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Prealgebra

9.6 Solve Geometry Applications: Volume and Surface Area

Prealgebra9.6 Solve Geometry Applications: Volume and Surface Area
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Find volume and surface area of rectangular solids
  • Find volume and surface area of spheres
  • Find volume and surface area of cylinders
  • Find volume of cones
Be Prepared 9.6

Before you get started, take this readiness quiz.

  1. Evaluate x3x3 when x=5.x=5.
    If you missed this problem, review Example 2.15.
  2. Evaluate 2x2x when x=5.x=5.
    If you missed this problem, review Example 2.16.
  3. Find the area of a circle with radius 72.72.
    If you missed this problem, review Example 5.39.

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Problem Solving Strategy for Geometry Applications
  1. Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
  2. Step 2. Identify what you are looking for.
  3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Step 5. Solve the equation using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure 9.28). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

This is an image of a wooden crate.
Figure 9.28 This wooden crate is in the shape of a rectangular solid.

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in Figure 9.29 has length 44 units, width 22 units, and height 33 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

A rectangular solid is shown. Each layer is composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this is an image of the top layer that says “The top layer has 8 cubic units.” The orange layer is shown and says “The middle layer has 8 cubic units.” The green layer is shown and says, “The bottom layer has 8 cubic units.”
Figure 9.29 Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 44 by 22 by 33 rectangular solid has 2424 cubic units.

Altogether there are 2424 cubic units. Notice that 2424 is the length×width×height.length×width×height.

The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.

The volume, V,V, of any rectangular solid is the product of the length, width, and height.

V=LWHV=LWH

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B,B, is equal to length×width.length×width.

B=L·WB=L·W

We can substitute BB for L·WL·W in the volume formula to get another form of the volume formula.

The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4×2×34×2×3 rectangular solid we started with. See Figure 9.29.

An image of a rectangular solid is shown. It is made up of cubes. It is labeled as 2 by 4 by 3. Beside the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.
Figure 9.30

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

Afront=L×WAside=L×WAtop=L×WAfront=4·3Aside=2·3Atop=4·2Afront=12Aside=6Atop=8Afront=L×WAside=L×WAtop=L×WAfront=4·3Aside=2·3Atop=4·2Afront=12Aside=6Atop=8

Notice for each of the three faces you see, there is an identical opposite face that does not show.

S=(front+back)+(left side+right side)+(top+bottom)S=(2·front)+(2·left side)+(2·top)S=2·12+2·6+2·8S=24+12+16S=52sq. unitsS=(front+back)+(left side+right side)+(top+bottom)S=(2·front)+(2·left side)+(2·top)S=2·12+2·6+2·8S=24+12+16S=52sq. units

The surface area SS of the rectangular solid shown in Figure 9.30 is 5252 square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure 9.31). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

S=2LH+2LW+2WHS=2LH+2LW+2WH
A rectangular solid is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.
Figure 9.31 For each face of the rectangular solid facing you, there is another face on the opposite side. There are 66 faces in all.

Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length L,L, width W,W, and height H:H:

A rectangular solid is shown. The sides are labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Painted Cube” will help you develop a better understanding of volume and surface area.

Example 9.47

For a rectangular solid with length 1414 cm, height 1717 cm, and width 99 cm, find the volume and surface area.

Try It 9.93

Find the volume and surface area of rectangular solid with the: length 88 feet, width 99 feet, and height 1111 feet.

Try It 9.94

Find the volume and surface area of rectangular solid with the: length 1515 feet, width 1212 feet, and height 88 feet.

Example 9.48

A rectangular crate has a length of 3030 inches, width of 2525 inches, and height of 2020 inches. Find its volume and surface area.

Try It 9.95

A rectangular box has length 99 feet, width 44 feet, and height 66 feet. Find its volume and surface area.

Try It 9.96

A rectangular suitcase has length 2222 inches, width 1414 inches, and height 99 inches. Find its volume and surface area.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

V=LWHS=2LH+2LW+2WHV=s·s·sS=2s·s+2s·s+2s·sV=s3S=2s2+2s2+2s2S=6s2V=LWHS=2LH+2LW+2WHV=s·s·sS=2s·s+2s·s+2s·sV=s3S=2s2+2s2+2s2S=6s2

So for a cube, the formulas for volume and surface area are V=s3V=s3 and S=6s2.S=6s2.

Volume and Surface Area of a Cube

For any cube with sides of length s,s,

An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.

