Chapter 12.3, Problem 2PSA

a.

To determine

To find: The lateral surface area and total surface area of a cone with radius 3 and slant height 8.

Answer to Problem 2PSA

The lateral area and total area of a cone are $75.42{\text{units}}^2\text{and}103.7{\text{units}}^2$ respectively.

Explanation of Solution

Given Information:

Radius of a cone $\left(r\right)=3\text{units}$

Slant height of a cone $\left(l\right)=8\text{units}$

Formula used:

Lateral surface area of a cone $\left(LSA\right)=\pi r\sqrt{h^2+r^2}$

  $"r"$ is the radius of a cone

  $"h"$ is the height of a cone

Slant height of a cone $\left(l\right)=\sqrt{h^2+r^2}$

Total surface area of a cone $\left(TSA\right)=\pi r^2+\pi rl$

Calculation:

We know that, slant height of a cone $\left(l\right)=\sqrt{h^2+r^2}$

And, Lateral surface area of a cone $\left(LSA\right)=\pi r\sqrt{h^2+r^2}$

  $"r"$ is the radius of a cone

  $"h"$ is the height of a cone

  $\therefore LSA=\pi rl$

As $r=3\text{and}l=8$

  $\begin{array}{l}\therefore LSA=\frac{22}{7}\times 3\times 8\\ \therefore LSA=75.42{\text{units}}^2\end{array}$

Now, total surface area of a cone $\left(TSA\right)=\pi r^2+\pi rl$

  $\begin{array}{l}\therefore TSA=\frac{22}{7}\times 3^2+75.42\\ \therefore TSA=28.28+75.42\\ \therefore TSA=103.7{\text{units}}^2\end{array}$

Hence, the lateral area and total area of a cone are $75.42{\text{units}}^2\text{and}103.7{\text{units}}^2$ respectively.

b.

To determine

To find: The lateral surface area and total surface area of a cylinder with radius 7 and height 10.

Answer to Problem 2PSA

The lateral area and total area of a cylinder are $440{\text{units}}^2\text{and}748{\text{units}}^2$ respectively.

Explanation of Solution

Given Information:

Radius of the cylinder $\left(r\right)=7\text{units}$

Height of the cylinder $\left(h\right)=10\text{units}$

Formula used:

Lateral surface area of a cylinder $\left(LSA\right)=2\pi rh$

Total surface area of a cylinder $\left(TSA\right)=2\pi rh+2\pi r^2$

  $"r"$ is the radius of a cylinder

  $"h"$ is the height of a cylinder

Calculation:

We know that,

Lateral surface area of a cylinder $\left(LSA\right)=2\pi rh$

As $r=7\text{and}h=10$

  $\begin{array}{l}\therefore LSA=2\times \frac{22}{7}\times 7\times 10\\ \therefore LSA=440{\text{units}}^2\end{array}$

Now, total surface area of a cylinder $\left(TSA\right)=2\pi rh+2\pi r^2$

  $\begin{array}{l}\therefore TSA=LSA+2\pi r^2\\ \therefore TSA=440+2\times \frac{22}{7}\times 7^2\\ \therefore TSA=440+308\\ \therefore TSA=748{\text{units}}^2\end{array}$

Hence, the lateral area and total area of a cylinder are $440{\text{units}}^2\text{and}748{\text{units}}^2$ respectively.

c.

To determine

To find: The lateral surface area and total surface area of a cylinder with radius 4 and height 5.

Answer to Problem 2PSA

The lateral area and total area of a cylinder are $125.71{\text{units}}^2\text{and}226.28{\text{units}}^2$ respectively.

Explanation of Solution

Given Information:

Radius of the cylinder $\left(r\right)=4\text{units}$

Height of the cylinder $\left(h\right)=5\text{units}$

Formula used:

Lateral surface area of a cylinder $\left(LSA\right)=2\pi rh$

Total surface area of a cylinder $\left(TSA\right)=2\pi rh+2\pi r^2$

  $"r"$ is the radius of a cylinder

  $"h"$ is the height of a cylinder

Calculation:

We know that,

Lateral surface area of a cylinder $\left(LSA\right)=2\pi rh$

As $r=4\text{and}h=5$

  $\begin{array}{l}\therefore LSA=2\times \frac{22}{7}\times 4\times 5\\ \therefore LSA=125.71{\text{units}}^2\end{array}$

Now, total surface area of a cylinder $\left(TSA\right)=2\pi rh+2\pi r^2$

  $\begin{array}{l}\therefore TSA=LSA+2\pi r^2\\ \therefore TSA=125.71+2\times \frac{22}{7}\times 4^2\\ \therefore TSA=125.71+100.57\\ \therefore TSA=226.28{\text{units}}^2\end{array}$

Hence, the lateral area and total area of a cylinder are $125.71{\text{units}}^2\text{and}226.28{\text{units}}^2$ respectively.

d.

To determine

To find: The lateral surface area and total surface area of a cone with radius 1 and slant height 3.

Answer to Problem 2PSA

The lateral area and total area of a cone are $9.42{\text{units}}^2\text{and}31.42{\text{units}}^2$ respectively.

Explanation of Solution

Given Information:

Radius of a cone $\left(r\right)=1\text{units}$

Slant height of a cone $\left(l\right)=3\text{units}$

Formula used:

Lateral surface area of a cone $\left(LSA\right)=\pi r\sqrt{h^2+r^2}$

Slant height of a cone $\left(l\right)=\sqrt{h^2+r^2}$

Total surface area of a cone $\left(TSA\right)=\pi r^2+\pi rl$

  $"r"$ is the radius of a cone

  $"h"$ is the height of a cone

Calculation:

We know that, slant height of a cone $\left(l\right)=\sqrt{h^2+r^2}$

And, Lateral surface area of a cone $\left(LSA\right)=\pi r\sqrt{h^2+r^2}$

  $"r"$ is the radius of a cone

  $"h"$ is the height of a cone

  $\therefore LSA=\pi rl$

As $r=1\text{and}l=3$

  $\begin{array}{l}\therefore LSA=\frac{22}{7}\times 1\times 3\\ \therefore LSA=9.42{\text{units}}^2\end{array}$

Now, total surface area of a cone $\left(TSA\right)=\pi r^2+\pi rl$

  $\begin{array}{l}\therefore TSA=\frac{22}{7}\times 1^2+9.42\\ \therefore TSA=3.14+28.28\\ \therefore TSA=31.42{\text{units}}^2\end{array}$

Hence, the lateral area and total area of a cone are $9.42{\text{units}}^2\text{and}31.42{\text{units}}^2$ respectively.