Chapter 12.3, Problem 1PSA

a.

To determine

To calculate: The total surface area of a sphere having radius 7 units.

Answer to Problem 1PSA

The total surface area of a sphere is $616{\text{units}}^2.$ .

Explanation of Solution

Given Information:

Radius of a sphere $\left(r\right)=6\text{units}$

Formula used:

Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

Calculation:

We know that, Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

Where $"r"$ is the radius of a sphere

  $\begin{array}{l}\therefore TSA=4\times \frac{22}{7}\times 7^2\mathrm{...}\left(\because r=7\right)\\ \therefore TSA=4\times 22\times 7\\ \therefore TSA=616{\text{units}}^2\end{array}$

Hence, total surface area of a sphere is $616{\text{units}}^2.$

b.

To determine

To find: The total surface area of a sphere having radius of 3 units.

Answer to Problem 1PSA

The total surface area of a sphere is $113.14{\text{units}}^2.$ .

Explanation of Solution

Given Information:

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

Formula used:

Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

Calculation:

We know that, Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

Where $"r"$ is the radius of a sphere

  $\begin{array}{l}\therefore TSA=4\times \frac{22}{7}\times 3^2\mathrm{...}\left(\because r=7\right)\\ \therefore TSA=4\times \frac{22}{7}\times 9\\ \therefore TSA=113.14{\text{units}}^2\end{array}$

Hence, total surface area of a sphere is $113.14{\text{units}}^2.$

c.

To determine

To find: The total surface area of a sphere having diameter of 6 units.

Answer to Problem 1PSA

The total surface area of a sphere with diameter 6 is $113.14{\text{units}}^2.$

Explanation of Solution

Given Information:

Diameter of a sphere $\left(d\right)=6\text{units}$

Formula used:

Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

Calculation:

As diameter of a sphere $\left(d\right)=6\text{units}$

Therefore, radius $\left(r\right)=\frac{d}{2}=3\text{units}$

We know that, Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

Where $"r"$ is the radius of a sphere

  $\begin{array}{l}\therefore TSA=4\times \frac{22}{7}\times 3^2\mathrm{...}\left(\because r=3\right)\\ \therefore TSA=4\times \frac{22}{7}\times 9\\ \therefore TSA=113.14{\text{units}}^2\end{array}$

Hence, total surface area of a sphere is $113.14{\text{units}}^2.$

d.

To determine

To find: The total surface area of a sphere having diameter of 5 units.

Answer to Problem 1PSA

The total surface area of a sphere is $78.57{\text{units}}^2.$

Explanation of Solution

Given Information:

Diameter of a sphere $\left(d\right)=5\text{units}$

Formula used:

Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

  $"r"$ is the radius of a sphere

Calculation:

As diameter of a sphere $\left(d\right)=5\text{units}$

Therefore, radius $\left(r\right)=\frac{d}{2}=\frac{5}{2}\text{units}$

We know that, Total surface area of a sphere $\left(TSA\right)=4\pi r^2$

Where $"r"$ is the radius of a sphere

  $\begin{array}{l}\therefore TSA=4\times \frac{22}{7}\times {\left(\frac{5}{2}\right)}^2\mathrm{...}\left(\because r=\frac{5}{2}\right)\\ \therefore TSA=4\times \frac{22}{7}\times \frac{25}{4}\\ \therefore TSA=\frac{22}{7}\times 25\\ \therefore TSA=78.57{\text{units}}^2\end{array}$

Hence, total surface area of a sphere is $78.57{\text{units}}^2.$