Example 9.49

A cube is 2.52.5 inches on each side. Find its volume and surface area.

Try It 9.97

For a cube with side 4.5 meters, find the volume and surface area of the cube.

Try It 9.98

For a cube with side 7.3 yards, find the volume and surface area of the cube.

Example 9.50

A notepad cube measures 22 inches on each side. Find its volume and surface area.

Try It 9.99

A packing box is a cube measuring 44 feet on each side. Find its volume and surface area.

Try It 9.100

A wall is made up of cube-shaped bricks. Each cube is 1616 inches on each side. Find the volume and surface area of each cube.

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate ππ with 3.14.3.14.

Volume and Surface Area of a Sphere

For a sphere with radius r:r:

An image of a sphere is shown. The radius is labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.

Example 9.51

A sphere has a radius 66 inches. Find its volume and surface area.

Try It 9.101

Find the volume and surface area of a sphere with radius 3 centimeters.

Try It 9.102

Find the volume and surface area of each sphere with a radius of 11 foot

Example 9.52

A globe of Earth is in the shape of a sphere with radius 1414 centimeters. Find its volume and surface area. Round the answer to the nearest hundredth.

Try It 9.103

A beach ball is in the shape of a sphere with radius of 99 inches. Find its volume and surface area.

Try It 9.104

A Roman statue depicts Atlas holding a globe with radius of 1.51.5 feet. Find the volume and surface area of the globe.

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 hh of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, hh , will be perpendicular to the bases.

An image of a cylinder is shown. There is a red arrow pointing to the radius of the top labeling it r, radius. There is a red arrow pointing to the height of the cylinder labeling it h, height.
Figure 9.32 A cylinder has two circular bases of equal size. The height is the distance between the bases.

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=BhV=Bh , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, BB , is the area of the rectangular base, length × width. For a cylinder, the area of the base, B,B, is the area of its circular base, πr2.πr2. Figure 9.33 compares how the formula V=BhV=Bh is used for rectangular solids and cylinders.

In (a), a rectangular solid is shown. The sides are labeled L, W, and H. Below this is V equals capital Bh, then V equals Base times h, then V equals parentheses lw times h, then V equals lwh. In (b), a cylinder is shown. The radius of the top is labeled r, the height is labeled h. Below this is V equals capital Bh, then V equals Base times h, then V equals parentheses pi r squared times h, then V equals pi times r squared times h.
Figure 9.33 Seeing how a cylinder is similar to a rectangular solid may make it easier to understand the formula for the volume of a cylinder.

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 Figure 9.34.

A cylindrical can of green beans is shown. The height is labeled h. Beside this are pictures of circles for the top and bottom of the can and a rectangle for the other portion of the can. Above the circles is C equals 2 times pi times r. The top of the rectangle says l equals 2 times pi times r. The left side of the rectangle is labeled h, the right side is labeled w.
Figure 9.34 By cutting and unrolling the label of a can of vegetables, we can see that the surface of a cylinder is a rectangle. The length of the rectangle is the circumference of the cylinder’s base, and the width is the height of the cylinder.

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

The top line says A equals l times red w. Below the l is 2 times pi times r. Below the w is a red h.

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

A rectangle is shown with circles coming off the top and bottom.

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

S=2πr2+2πrhS=2πr2+2πrh

Volume and Surface Area of a Cylinder

For a cylinder with radius rr and height h:h:

A cylinder is shown. The height is labeled h and the radius of the top is labeled r. Beside it is Volume: V equals pi times r squared times h or V equals capital B times h. Below this is Surface Area: S equals 2 times pi times r squared plus 2 times pi times r times h.

Example 9.53

A cylinder has height 55 centimeters and radius 33 centimeters. Find the volume and surface area.

Try It 9.105

Find the volume and surface area of the cylinder with radius 4 cm and height 7cm.

Try It 9.106

Find the volume and surface area of the cylinder with given radius 2 ft and height 8 ft.

Example 9.54

Find the volume and surface area of a can of soda. The radius of the base is 44 centimeters and the height is 1313 centimeters. Assume the can is shaped exactly like a cylinder.

Try It 9.107

Find the volume and surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.

Try It 9.108

Find the volume and surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See Figure 9.35.

An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r.
Figure 9.35 The height of a cone is the distance between its base and the vertex.

Earlier in this section, we saw that the volume of a cylinder is V=πr2h.V=πr2h. We can think of a cone as part of a cylinder. Figure 9.36 shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

An image of a cone is shown. There is a cylinder drawn around it.
Figure 9.36 The volume of a cone is less than the volume of a cylinder with the same base and height.

