22, 24 and 28. Suppose a similar rectangular pyramid to the one shown has a scale factor of 2/3. What would be the volume of the new pyramid?

Answer 1

To find the volume of a pyramid that is similar to another pyramid but has a different scale factor, you can use the formula that relates the volumes of similar solids. The volume of similar solids scales with the cube of the scale factor.

The volume $V$ of a pyramid is given by the formula:

$$V=\frac{1}{3}\times B\times h$$

where $B$ is the area of the base and $h$ is the height of the pyramid.

Let's compute the volume of the original pyramid with dimensions 22, 24, and 28. Assuming these dimensions represent the base's dimensions (22 and 24) and the height (28), we first find the area of the base.

The area of the base $B$ can be calculated as:

$$B=\text{length}\times \text{width}=22\times 24=528$$

Now we can find the volume of the original pyramid:

$$V=\frac{1}{3}\times B\times h=\frac{1}{3}\times 528\times 28$$

Calculating this step-by-step:

1. Calculate $528\times 28$:

$$528\times 28=14784$$

2. Now divide by 3:

$$\frac{14784}{3}=4928$$

So, the volume $V$ of the original pyramid is:

$$V=4928$$

Now, to find the volume of the new pyramid which has a scale factor of $\frac{2}{3}$:

The volume of similar pyramids scales with the cube of the scale factor. Therefore, we calculate the new volume $V^{\prime }$ as follows:

$$V^{\prime }=V\times {\left(\frac{2}{3}\right)}^3$$

Calculating ${\left(\frac{2}{3}\right)}^3$:

$${\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3}=\frac{8}{27}$$

Now multiply the original volume by this factor:

$$V^{\prime }=4928\times \frac{8}{27}$$

Calculating this:

$$V^{\prime }=\frac{4928\times 8}{27}=\frac{39424}{27}\approx 1461.63$$

Thus, the volume of the new pyramid is approximately:

$$\boxed{{\displaystyle 1461.63}}$$