m81328 Solve Geometry Applications: Volume and Surface Area 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 39537675-6d2a-48bb-9917-7a43d2c116fc Before you get started, take this readiness quiz. Evaluate x3 when x=5.If you missed this problem, review . 125 Evaluate 2x when x=5.If you missed this problem, review . 32 Find the area of a circle with radius 72.If you missed this problem, review . 772 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. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information. Identify what you are looking for. Name what you are looking for. Choose a variable to represent that quantity. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Solve the equation using good algebra techniques. Check the answer in the problem and make sure it makes sense. Answer the question with a complete sentence.

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See ). 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 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 has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 4 by 2 by 3 rectangular solid has 24 cubic units.

Altogether there are 24 cubic units. Notice that 24 is the length×width×height. The volume, V, of any rectangular solid is the product of the length, width, and height. V=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, is equal to length×width. B=L·W We can substitute B for L·W in the volume formula to get another form of the volume formula. We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4×2×3 rectangular solid we started with. See .

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=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. units The surface area S of the rectangular solid shown in is 52 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 two dimensions, either length and width, length and height, or width and height (see ). 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+2WH

For each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all.

For a rectangular solid with length L, width W, and height H: Doing the Manipulative Mathematics activity “Painted Cube” will help you develop a better understanding of volume and surface area. For a rectangular solid with length 14 cm, height 17 cm, and width 9 cm, find the ⓐ volume and ⓑ surface area. 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 rectangular solid Step 3. Name. Choose a variable to represent it. Let V= volume Step 4. Translate.Write the appropriate formula.Substitute. V=LWHV=14⋅9⋅17 Step 5. Solve the equation. V=2,142 Step 6. CheckWe leave it to you to check your calculations. Step 7. Answer the question. The volume is 2,142 cubic centimeters. ⓑ Step 2. Identify what you are looking for. the surface area of the solid Step 3. Name. Choose a variable to represent it. Let S= surface area Step 4. Translate.Write the appropriate formula.Substitute. S=2LH+2LW+2WHS=2(14⋅17)+2(14⋅9)+2(9⋅17) Step 5. Solve the equation. S=1,034 Step 6. Check: Double-check with a calculator. Step 7. Answer the question. The surface area is 1,034 square centimeters. Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length 8 feet, width 9 feet, and height 11 feet. ⓐ 792 cu. ftⓑ 518 sq. ft Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length 15 feet, width 12 feet, and height 8 feet. ⓐ 1,440 cu. ftⓑ 792 sq. ft A rectangular crate has a length of 30 inches, width of 25 inches, and height of 20 inches. Find its ⓐ volume and ⓑ surface area. 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 crate Step 3. Name. Choose a variable to represent it. let V= volume Step 4. Translate.Write the appropriate formula.Substitute. V=LWHV=30⋅25⋅20 Step 5. Solve the equation. V=15,000 Step 6. Check: Double check your math. Step 7. Answer the question. The volume is 15,000 cubic inches. ⓑ Step 2. Identify what you are looking for. the surface area of the crate Step 3. Name. Choose a variable to represent it. let S= surface area Step 4. Translate.Write the appropriate formula.Substitute. S=2LH+2LW+2WHS=2(30⋅20)+2(30⋅25)+2(25⋅20) Step 5. Solve the equation. S=3,700 Step 6. Check: Check it yourself! Step 7. Answer the question. The surface area is 3,700 square inches. A rectangular box has length 9 feet, width 4 feet, and height 6 feet. Find its ⓐ volume and ⓑ surface area. ⓐ 216 cu. ftⓑ 228 sq. ft A rectangular suitcase has length 22 inches, width 14 inches, and height 9 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 2,772 cu. in.ⓑ 1,264 sq. in.

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=6s2So for a cube, the formulas for volume and surface area are V=s3 and S=6s2. For any cube with sides of length s, A cube is 2.5 inches on each side. Find its ⓐ volume and ⓑ surface area. 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 cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.Write the appropriate formula. V=s3 Step 5. Solve. Substitute and solve. V=(2.5)3V=15.625 Step 6. Check: Check your work. Step 7. Answer the question. The volume is 15.625 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=6s2 Step 5. Solve. Substitute and solve. S=6⋅(2.5)2S=37.5 Step 6. Check: The check is left to you. Step 7. Answer the question. The surface area is 37.5 square inches. For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube. ⓐ 91.125 cu. mⓑ 121.5 sq. m For a cube with side 7.3 yards, find the ⓐ volume and ⓑ surface area of the cube. ⓐ 389.017 cu. yd.ⓑ 319.74 sq. yd. A notepad cube measures 2 inches on each side. Find its ⓐ volume and ⓑ surface area. 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 cube Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.Write the appropriate formula. V=s3 Step 5. Solve the equation. V=23V=8 Step 6. Check: Check that you did the calculations correctly. Step 7. Answer the question. The volume is 8 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=6s2 Step 5. Solve the equation. S=6⋅22S=24 Step 6. Check: The check is left to you. Step 7. Answer the question. The surface area is 24 square inches. A packing box is a cube measuring 4 feet on each side. Find its ⓐ volume and ⓑ surface area. ⓐ 64 cu. ftⓑ 96 sq. ft A wall is made up of cube-shaped bricks. Each cube is 16 inches on each side. Find the ⓐ volume and ⓑ surface area of each cube. ⓐ 4,096 cu. in.ⓑ 1536 sq. in.

