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Pre-Algebra Interactive Chalkboard

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Lesson 11-1Three-Dimensional Figures

Lesson 11-2Volume: Prisms and Cylinders

Lesson 11-3Volume: Pyramids and Cones

Lesson 11-4Surface Area: Prisms and Cylinders

Lesson 11-5Surface Area: Pyramids and Cones

Lesson 11-6Similar Solids

Lesson 11-7Precision and Significant Digits

Example 1 Identify Prisms and Pyramids

Example 2 Identify Diagonals and Skew Lines

Example 3 Analyze Real-World Drawings

Identify the solid. Name the bases, faces, edges, and vertices.

Answer: This figure has two parallel congruent bases that are rectangles, GHJK and LMNP, so it is a rectangular pyramid.

faces: GHJK, LMNP, GHML, HJNM, JKPN, GKPL

edges:

vertices: G, H, J, K, L, M, N, P

Identify the solid. Name the bases, faces, edges, and vertices.

Answer: This figure has one triangular base, DEF, so it is atriangular pyramid.

faces: DEF, DEG, DFG, EFG

edges:

vertices: D, E, F, G

Identify each solid. Name the bases, faces, edges, and vertices.

a.

Answer: rectangular pyramid

base: BCDE

faces: ABC, ACD, ADE, AEB, BCDE

edges:

vertices: A, B, C, D, E

Identify each solid. Name the bases, faces, edges, and vertices.

b.

Answer: rectangular prism

bases: GHJK, LMNP or GKPL,HJNM or GHML, KJNP

faces: GHJK, LMNP, GHML, HJNM, JKPN, GKPL

edges:

vertices: G, H, J, K, L, M, N, P

Identify a diagonal and name all segments that are skew to it.

Answer: is a diagonal because vertex Q and vertex W do not intersect any of the same faces;

Identify a diagonal and name all segments that are skew to it.

Answer:

Architecture An architect’s sketch shows the plans for a new office building.

Find the area of the ground floor if each unit on the drawing represents 55 feet.

The drawing is 6 5, so the actual dimensions are 6(55) 5(55) or 330 feet by 275 feet.

Formula for area

Answer: The area of the ground floor is 90,750 square feet.

How many floors are in the office building if each floor is 12 feet high? Assume each unit on the drawing represents 40 feet.

You can see from the side view that the height of the building is 3 units.

total height:

number of floors:

Answer: There are 10 floors in the office building.

Architecture An architect’s sketch shows the plans for a new office building.

a. Find the area of the ground floor if each unitrepresents 75 feet.

b. How many floors are in the office building if eachfloor is 15 feet high? Assume each unit on the drawing represents 45 feet.

Answer: 168,750 ft2

Answer: 9 floors

Example 1 Volume of a Rectangular Prism

Example 2 Volume of a Triangular Prism

Example 3 Height of a Prism

Example 4 Volume of a Complex Solid

Example 5 Volume of a Cylinder

Find the volume of the prism.

Formula for volume of a prism

The base is a rectangle, so

Simplify.

Answer: The volume is 3200 cubic centimeters.

Find the volume of the prism.

Answer: The volume is 45 ft3.

Find the volume of the triangular prism.


The height of the prism is 3 in.

B = area of base or .

Simplify.

Answer: The volume is 15 cubic inches.

Find the volume of the triangular prism.

Answer: The volume is 15 ft3.

Baking Cake batter is poured into a pan that is a rectangular prism whose base is an 8-inch square. If the cake batter occupies 192 cubic inches, what will be the height of the batter?


Formula for volume of a rectangular prism

Simplify.

Divide each side by 64.

Answer: The height of the batter is 3 inches.

Swimming Pool A swimming pool is filled with 960 cubic feet of water. The pool is a rectangular prism 20 feet long and 12 feet wide and is the same depth throughout. Find the depth of the water.

Answer: The water is 4 feet deep.

Multiple-Choice Test Item

Find the volume of the solid.

A 262 m3 B 918 m3

C 972 m3 D 1458 m3

Read the Test ItemThe solid is made up of a rectangular prism and a triangular prism. The volume of the solid is the sum of both volumes.

Solve the Test Item

Step 1 The volume of the rectangular prism is 12(9)(9)or 972 m3.

Step 2 In the triangular prism, the area of the base is

and the height is 12. Therefore, the

volume is

Step 3 Add the volumes.

Answer: The answer is D.

Multiple-Choice Test Item

Find the volume of the solid.

A 932 in3

B 896 in3

C 1432 in3

D 718 in3

Answer: The answer is B.

Find the volume of the cylinder. Round to the nearest tenth.

Formula for volume of a cylinder

Replace r with 7 and h with 14.

Simplify.

Answer: The volume is about 2155.1 cubic feet.

