9.6 Solve geometry applications: volume and surface area  (Page 4/11)

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate $\pi$ with $3.14.$

Volume and surface area of a sphere

For a sphere with radius $r\text{:}$

A sphere has a radius $6$ inches. Find its ⓐ volume and ⓑ surface area.

Solution

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and label it with the given information.

ⓐ
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula.	$V=\frac{4}{3}\pi r^3$
Step 5. Solve.	$V\approx \frac{4}{3}(3.14)6^3$ $V\approx 904.32\text{cubic inches}$
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question.	The volume is approximately 904.32 cubic inches.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the cube
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula.	$S=4\pi r^2$
Step 5. Solve.	$S\approx 4(3.14)6^2$ $S\approx 452.16$
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question.	The surface area is approximately 452.16 square inches.

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A globe of Earth is in the shape of a sphere with radius $14$ centimeters. Find its ⓐ volume and ⓑ surface area. Round the answer to the nearest hundredth.

Solution

Step 1. Read the problem. Draw a figure with the given information and label it.

ⓐ
Step 2. Identify what you are looking for.	the volume of the sphere
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\frac{4}{3}\pi r^3$ $V\approx \frac{4}{3}(3.14){14}^3$
Step 5. Solve.	$V\approx \mathrm{11,488.21}$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 11,488.12 cubic inches.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the sphere
Step 3. Name. Choose a variable to represent it.	let S = surface area
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$S=4\pi r^2$ $S\approx 4(3.14){14}^2$
Step 5. Solve.	$S\approx 2461.76$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The surface area is approximately 2461.76 square inches.

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Find the volume and surface area of a cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder     is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height $h$ of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, $h$ , will be perpendicular to the bases.