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 π with 3.14. For a sphere with radius r: A sphere has a radius 6 inches. Find its ⓐ volume and ⓑ surface area. 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=43πr3 Step 5. Solve. V≈43(3.14)63V≈904.32cubic 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 sphere Step 3. Name. Choose a variable to represent it. let S = surface area Step 4. Translate.Write the appropriate formula. S=4πr2 Step 5. Solve. S≈4(3.14)62S≈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. Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters. ⓐ 113.04 cu. cmⓑ 113.04 sq. cm Find the ⓐ volume and ⓑ surface area of each sphere with a radius of 1 foot ⓐ 4.19 cu. ftⓑ 12.56 sq. ft A globe of Earth is in the shape of a sphere with radius 14 inches. Find its ⓐ volume and ⓑ surface area. Round the answer to the nearest hundredth. 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 π) V=43πr3V≈43(3.14)143 Step 5. Solve. V≈11,488.21 Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The volume is approximately 11,488.21 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 π) S=4πr2S≈4(3.14)142 Step 5. Solve. S≈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. A beach ball is in the shape of a sphere with radius of 9 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 3052.08 cu. in.ⓑ 1017.36 sq. in. A Roman statue depicts Atlas holding a globe with radius of 1.5 feet. Find the ⓐ volume and ⓑ surface area of the globe. ⓐ 14.13 cu. ft ⓑ 28.26 sq. ft

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.

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=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, πr2. compares how the formula V=Bh is used for rectangular solids and cylinders.

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 .

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 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 S=2πr2+2πrh For a cylinder with radius r and height h: A cylinder has height 5 inches and radius 3 inches. Find the ⓐ volume and ⓑ surface area. 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 π) V=πr2hV≈(3.14)32⋅5 Step 5. Solve. V≈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 π) S=2πr2+2πrhS≈2(3.14)32+2(3.14)(3)5 Step 5. Solve. S≈150.72 Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The surface area is approximately 150.72 square inches. Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm. ⓐ 351.68 cu. cmⓑ 276.32 sq. cm Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft. ⓐ 100.48 cu. ftⓑ 125.6 sq. ft 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. 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 π) V=πr2hV≈(3.14)42⋅13 Step 5. Solve. V≈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 π) S=2πr2+2πrhS≈2(3.14)42+2(3.14)(4)13 Step 5. Solve. S≈427.04 Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The surface area is approximately 427.04 square centimeters. 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. ⓐ 3,818.24 cu. cmⓑ 1,356.48 sq. cm 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. ⓐ 91.5624 cu. ftⓑ 113.6052 sq. ft

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 .

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. We can think of a cone as part of a cylinder. 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.

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 Since the base of a cone is a circle, we can substitute the formula of area of a circle, πr2 , for B to get the formula for volume of a cone. In this book, we will only find the volume of a cone, and not its surface area. For a cone with radius r and height h. Find the volume of a cone with height 6 inches and radius of its base 2 inches. 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 cone Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate.Write the appropriate formula.Substitute. (Use 3.14 for π) V=13πr2hV≈133.14(2)2(6) Step 5. Solve. V≈25.12 Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The volume is approximately 25.12 cubic inches. Find the volume of a cone with height 7 inches and radius 3 inches 65.94 cu. in. Find the volume of a cone with height 9 centimeters and radius 5 centimeters 235.5 cu. cm 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 8 inches tall and 5 inches in diameter? Round the answer to the nearest hundredth. Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone. Step 2. Identify what you are looking for. the volume of the cone Step 3. Name. Choose a variable to represent it. let V = volume Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for π, and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.) V=13πr2hV≈133.14(2.5)2(8) Step 5. Solve. V≈52.33 Step 6. Check: We leave it to you to check your calculations. Step 7. Answer the question. The volume of the wrap is approximately 52.33 cubic inches. How many cubic inches of candy will fit in a cone-shaped piñata that is 18 inches long and 12 inches across its base? Round the answer to the nearest hundredth. 678.24 cu. in. What is the volume of a cone-shaped party hat that is 10 inches tall and 7 inches across at the base? Round the answer to the nearest hundredth. 128.2 cu. in.