Find the volume of the cylinder. Round to the nearest tenth.

diameter of base 10 m, height 2 m

Formula for volume of a cylinder

Replace r with 5 and h with 2.

Simplify.

Answer: The volume is about 157.1 cubic meters.

Since the diameter is 10 m, the radius is 5 m.

Find the volume of each cylinder. Round to the nearest tenth.

a.

b. diameter of base 8 cm, height 6 cm

Answer: 351.9 in3

Answer: 301.6 cm3

Example 1 Volumes of Pyramids

Example 2 Volume of a Cone

Example 3 Use Volume to Solve Problems

Find the volume of the pyramid. If necessary, round to the nearest tenth.

Formula for volume of a pyramid

The base is a square, so

The height of the pyramid is 12 inches.

Simplify.

Answer: The volume is 900 cubic inches.

Find the volume of the pyramid. If necessary, round to the nearest tenth.

base area 19 cm2, height 21 cm

Formula for volume of a pyramid

Replace B with 19 and h with 21.

Answer: The volume is 133 cubic centimeters.

Simplify.

Find the volume of each pyramid. If necessary, round to the nearest tenth.

a.

b. base area 32 cm2, height 9 cm

Answer: 112 in3

Answer: 96 cm3

Find the volume of the cone. Round to the nearest tenth.

Formula for volume of a cone

Replace r with 5.5 and h with 8.

Simplify.

Answer: The volume is about 253.4 cubic meters.

Find the volume of the cone. Round to the nearest tenth.

Answer: 422.7 in3

Landscaping When mulch was dumped from a truck, it formed a cone-shaped mound with a diameter of 15 feet and a height of 8 feet.

What is the volume of the mulch?

Formula for volume of a cone

Since d = 15, replace r with 7.5. Replace h with 8.

Answer: The volume of the mulch is about 471 cubic feet.

Landscaping When mulch was dumped from a truck, it formed a cone-shaped mound with a diameter of 15 feet and a height of 8 feet.

How many square feet can be covered with this mulch if 1 cubic foot covers 4 square feet of ground?

Answer: 1884 square feet can be covered with this mulch.

Playground A load of wood chips for a playground was dumped and formed a cone-shaped mound with a diameter of 10 feet and a height of 6 feet.

a. What is the volume of the wood chips?

b. How many square feet of the playground can be covered with wood chips if 1 cubic foot of wood chips can cover 3 square feet of the playground?

Answer: about 157 ft3

Answer: 471 ft2

Example 1 Surface Area of a Rectangular Prism

Example 2 Surface Area of a Triangular Prism

Example 3 Surface Area of a Cylinder

Example 4 Compare Surface Areas

Find the surface area of the rectangular prism.

Write the formula.

Substitution

Simplify.

Answer: The surface area of the rectangular prism is 1868 square centimeters.

Find the surface area of the rectangular prism.

Answer: 444 in2

Find the surface area of the triangular prism.

Find the area of each face.

Bottom

Left side

Right side

Two bases

Add to find the total surface area.

Answer: The surface area of the triangular prism is 336 square meters.

Find the surface area of the triangular prism.

Answer: 96 ft2

Find the surface area of the cylinder. Round to the nearest tenth.

Formula for surface area of a cylinder

Replace r with 2.5 andh with 8.

Simplify.

Answer: The surface area of the cylinder is about 164.9 square meters.

Find the surface area of the cylinder. Round to the nearest tenth.

Answer: 1504.4 in2

Cereals A company packages its cereal in a rectangular prism that is 2.5 inches by 7 inches by 12 inches. It is considering packaging it in a cylinder-shaped container having a 6-inch diameter and aheight of 7.5 inches. Which uses the least amount of packaging?

Surface area of rectangular prism

top/bottom front/back sides

Surface area of cylinder

top/bottom curved surface

Answer: Since 197.9 square inches < 263 square inches, the cylinder uses less packaging.

Candy A candy company is deciding between two types of packaging for its gumballs. The first option is a rectangular prism that is 6 inches by 4 inches by 1.5 inches. The second option is a cylinder having a radius of 2 inches and a height of 5 inches. Which option requires less packaging?

Answer: The rectangular prism requires less packaging. 78 < 88.0

Example 1 Surface Area of a Pyramid

Example 2 Use Surface Area to Solve a Problem

Example 3 Surface Area of a Cone

Find the surface area of the square pyramid.

Find the lateral area and the base area.

Area of each lateral face

Area of a triangle

Replace b with 8 and h with 8.9.

Simplify.

There are 4 faces, so the lateral area is 4(35.6) or 142.4 square feet.

Area of base

Replace s with 8 and simplify.

Area of a square

The surface area of a pyramid equals

the lateral area plus

the area of the base.

S 142.4 64

Answer: The surface area of the square pyramid is 206.4 square feet.

