Chapter 12.5, Problem 19WE

(a)

To determine

To find: The ratio of area of the shaded circles.

Answer to Problem 19WE

The ratio of area of the shaded circles is $\frac{9}{16}$ .

Explanation of Solution

Given information:

The givenfigure is shown below.

  

Calculation:

Let’s the area of middle circle is $A_1$ and the bottom circle is $A_2$ .

  $\begin{array}{c}\frac{A_1}{A_2}={\left(\frac{3}{4}\right)}^2\\ =\frac{9}{16}\end{array}$

Therefore,the ratio of area of the shaded circles is $\frac{9}{16}$ .

(b)

To determine

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

Answer to Problem 19WE

The ratio of lateral area of the top part of the cone to the lateral area of whole cone is $\frac{9}{16}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

Lateral area of cone in terms of radius and slant height is $\pi rl$ . The lateral area of cone is proportional to the radius and slant height.

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

Both the cones are similar so, ratio of radius and slant height would be similar.

  $\frac{r_1}{r_2}=\frac{l_1}{l_2}$

The ratio of lateral area of the top part of the cone to the lateral area of whole cone is,

  $\begin{array}{c}\frac{L{A}_1}{L{A}_2}={\left(\frac{3}{4}\right)}^2\\ =\frac{9}{16}\end{array}$

Therefore, the ratio of lateral area of the top part of the cone to the lateral area of whole cone is $\frac{9}{16}$ .

(c)

To determine

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

Answer to Problem 19WE

The ratio of lateral area of the top part of the cone to the lateral area of whole cone is $\frac{9}{7}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

The lateral area of the bottom cone is the whole cone area minus the top cone area.

  $\begin{array}{c}\frac{L{A}_1}{L{A}_2}=\frac{9}{16-9}\\ =\frac{9}{7}\end{array}$

Therefore, the ratio of lateral area of the top part of the cone to the lateral area of bottom cone is $\frac{9}{7}$ .

(d)

To determine

To find: The ratio of the volume of the top part of the cone to the volume of wholecone.

Answer to Problem 19WE

The ratio of the volume of the top part of the cone to the volume of whole cone is $\frac{27}{64}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

The volume of cone in terms of radius and slant height is $\frac{1}{3}\pi r^{2}h$ . The volume of right cone is proportion to the square of radius and height.

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

  $\frac{r_1}{r_2}=\frac{h_1}{h_2}$

The ratio of the volume of the top part of the cone is to the volume of whole cone.

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{3}{4}\right)}^3\\ =\frac{27}{64}\end{array}$

Therefore, the ratio of the volume of the top part of the cone to the volume of whole cone is $\frac{27}{64}$ .

(e)

To determine

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

Answer to Problem 19WE

The ratio of the volume of the top part of the cone to the volume of bottom cone is $\frac{27}{37}$ .

Explanation of Solution

Given information:

The given figure is shown below.

  

Calculation:

The ratio of the volume of the top part of the cone is to the volume of bottom cone.

  $\begin{array}{c}\frac{V_1}{V_2}=\frac{27}{64-27}\\ =\frac{27}{37}\end{array}$

Therefore, the ratio of the volume of the top part of the cone to the volume of bottom cone is $\frac{27}{37}$ .