Chapter 12.4, Problem 23WE

a.

To determine

To find: height of the cone.

Answer to Problem 23WE

Cans needed to paint the ceiling is 6.

Explanation of Solution

Given information:

A solid metal ball with radius 8 cm is melted down and recast as a solid cone with the same radius.

Calculation:

If the solid metal ball is melted down into cone then the volume of metal ball is same as volume of the cone.

The Volume sphere is given by

  $V=\frac{4}{3}\pi r^3$ where $r$ is the radius of the sphere.

And, volume of cone is given by

  $V=\frac{1}{3}\pi r^{2}h$ where $r$ is the radius of the cone and $h$ is the height.

Therefore,

  $\begin{array}{l}\frac{1}{3}\pi r^{2}h=\frac{4}{3}\pi r^3\\ \frac{1}{3}\pi {\left(8\right)}^{2}h=\frac{4}{3}\pi {\left(8\right)}^3\\ h=\frac{\frac{4}{3}\pi {\left(8\right)}^3}{\frac{1}{3}\pi {\left(8\right)}^2}\\ h=4\times 8\\ h=32\text{ cm}\end{array}$

Hence,

Height of the cone will be 32 cm.

a.

To determine

To find: height of the cone.

Answer to Problem 23WE

The lateral surface area of cone is about 3% more than the area of the sphere is shown below.

Explanation of Solution

Given information:

A solid metal ball with radius 8 cm is melted down and recast as a solid cone with the same radius.

Calculation:

Lateral surface area of the cone is given by

  $A=\pi r\sqrt{r^2+h^2}$ where r is the radius and h is the height of the cone.

Also, surface area of the sphere is given by

  $A=4\pi r^2$ where r is the radius of the sphere.

Here, radius of the cone is $r=8$ cm and height from part (a) is $h=32$ .

Therefore, area of the cone is

  $\begin{array}{l}{A}_c=\pi r\sqrt{r^2+h^2}\\ A_c=\pi \left(8\right)\sqrt{{\left(8\right)}^2+{\left(32\right)}^2}\\ A_c=\pi \left(8\right)\sqrt{1088}\\ A_c=829.33{\text{ cm}}^2\end{array}$

Also, area the sphere is

  $\begin{array}{l}{A}_s=4\pi {\left(8\right)}^2\\ A_s=4\pi \left(64\right)\\ A_s=804.57{\text{ cm}}^2\end{array}$

Now,

  $\begin{array}{l}\text{Difference in area}=\left(829.33-804.57\right){\text{ cm}}^2\\ \text{Difference in area}=24.76{\text{ cm}}^2\end{array}$

So, the lateral surface area of the cone more than the area of the sphere by

  $\frac{24.76}{804.57}\times 100\%=3\%$

Hence proved.