Chapter 12.3, Problem 4PSA

a.

To determine

To find: The total surface area of a solid which is the combination of a cone and a cylinder.

Answer to Problem 4PSA

The total area of the solid is $106.85{\text{units}}^2.$

Explanation of Solution

Given Information:

Radius of a cone $\left(r\right)=2\text{units}$

Radius of a cylinder = Radius of a cone $\left(r\right)=2\text{units}$

Slant height of a cone $\left(l\right)=5\text{units}$

Height of a cone $\left(h\right)=5\text{units}$

Formula used:

Lateral surface area of a cone $\left(LSA\right)=\pi rl$

  $"r"$ is the radius of a cone

  $"l"$ is the slant height of a cone

Lateral surface area of a cylinder $\left(LS{A}_1\right)=2\pi rh$

Area of a circle $\left(A\right)=\pi r^2$

Calculation:

The given solid is formed by combining a cone and a cylinder

So, its total area $\left(TA\right)=$ Lateral area of cone + Lateral area of cylinder + Area of a circle

Lateral surface area of a cone $\left(LSA\right)=\pi rl$

  $"r"$ is the radius of a cone

  $"l"$ is the slant height of a cone

Lateral surface area of a cylinder $\left(LS{A}_1\right)=2\pi rh$

Area of a circle $\left(A\right)=\pi r^2$

  $\begin{array}{l}\therefore TA=\pi rl+2\pi rh+\pi r^2\\ \therefore TA=\pi r\left(l+2h+r\right)\\ \therefore TA=\frac{22}{7}\times 2\left(5+2\times 5+2\right)\\ \therefore TA=\frac{22}{7}\times 2\times 17\\ \therefore TA=106.85\end{array}$

Hence, the total area of the solid is $106.85{\text{units}}^2.$

b.

To determine

To find: total area of a hemisphere with radius 6.

Answer to Problem 4PSA

The total area of the hemisphere is $339.42{\text{units}}^2.$

Explanation of Solution

Given Information:

Radius of the hemisphere $\left(r\right)=6\text{units}$

Formula used:

Total surface area of a hemisphere $\left(TA\right)=3\pi r^2$

  $"r"$ is the radius of a hemisphere

Calculation:

We know that,

Total surface area of a hemisphere $\left(TA\right)=3\pi r^2$

  $"r"$ is the radius of a hemisphere

As $r=6\text{units}$

  $\begin{array}{l}\therefore TA=3\times \frac{22}{7}\times 6^2\\ \therefore TA=339.42{\text{units}}^2\end{array}$

Hence, the total area of the hemisphere is $339.42{\text{units}}^2.$