Chapter 12.5, Problem 12CE

(a)

To determine

To find: The ratio of perimeter of $\Delta PQR$ to the perimeter of $\Delta ABC$ .

Answer to Problem 12CE

The ratio of perimeter of $\Delta PQR$ to the perimeter of $\Delta ABC$ is $\frac{1}{2}$ .

Explanation of Solution

Given information:

The givenfigure is shown below.

  

Calculation:

The plane $PQR$ bisects the altitude. So, the ratio of perimeter of $\Delta PQR$ to the perimeter of $\Delta ABC$ is,

  $\frac{P_1}{P_2}=\left(\frac{1}{2}\right)$

Therefore, the ratio of perimeter of $\Delta PQR$ to the perimeter of $\Delta ABC$ is $\frac{1}{2}$ .

(b)

To determine

To find: The ratio of the lateral area of the top part of the pyramid to the lateral area of the whole pyramid.

Answer to Problem 12CE

The ratio of the lateral area of the top part of the pyramid to the lateral area of the whole pyramid is $\frac{1}{4}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

The lateral area of the pyramid in terms of base edge and height is $\frac{1}{2}\rho l$ .

This means lateral area of pyramid is proportional to the base edge and slant height.

Consider the lateral area of cone is $L{A}_1$ and $L{A}_2$ .

Both the pyramid is similar. So, the ratio of base edge and slant height would be similar.

  $\frac{{\rho }_1}{{\rho }_2}=\frac{l_1}{l_2}$

The ratio of lateral area of the top part of the pyramid to the lateral area of whole pyramid,

  $\begin{array}{c}\frac{L{A}_1}{L{A}_2}={\left(\frac{1}{2}\right)}^2\\ =\frac{1}{4}\end{array}$

Therefore, the ratio of lateral area of the top part of the pyramid to the lateral area of whole pyramid is $\frac{1}{4}$ .

(c)

To determine

To find: The ratio of lateral area of the top part of the pyramid to the lateral area of bottom part.

Answer to Problem 12CE

The ratio of lateral area of the top part of the pyramid to the lateral area of bottom part is $1$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

The ratio of lateral area of the top part of the pyramid to the lateral area of bottom part,

  $\begin{array}{c}\frac{L{A}_1}{L{A}_2}={\left(1\right)}^2\\ =1\end{array}$

Therefore, the ratio of lateral area of the top part of the pyramid to the lateral area of bottom part is $1$ .

(d)

To determine

To find: The ratio of the volume of the top part of the pyramid to the volume of bottom part.

Answer to Problem 12CE

The ratio of the volume of the top part of the pyramid to the volume of bottom part is $\frac{1}{8}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

Volume of pyramid in terms of base edge and height is $\frac{1}{3}Bh$ .

This means volume of pyramid is proportion to the base are and height.Both the pyramid is similar.

Consider the volume of cone is $V_1$ and $V_2$ .

  $\frac{{\rho }_1}{{\rho }_2}=\frac{l_1}{l_2}$

The ratio of the volume of the top part of the pyramid to the volume of bottom part,

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{1}{2}\right)}^3\\ =\frac{1}{8}\end{array}$

Therefore, the ratio of the volume of the top part of the pyramid to the volume of bottom part is $\frac{1}{8}$ .