, calculate the area of the base of the pyramid. The base is a square with a side length of 6 feet, so the area of the base \(B\) is given by \(B = 6 \times 6 = 36 \text{ square feet}\). Then, use the formula for the volume of a pyramid \(\frac{1}{3} B h\) to find the volume: \[ V_{\text{pyramid}} = \frac{1}{3} \times 36 \times 20 = 240 \text{ cubic feet} \]

Use the volume of the pyramid to find the height of the cone

Since the volume of the cone is the same as the volume of the pyramid, the volume \( V_{\text{cone}} = 240 \text{ cubic feet} \). The formula for the volume of a cone is also \(\frac{1}{3} B h\), but with the base area \(B\) and height \(h_{\text{cone}}\). Rearrange the formula to solve for the height of the cone: \[ 240 = \frac{1}{3} B \times h_{\text{cone}} \]

Determine the base area of the cone and solve for the height

To solve for the height of the cone \(h_{\text{cone}}\), note that we need the base area \(B\) of the cone. Since it's not provided, assume that the base area of the cone remains same as the pyramid, which is \(36 \text{ square feet}\). Plug this into the rearranged formula: \[ 240 = \frac{1}{3} \times 36 \times h_{\text{cone}} \] \[ 240 = 12 \times h_{\text{cone}} \] \[ h_{\text{cone}} = \frac{240}{12} = 20 \text{ feet} \]

Choose the correct answer

Compare the calculated height of the cone (20 feet) with the provided options. Since none of the options match, there must be an error in either the problem statement or assumptions. Recheck the given choices.

Key concepts

Volume Calculation

Calculating the volume of geometric shapes is a core concept in geometry. For certain shapes like pyramids and cones, the formula for volume is quite straightforward. You need to know the area of the base (\text{B}) and the height (\text{h}) of the shape. The basic formula given is \(\frac{1}{3} B h\) for both pyramids and cones. Essentially, you multiply the area of the base with the height of the shape and then multiply by \(\frac{1}{3}\). This formula helps in finding the space occupied within these shapes. Make sure to memorize this formula as it often comes in handy while solving geometry problems.

Pyramid Volume

To calculate the volume of a pyramid, you start by finding the area of its base. If the base is a square, the area can be calculated by squaring the length of one side. For example:

• If the side length is 6 feet, the area (B) of the base is \(6 \times 6 = 36 \text{ square feet}\)

Once you have the base area, you use the volume formula: \(\frac{1}{3} B h\)
For a pyramid with height of 20 feet:
\(\frac{1}{3} \times 36 \times 20 = 240 \text{ cubic feet}\)
This calculation shows the total volume inside the pyramid.

Cone Volume

The process of calculating the volume of a cone is similar to that of a pyramid. You utilize the same formula, \(\frac{1}{3} B h\), where B represents the base area and h is the height. For a cone, the base is usually a circle, so you often need to know the radius to find the base area (B). However, in this example, let’s assume the base area is identical to that of the pyramid (36 square feet). Using the volume of the same pyramid (240 cubic feet), you can determine the height of the cone:

• 240 = \(\frac{1}{3} \times 36 \times h_{\text{cone}}\)
• 240 = 12 \(h_{\text{cone}}\)
• h_{\text{cone}} = \frac{240}{12}\ = 20 feet

This calculation allows you to find the height of the cone that shares the same volume as the pyramid.

Geometry Problems

Solving geometry problems often requires a clear understanding of shape properties and volume formulas. Here are some tips to help you navigate these problems more smoothly:

• Familiarize yourself with key volume formulas such as \(\frac{1}{3} B h\) for pyramids and cones.
• Always recall the base area calculations for different shapes (squares, circles, etc.).
• If an exercise involves comparing volumes, make sure you clearly understand the volume formulas involved and the relationship between base area and height.
• Recheck the given options and your calculations to avoid any missteps.

By following these steps, you can develop a solid approach to solving even the trickiest geometry volume problems, just like in this example where we compared the volumes of a pyramid and cone.