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

Volume and Surface Area of a Rectangular Solid V=LWH S=2LH+2LW+2WH Volume and Surface Area of a Cube V=s3 S=6s2 Volume and Surface Area of a Sphere V=43πr3 S=4πr2 Volume and Surface Area of a Cylinder V=πr2h S=2πr2+2πrh Volume of a Cone For a cone with radius r and height h: Volume: V=13πr2h

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. length 2 meters, width 1.5 meters, height 3 meters ⓐ 9 cu. mⓑ 27 sq. m length 5 feet, width 8 feet, height 2.5 feet length 3.5 yards, width 2.1 yards, height 2.4 yards ⓐ 17.64 cu. yd.ⓑ 41.58 sq. yd. length 8.8 centimeters, width 6.5 centimeters, height 4.2 centimeters In the following exercises, solve. Moving van A rectangular moving van has length 16 feet, width 8 feet, and height 8 feet. Find its ⓐ volume and ⓑ surface area. ⓐ 1,024 cu. ftⓑ 640 sq. ft Gift box A rectangular gift box has length 26 inches, width 16 inches, and height 4 inches. Find its ⓐ volume and ⓑ surface area. Carton A rectangular carton has length 21.3 cm, width 24.2 cm, and height 6.5 cm. Find its ⓐ volume and ⓑ surface area. ⓐ 3,350.49 cu. cmⓑ 1,622.42 sq. cm Shipping container A rectangular shipping container has length 22.8 feet, width 8.5 feet, and height 8.2 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. 5 centimeters ⓐ 125 cu. cmⓑ 150 sq. cm 6 inches 10.4 feet ⓐ 1124.864 cu. ft.ⓑ 648.96 sq. ft 12.5 meters In the following exercises, solve. Science center Each side of the cube at the Discovery Science Center in Santa Ana is 64 feet long. Find its ⓐ volume and ⓑ surface area. ⓐ 262,144 cu. ftⓑ 24,576 sq. ft Museum A cube-shaped museum has sides 45 meters long. Find its ⓐ volume and ⓑ surface area. Base of statue The base of a statue is a cube with sides 2.8 meters long. Find its ⓐ volume and ⓑ surface area. ⓐ 21.952 cu. mⓑ 47.04 sq. m 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. 3 centimeters ⓐ 113.04 cu. cmⓑ 113.04 sq. cm 9 inches 7.5 feet ⓐ 1,766.25 cu. ftⓑ 706.5 sq. ft 2.1 yards In the following exercises, solve. Round answers to the nearest hundredth. Exercise ball An exercise ball has a radius of 15 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 14,130 cu. in.ⓑ 2,826 sq. in. Balloon ride The Great Park Balloon is a big orange sphere with a radius of 36 feet . Find its ⓐ volume and ⓑ surface area. Golf ball A golf ball has a radius of 4.5 centimeters. Find its ⓐ volume and ⓑ surface area. ⓐ 381.51 cu. cmⓑ 254.34 sq. cm Baseball A baseball has a radius of 2.9 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. radius 3 feet, height 9 feet ⓐ 254.34 cu. ftⓑ 226.08 sq. ft radius 5 centimeters, height 15 centimeters radius 1.5 meters, height 4.2 meters ⓐ 29.673 cu. mⓑ 53.694 sq. m radius 1.3 yards, height 2.8 yards In the following exercises, solve. Round answers to the nearest hundredth. Coffee can A can of coffee has a radius of 5 cm and a height of 13 cm. Find its ⓐ volume and ⓑ surface area. ⓐ 1,020.5 cu. cmⓑ 565.2 sq. cm Snack pack A snack pack of cookies is shaped like a cylinder with radius 4 cm and height 3 cm. Find its ⓐ volume and ⓑ surface area. Barber shop pole A cylindrical barber shop pole has a diameter of 6 inches and height of 24 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 678.24 cu. in.ⓑ 508.68 sq. in. Architecture A cylindrical column has a diameter of 8 feet and a height of 28 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. height 9 feet and radius 2 feet 37.68 cu. ft height 8 inches and radius 6 inches height 12.4 centimeters and radius 5 cm 324.47 cu. cm height 15.2 meters and radius 4 meters In the following exercises, solve. Round answers to the nearest hundredth. Teepee What is the volume of a cone-shaped teepee tent that is 10 feet tall and 10 feet across at the base? 261.67 cu. ft Popcorn cup What is the volume of a cone-shaped popcorn cup that is 8 inches tall and 6 inches across at the base? Silo What is the volume of a cone-shaped silo that is 50 feet tall and 70 feet across at the base? 64,108.33 cu. ft Sand pile What is the volume of a cone-shaped pile of sand that is 12 meters tall and 30 meters across at the base?

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 30 feet tall with base radius 1 foot. The smaller cone is 10 feet tall with base radius of 0.5 feet. To the nearest tenth, ⓐ find the volume of the large cone. ⓑ find the volume of the small cone. ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone. ⓐ 31.4 cu. ft ⓑ 2.6 cu. ft ⓒ 28.8 cu. ft Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of 2.5 inches. A waffle cone is 7 inches tall and has a diameter of 3.25 inches. To the nearest hundredth, ⓐ find the volume of the regular ice cream cone. ⓑ find the volume of the waffle cone. ⓒ how much more ice cream fits in the waffle cone compared to the regular cone?

The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape. Answers will vary. Which has a larger volume, a cube of sides of 8 feet or a sphere with a diameter of 8 feet? Explain your reasoning.

ⓐ 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?

cone A cone is a solid figure with one circular base and a vertex. cube A cube is a rectangular solid whose length, width, and height are equal. cylinder A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.