Interactive Chalkboard

Pre-Algebra Interactive Chalkboard
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GLENCOE DIVISION
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8787 Orion Place
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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 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 unit
represents 75 feet.
Answer: 168,750 ft2
b. How many floors are in the office building if each
floor is 15 feet high? Assume each unit on the
drawing represents 45 feet.
Answer: 9 floors
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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
B = area of base or
.
The height of the prism is 3 in.
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
C 972 m3
B 918 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
B
C
D
932 in3
896 in3
1432 in3
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
Since the diameter is 10 m, the radius is 5 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.
Find the volume of each cylinder. Round to the
nearest tenth.
a.
Answer: 351.9 in3
b. diameter of base 8 cm, height 6 cm
Answer: 301.6 cm3
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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.
Simplify.
Answer: The volume is 133 cubic centimeters.
Find the volume of each pyramid. If necessary, round
to the nearest tenth.
a.
Answer: 112 in3
b. base area 32 cm2, height 9 cm
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?
Answer: about 157 ft3
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: 471 ft2
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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 and
h 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 cylindershaped container having a 6-inch diameter and a
height 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
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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
Area of a square
Replace s with 8 and simplify.
The surface
area of a
pyramid
S
equals
the lateral
area
plus
the area of
the base.
142.4
Answer: The surface area of the square pyramid is
206.4 square feet.
64
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 and
h 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 for
surface area of a cone
Replace r with 3.5 and
with 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
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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.
Determine whether the
pair of solids is 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.
a.
Answer: yes
Determine whether the pair of solids is similar.
b.
Answer: no
The cylinders to
the right are similar.
Find the radius
of 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
Solve
.
Write 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
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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
Answer: 4
b. 0.000003 mile
Answer: 1
c. 2300 centimeters
Answer: 2
d. 2.08 meters
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
0.044
0.024
+ 0.103
0.771
1 decimal place
3 decimal places
3 decimal places
3 decimal places
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?
To find the area,
multiply the length
and the width.
12.25
x 14
171.5
4 significant digits
2 significant digits
4 significant digits
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
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will find extra examples for each lesson in the
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