Chapter 10, Problem 12CST

a

To determine

To find the value of $x$

Answer to Problem 12CST

  $16in$

Explanation of Solution

Given information:

For small cone

Base $=4in$ , Perpendicular $=x$

For larger cone

Base $=7in$ , Perpendicular $=12in$

Formula used:

  $\tan \theta =\frac{base}{perpendicular}$

Calculation:

For smaller cone

  $\tan \theta =\frac{4}{x}$

For larger cone,

  $\tan \theta =\frac{7}{12+x}$

Comparing these equation ,

  $\frac{4}{x}=\frac{7}{12+x}$

Using cross multiplication method

  $48+4x=7x$

Calculating the above expression

  $4x=48$

The final answer is

  $x=16in$

b.

To determine

To find the lateral surface area of the smaller and larger cones

Answer to Problem 12CST

For smaller cone, area is

  $207i{n}^2$

For larger cone, area is

  $634i{n}^2$

Explanation of Solution

Given information:

For small cone

  $radius=4in$ , $height=16in$

For larger cone

  $radius=7in$ , $height=28in$

Formula used:

To calculate the slant height of both the cones

  $hypotenus{e}^2=heigh{t}^2+radiu{s}^2$

To calculate lateral surface areas of the cone

  $lateralsurfacearea=\pi rl$

Calculation:

For smaller cone, slant height is

  $l^2={16}^2+4^2$

Squaring and adding the above expression

  $l=16.5in$

Now, lateral surface area of the smaller cone is

  $area=\pi rl$

Substituting values of radius and slant height

  $area=3.14\times 4\times 16.5$

So the lateral surface area of the smaller cone is

  $area=207i{n}^2$

For larger cone, slant height is

  $l^2={28}^2+7^2$

Squaring and adding the above expression

  $l=28.86$

Now, lateral surface area of the larger cone is

  $area=\pi rl$

Substituting values of radius and slant height

  $area=3.14\times 7\times 28.8$

Lateral surface area of the larger cone is

  $634i{n}^2$

c.

To determine

To find the lateral surface area of the lampshade

Answer to Problem 12CST

  $427i{n}^2$

Explanation of Solution

Given information:

Surface area of the larger cone

  $634i{n}^2$

Surface area of the smaller cone

  $207i{n}^2$

Formula used:

Lateral surface area of lampshade $=$ lateral surface area of larger cone $-$ lateral surface area of the smaller cone

Calculation:

Substituting the values of smaller and lager cones in the formula

  $area=$

  $634-207$

So the lateral surface area of the lampshade is

  $area=427i{n}^2$