lesson 1 contents example 1identify prisms and pyramids example 2identify diagonals and skew lines...
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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 a triangular 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.
Formula for volume of a 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 prism
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 Item The 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.
Find the cross products.
Simplify.
Answer: The corresponding measures are proportional, so the pyramids are similar.
Determine whether the pair of solids is similar.
Determine whether the pair of solids is similar.
a.
Answer: yes
Determine whether the pair of solids is similar.
b.
Answer: no
The cylinders to the right are similar. Find the radiusof cylinder A.
Substitute the known values.
Find the cross products.
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 1 and 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.
Answer: 5 significant digits
Determine the number of significant digits in 0.06300 liter.
Answer: 4 significant digits
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 place0.044 3 decimal places0.024 3 decimal places
+ 0.103 3 decimal places
0.771The 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