Find the surface area of the square pyramid.

Answer: 42 m2

Canopies A canopy is in the shape of a square pyramid that is 3.4 meters on each side. The slant height is 2 meters. How much canvas is used for the canopy?

Find the lateral area only, since there is no bottom to the canopy.

Area of each lateral face

Formula for area of a triangle

Replace b with 3.4 andh with 2.

Simplify.

One lateral face has an area of 3.4 square meters. There are 4 lateral faces, so the lateral area is 4(3.4) or 13.6 square meters.

Answer: 13.6 square meters of canvas was used to cover the canopy.

Tent A tent is in the shape of a square pyramid that is 8 feet on each side. The slant height is 10 feet. Find the surface area of the tent.

Answer: 160 ft2

Find the surface area of the cone. Round to the nearest tenth.

Formula forsurface area of a cone

Replace r with 3.5 andwith 10.

Simplify.

Answer: The surface area of the cone is about 148.4 square feet.

Find the surface area of the cone. Round to the nearest tenth.

Answer: 587.4 cm2

Example 1 Identify Similar Solids

Example 2 Find Missing Measures

Example 3 Use Similar Solids to Solve a Problem

Determine whether the pair of solids is similar.

Write a proportion comparing radii and heights.

Find the cross products.

Simplify.

Answer: The radii and heights are not proportional, so the cylinders are not similar.

Write a proportion comparing corresponding edge lengths.


Simplify.

Answer: The corresponding measures are proportional, so the pyramids are similar.



a.

Answer: yes


b.

Answer: no

The cylinders to the right are similar. Find the radiusof cylinder A.

Substitute the known values.


Simplify.

Divide each side by 6.

Answer: The radius of cylinder A is 6 centimeters.

The rectangular prisms below are similar. Find the height of prism B.

Answer: 4.5 in.

Doll Houses Lita made a model of her fish tank for

her doll house. The model is exactly the size of

the original fish tank, whose dimensions are

120 30 38 cm. What is the volume of the model?

Explore You know the scale factor

and the volume of the fish tank is

Plan Since the volumes have a ratio of

, replace a with 1

and b with 25 in .

SolveWrite the ratio of volumes.

Replace a with 1and b with 25.

Simplify.

So, the volume of the tank is 15,625 times the volume of the model.

Answer: The volume of the model is or about 8.8 cubic centimeters.

Examine Check your answer by finding the dimensions of the model.

Next, find the volume of the model using these dimensions.

Trains A scale model of a railroad boxcar is in the

shape of a rectangular prism and is the size of the

actual boxcar. The scale model has a volume of 72

cubic inches. What is the volume of the actual boxcar?

Answer: 9,000,000 in3

Example 1 Identify Precision Units

Example 2 Identify Significant Digits

Example 3 Add Measurements

Example 4 Multiply Measurements

Identify the precision unit of the thermometer shown on the right.

Answer: The precision unit is 5°F.

Identify the precision unit of the ruler shown on the right.

Answer:

Determine the number of significant digits in 1040 miles.

Answer: 3 significant digits

Determine the number of significant digits in 0.003 centimeter.

Answer: 1 significant digit

Determine the number of significant digits in 90.051 kilograms.


Determine the number of significant digits in 0.06300 liter.


Determine the number of significant digits in each measure.

a. 34.70 inches

b. 0.000003 mile

c. 2300 centimeters

d. 2.08 meters

Answer: 4

Answer: 1

Answer: 2

Answer: 3

The sides of a quadrilateral measure 0.6 meter, 0.044 meter, 0.024 meter, and 0.103 meter. Use the correct number of significant digits to find the perimeter.

0.6 1 decimal place

0.044 3 decimal places

0.024 3 decimal places

+ 0.103 3 decimal places

0.771

The least precise measurement, 0.6, has one decimal place. So, round 0.771 to one decimal place, 0.8.

Answer: The perimeter of the quadrilateral is about 0.8 meter.

The sides of a triangle measure 2.04 centimeters, 3.2 centimeters, and 2.625 centimeters. Use the correct number of significant digits to find the perimeter.

Answer: 7.9 cm

What is the area of the bedroom shown here?

12.25 4 significant digits

171.5 4 significant digits

x 14 2 significant digits

To find the area, multiply the length and the width.

The answer cannot have more significant digits than the measurements of the length and width. So, round 171.5 square feet to 2 significant digits.

Answer: The area of the bedroom is about 170 square feet.

Suppose a bedroom was 13.75 feet wide and 12.5 feet long. What would be the area of the bedroom?

Answer: 171 ft2

Explore online information about the information introduced in this chapter.

Click on the Connect button to launch your browser and go to the Pre-Algebra Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.pre-alg.com/extra_examples.

glencoe division · playground a load of wood chips for a playground was dumped and formed a...

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