9.6 Solve geometry applications: volume and surface area  (Page 5/11)

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, $V=Bh$ , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, $B$ , is the area of the rectangular base, length × width. For a cylinder, the area of the base, $B,$ is the area of its circular base, $\pi r^2.$ [link] compares how the formula $V=Bh$ is used for rectangular solids and cylinders.

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See [link] .

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length $L$ of the rectangular label. The height of the cylinder is the width $W$ of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius $r$ and height $h,$ is

Volume and surface area of a cylinder

For a cylinder with radius $r$ and height $h:$

A cylinder has height $5$ centimeters and radius $3$ centimeters. Find the ⓐ volume and ⓑ surface area.

Solution

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 cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\pi r^{2}h$ $V\approx (3.14)3^2\cdot 5$
Step 5. Solve.	$V\approx 141.3$
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question.	The volume is approximately 141.3 cubic inches.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the cylinder
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 $\pi$ )	$S=2\pi r^2+2\pi rh$ $S\approx 2(3.14)3^2+2(3.14)(3)5$
Step 5. Solve.	$S\approx 150.72$
Step 6. Check: We leave it to you to check your arithmetic.
Step 7. Answer the question.	The surface area is approximately 150.72 square inches.

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Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is $4$ centimeters and the height is $13$ centimeters. Assume the can is shaped exactly like a cylinder.

Solution

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 cylinder
Step 3. Name. Choose a variable to represent it.	let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for $\pi$ )	$V=\pi r^{2}h$ $V\approx (3.14)4^2\cdot 13$
Step 5. Solve.	$V\approx 653.12$
Step 6. Check: We leave it to you to check.
Step 7. Answer the question.	The volume is approximately 653.12 cubic centimeters.

ⓑ
Step 2. Identify what you are looking for.	the surface area of the cylinder
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 $\pi$ )	$S=2\pi r^2+2\pi rh$ $S\approx 2(3.14)4^2+2(3.14)(4)13$
Step 5. Solve.	$S\approx 427.04$
Step 6. Check: We leave it to you to check your arithmetic.
Step 7. Answer the question.	The surface area is approximately 427.04 square centimeters.

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