Chapter 12.3, Problem 28PPS

a.

To determine

To find:The effect of change on surface area of the given cone when radius and slant height are doubled.

Answer to Problem 28PPS

The surface area becomes $4$ times that is $432\pi$ cm².

Explanation of Solution

M respectively.

Given:

The radius and slant height of the cone are 6 cm and 12 c

Formula/ concept used:

The surface area of the cone is $S=\pi rl+\pi r^2$ , where r and l are radius and slant height of cone.

Calculation:

The surface area of the cone is $S=\pi rl+\pi r^2$

  $S\Rightarrow \pi \left(6\right)\left(12\right)+\pi {\left(6\right)}^2=108\pi$ .

If the radius and slant height is doubled then new surface area would be $S^{\prime }$ (say)

  $S^{\prime }=\pi \left(2r\right)\left(2l\right)+\pi {\left(2r\right)}^2$

  $S^{\prime }=4\pi rl+4\pi r^2$

  $S^{\prime }=4\left(\pi rl+4\pi r^2\right)$

  $S^{\prime }=4S$

Thus, the surface area becomes $4$ times that is $108\pi \times 4=432\pi$ cm².

Conclusion:

The surface area becomes $4$ times that is $432\pi$ cm².

b.

To determine

The effect of change on surface area of the given cone when radius and slant height are divided by 3.

Answer to Problem 28PPS

The surface area becomes $\frac{1}{9}$ times that is $12\pi$ cm².

Explanation of Solution

M respectively.

Given:

The radius and slant height of the cone are 6 cm and 12 c

Formula/ concept used:

The surface area of the cone is $S=\pi rl+\pi r^2$ , where r and l are radius and slant height of cone.

Calculation:

The surface area of the cone is $S=\pi rl+\pi r^2$

  $S\Rightarrow \pi \left(6\right)\left(12\right)+\pi {\left(6\right)}^2=108\pi$ .

If the radius and slant height are divided by 3 then new surface area would be $S^{\prime }$ (say)

  $S^{\prime }=\pi \left(\frac{r}{3}\right)\left(\frac{l}{3}\right)+\pi {\left(\frac{r}{3}\right)}^2$

  $S^{\prime }=\frac{\pi rl}{9}+\frac{\pi r^2}{9}$

  $S^{\prime }=\frac{1}{9}\left(\pi rl+4\pi r^2\right)$

  $S^{\prime }=\frac{S}{9}=\frac{108\pi }{9}=12\pi$

Thus, the surface area becomes $\frac{1}{9}$ times that is $12\pi$ cm².

Conclusion:

The surface area becomes $\frac{1}{9}$ times that is $12\pi$ cm².