Chapter 12.5, Problem 35PPS

a.

To determine

To draw two pyramids with different bases and same height.

Answer to Problem 35PPS

A pentagonal pyramid and a rectangular pyramid.

Explanation of Solution

Given:

Height of the pyramids $=10$ centimeters

Base area of pyramids $=24$ square centimeters

The two pyramids with height as $10$ centimeters and base area as $24$ square centimeters are:

1. Pentagonal pyramid:

  

Height $=10$ centimeters

Area of pentagonal base $=24$ square centimeters

Side of the pentagon $=3.7$ centimeters

2. Rectangular pyramid:

  

Height $=10$ centimeters

Area of rectangular base $=24$ square centimeters

Length of the rectangle $=6$ centimeters

Breadth of the rectangle $=4$ centimeters

Conclusion:

A pentagonal pyramid and a rectangular pyramid can have height as 10 centimeters and base area as 24 square centimeters.

b.

To determine

To determine the analysis of the volumes of the pyramids

Explanation of Solution

Given:

Pyramids from sub-part (a)

Height $=10$ centimeters

Base area $=24$ square centimeters

Formula used:

Volume of pyramid, $V=\frac{1}{3}Bh$

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

Calculation:

Volume of rectangular pyramid $=\frac{1}{3}Bh$

  $\Rightarrow \frac{1}{3}\times 24\times 10$

  $\Rightarrow 80$ cubic centimeters

Volume of pentagonal pyramid $=\frac{1}{3}Bh$

  $\Rightarrow \frac{1}{3}\times 24\times 10$

  $\Rightarrow 80$ cubic centimeters

Since height and base area are same in both the cases the volume will also be same.

Conclusion:

Therefore, the volume will be same of both the pyramids as long as heights and base areas of both the pyramids are same.

c.

To determine

To explain ow multiplying the base area or height of the pyramid affects the volume of the pyramid

Answer to Problem 35PPS

  $\Rightarrow 5V=V\text{'}$

Explanation of Solution

Given:

Two pyramids

Pyramid 1:

Height $=h$ and base area $=B$

Pyramid 2

Height $=h$ and base area $=5B$

Formula used:

Volume of pyramid, $V=\frac{1}{3}Bh$

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

Calculation:

Volume of Pyramid 1:

  $V=\frac{1}{2}Bh$ ….(i).

Volume of Pyramid 2:

  $V^{\text{'}}=\frac{1}{3}5Bh$ ….(ii).

Dividing equation (i) by equation (ii),

  $\frac{V}{V^{\text{'}}}=\frac{\frac{1}{3}Bh}{\frac{1}{3}5Bh}$

On simplifying,

  $\Rightarrow \frac{V}{V\text{'}}=\frac{1}{5}$

  $\Rightarrow 5V=V\text{'}$

Conclusion:

Therefore, the volume of pyramid 1 will be 5 times the volume of the pyramid 2.