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

The formula V equals one-third times capital B times h is shown.

Since the base of a cone is a circle, we can substitute the formula of area of a circle, πr2πr2 , for BB to get the formula for volume of a cone.

The formula V equals one-third times pi times r squared times h is shown.

In this book, we will only find the volume of a cone, and not its surface area.

Volume of a Cone

For a cone with radius rr and height hh.

An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h.

Example 9.55

Find the volume of a cone with height 66 inches and radius of its base 22 inches.

Try It 9.109

Find the volume of a cone with height 77 inches and radius 33 inches

Try It 9.110

Find the volume of a cone with height 99 centimeters and radius 55 centimeters

Example 9.56

Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is 88 inches tall and 55 inches in diameter? Round the answer to the nearest hundredth.

Try It 9.111

How many cubic inches of candy will fit in a cone-shaped piñata that is 1818 inches long and 1212 inches across its base? Round the answer to the nearest hundredth.

Try It 9.112

What is the volume of a cone-shaped party hat that is 1010 inches tall and 77 inches across at the base? Round the answer to the nearest hundredth.

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

A table is shown that summarizes all of the formulas in the chapter. The first cell is for Supplementary and Complementary Angles, and says that the measure of angle A plus the measure of angle B equals 180 degrees for supplementary angles A and B and the measure of angle C plus the measure of angle D equals 90 degrees for complementary angles C and D. There is an image of two angles A and B that together form a straight line and two angles C and D that together form a right angle. The next cell says Rectangular Solid and shows the formulas Volume equals LWH and Surface Area equals 2LH plus 2LW plus 2WH. An image of a rectangular solid with sides L, W, and H is shown. The next cell says Triangle. An image of a triangle is shown with sides a, b, and c, vertices A, B, and C, and height h. It says, “For triangle ABC, angle measures measure of angle A plus measure of angle B plus measure of angle C equal 180 degrees. Below this is Perimeter, P equals a plus b plus c. Below this is Area, A equals one-half bh. The next cell says Cube and shows an image of a cube with sides s. It says Volume V equals s cubed and Surface Area S equals 6 times s squared. The next cell says Similar Triangles. It shows two similar triangles ABC and XYZ. It says if triangle ABC is similar to triangle XYZ, then measure of angle A equals measure of angle X, measure of angle B equals measure of angle Y, and measure of angle C equals measure of angle Z. It then says a over x equals b over y equal c over z. The next cell says Sphere and shows an image of a sphere with radius r. It says volume V equals four-thirds times pi times r and Surface Area S equals 4 times pi times r squared. The next cell says Circle. There is an image with two radii labeled r and the diameter labeled d. It says Circumference C equals 2 pi times r and C equals pi times d. It says Area equals pi times r squared. The next cell says Cylinder and shows an image of a cylinder with height h and radius of the base r. It says Volume V equals pi times r squared times h. Below this is V equals Bh. Below that is Surface Area S equals 2 times pi times r squared plus 2 times pi times rh. The next cell says Rectangle and shows an image of a rectangle with sides W and L. It says Perimeter P equals 2L plus 2W, then Area A equals LW. The next cell says Cone and shows an image of a cone with height h and radius of the base r. It says Volume V equals one-third times pi times r squared times h. The last cell says Trapezoid and shows an image of a trapezoid with bases little b and capital B, and height h. It says Area A equals one-half times h times parentheses little b plus capital B. This image shows a row with three columns. The first column says Rectangular solid with the formula below that says volume: V equals LWH. Under this, it says Surface Area: S equals 2LH plus 2LW plus 2WH. An image shows an image of a rectangular solid with the sides labeled L , W and H. The middle column says Rectangle. Under this it says Perimeter P equals 2L plus 2W, then Area A equals LW.  An image of a rectangle with sides W and L. The right column says Cube. Under this it says “Volume: V equals s to the third power.” Under this is says “Surface area: S equals 6 times s squared. Below it is an image of a cube with three sides labeled “s”.

Media Access Additional Online Resources

Section 9.6 Exercises

Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the volume and the surface area of the rectangular solid with the given dimensions.

263.

length 22 meters, width 1.51.5 meters, height 33 meters

264.

length 55 feet, width 88 feet, height 2.52.5 feet

265.

length 3.53.5 yards, width 2.12.1 yards, height 2.42.4 yards

266.

length 8.88.8 centimeters, width 6.56.5 centimeters, height 4.24.2 centimeters

In the following exercises, solve.

267.

Moving van A rectangular moving van has length 1616 feet, width 88 feet, and height 88 feet. Find its volume and surface area.

268.

Gift box A rectangular gift box has length 2626 inches, width 1616 inches, and height 44 inches. Find its volume and surface area.

269.

Carton A rectangular carton has length 21.321.3 cm, width 24.224.2 cm, and height 6.56.5 cm. Find its volume and surface area.

270.

Shipping container A rectangular shipping container has length 22.822.8 feet, width 8.58.5 feet, and height 8.28.2 feet. Find its volume and surface area.

In the following exercises, find the volume and the surface area of the cube with the given side length.

271.

55 centimeters

272.

66 inches

273.

10.410.4 feet

274.

12.512.5 meters

In the following exercises, solve.

275.

Science center Each side of the cube at the Discovery Science Center in Santa Ana is 6464 feet long. Find its volume and surface area.

276.

Museum A cube-shaped museum has sides 4545 meters long. Find its volume and surface area.

277.

Base of statue The base of a statue is a cube with sides 2.82.8 meters long. Find its volume and surface area.

278.

Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its volume and surface area.

Find the Volume and Surface Area of Spheres

In the following exercises, find the volume and the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

279.

33 centimeters

280.

99 inches

281.

7.57.5 feet

282.

2.12.1 yards

In the following exercises, solve. Round answers to the nearest hundredth.

283.

Exercise ball An exercise ball has a radius of 1515 inches. Find its volume and surface area.

284.

Balloon ride The Great Park Balloon is a big orange sphere with a radius of 3636 feet . Find its volume and surface area.

285.

Golf ball A golf ball has a radius of 4.54.5 centimeters. Find its volume and surface area.

286.

Baseball A baseball has a radius of 2.92.9 inches. Find its volume and surface area.

Find the Volume and Surface Area of a Cylinder

In the following exercises, find the volume and the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

287.

radius 33 feet, height 99 feet

288.

radius 55 centimeters, height 1515 centimeters

289.

radius 1.51.5 meters, height 4.24.2 meters

290.

radius 1.31.3 yards, height 2.82.8 yards

In the following exercises, solve. Round answers to the nearest hundredth.

291.

Coffee can A can of coffee has a radius of 55 cm and a height of 1313 cm. Find its volume and surface area.

292.

Snack pack A snack pack of cookies is shaped like a cylinder with radius 44 cm and height 33 cm. Find its volume and surface area.

293.

Barber shop pole A cylindrical barber shop pole has a diameter of 66 inches and height of 2424 inches. Find its volume and surface area.

294.

Architecture A cylindrical column has a diameter of 88 feet and a height of 2828 feet. Find its volume and surface area.

Find the Volume of Cones

In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

295.

height 99 feet and radius 22 feet

296.

height 88 inches and radius 66 inches

297.

height 12.412.4 centimeters and radius 55 cm

298.

height 15.215.2 meters and radius 44 meters

In the following exercises, solve. Round answers to the nearest hundredth.

299.

Teepee What is the volume of a cone-shaped teepee tent that is 1010 feet tall and 1010 feet across at the base?

300.

Popcorn cup What is the volume of a cone-shaped popcorn cup that is 88 inches tall and 66 inches across at the base?

301.

Silo What is the volume of a cone-shaped silo that is 5050 feet tall and 7070 feet across at the base?

302.

Sand pile What is the volume of a cone-shaped pile of sand that is 1212 meters tall and 3030 meters across at the base?

Everyday Math

303.

Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is 3030 feet tall with base radius 11 foot. The smaller cone is 1010 feet tall with base radius of 0.50.5 feet. To the nearest tenth,

  1. find the volume of the large cone.

  2. find the volume of the small cone.

  3. find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.

    An image of a cone is shown. There is a dark dotted line at the top indicating a smaller cone.
304.

Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of 2.52.5 inches. A waffle cone is 77 inches tall and has a diameter of 3.253.25 inches. To the nearest hundredth,

  1. find the volume of the regular ice cream cone.

  2. find the volume of the waffle cone.

  3. how much more ice cream fits in the waffle cone compared to the regular cone?

Writing Exercises

305.

The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.

306.

Which has a larger volume, a cube of sides of 88 feet or a sphere with a diameter of 88 feet? Explain your reasoning.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

After reviewing this checklist, what will you do to become confident for all objectives?

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