Transcript 2. - Crestwood Local Schools
Area: Parallelograms PRE-ALGEBRA LESSON 10-1
It takes 126 ft of fence to enclose a field that is twice as long as it is wide. What is the area of the field?
882 ft 2
Area: Parallelograms PRE-ALGEBRA LESSON 10-1 (For help, go to Lesson 3-4.)
Use
A
=
w
and find the third value.
1. A = 54 in.
2 ,
w
= 6 in.
3. A = 25 cm 2 , = 2.5 cm
2.
4.
= 35 m,
w
= 7 m = 7.2 ft,
w
= 7.2 ft
Check Skills You’ll Need
Area: Parallelograms PRE-ALGEBRA LESSON 10-1 Solutions
1. A =
w
54 = 6 = 9 in.
3. A =
w
25 = 2.5
w w
= 10 cm 2. A
A A
=
w
= 35 • 7 = 245 m 2 4. A
A A
=
w
= 7.2 • 7.2
= 51.84 ft 2
Area: Parallelograms PRE-ALGEBRA LESSON 10-1
Find the area of the rectangle.
Step 1
Change the units so that they are the same.
150 cm = 1.5 m Change 150 centimeters to meters.
Step 2
Find the area.
A
=
bh
= ( 4 )( 1.5
) = 6 The area of the rectangle is 6 m 2 .
Use the formula for area of a rectangle.
Replace
b
and 1.5.
and
h
with the dimensions 4 Simplify.
Quick Check
Area: Parallelograms PRE-ALGEBRA LESSON 10-1 a.
Find the area of each parallelogram.
b.
A
=
bh
= (8)(2) = 16 The area is 16 m 2 .
area formula Substitute.
Simplify.
A
=
bh
= (2.5)(6) = 15 The area is 15 in.
2 .
Quick Check
Area: Parallelograms PRE-ALGEBRA LESSON 10-1
Find each area.
1.
rectangle
ABFE
40 cm 2
2.
parallelogram
ACFD
48 cm 2
3.
a rectangle with a base of 50 cm and a height of 5 cm 250 cm 2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
1 Find the area of a rectangle that is 3 ft wide and twice as high.
2 24 ft 2 2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
Find each product.
1.
3.
1 2 1 2 • 16 • 5 • 15
2.
4.
1 2 • 14 • 6 1 2
(For help, go to Lesson 5-4.)
Check Skills You’ll Need
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Solutions 1.
1 2 1 2 1 • 16 • 16 8 = 8
3.
1 2 • 5 • 15 1 2 • 75 = 37 1 2
2.
1 2 • 14 • 6 1 2 1 7 • 14 7 • 6 • 6 = 42
4.
1 2 • 2 • 8 2 1 2 • 5 2 • 8 1 2 1 • • 8 2 1 2 5 • 2 = 10
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
Find the area of the triangle.
A
1 =
bh
2 = • 13 • 6 2 = 39 The area is 39 in.
2 .
Use the formula for area of a triangle.
Replace
b
with 13 and
h
with 6.
Simplify.
Quick Check
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
Find the area of the figure.
Area of triangle
A
1 =
bh
2 = • 45 • 20 2 = 450 Add to find the total: 450 + 1,350 = 1,800.
The area of the figure is 1,800 cm 2 .
Area of rectangle
A
=
bh
= 45 • 30 = 1,350
Quick Check
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
Suppose that, through the years, a layer of silt and mud settled in the bottom of the Erie Canal. Below is the resulting cross section of the canal. Find the area of the trapezoidal cross section.
A
= 1 2
A
= 1 2 = 1 2
h
(
b
1
• •
+ 3(31 + 40) 3(71)
b
2 ) = 1 2
•
213 = 106.5
Use the formula for the area of a trapezoid.
Replace
h
Simplify.
with 3,
b
1 The area of the cross section is 106.5 ft 2 .
with 31, and
b
2 with 40.
Quick Check
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2
Find each area.
1.
trapezoid
PQRU
192 ft 2
3.
triangle
QRS
28 ft 2
2.
triangle
PTU
20 ft 2
4.
trapezoid
PQSU
164 ft 2
Area: Circles PRE-ALGEBRA LESSON 10-3
Explain how this pattern works.
(15,873 7) 2 = (222,222) (15,873 7) 3 = (333,333) (15,873 7) 4 = (444,444) because (15,873 7) 1 = (111,111)
Area: Circles PRE-ALGEBRA LESSON 10-3
Simplify each expression.
1.
3.14 • 4 2
3.
3.14 • 9 2
2.
3.14 • 5 2
4.
3.14 • 0.5
2
(For help, go to Lesson 4-7.)
Check Skills You’ll Need
Area: Circles PRE-ALGEBRA LESSON 10-3 Solutions 1.
3.14 • 4 2 = 3.14 • 4 • 4 = 50.24
3.
3.14 • 9 2 = 3.14 • 9 • 9 = 254.34
2.
3.14 • 5 2 = 3.14 • 5 • 5 = 78.5
4.
3.14 • 0.5
2 = 3.14 • 0.5 • 0.5
= 0.785
Area: Circles PRE-ALGEBRA LESSON 10-3
Find the exact area of a circle with diameter 20 in.
A
=
r
2 = (10) 2 = 100
r
1 =
d
;
r
2 = 10 Simplify.
The area is 100 in.
2 .
Quick Check
Area: Circles PRE-ALGEBRA LESSON 10-3
A TV station’s weather radar can detect precipitation in a circular region having a diameter of 100 mi. Find the area of the region.
A
=
r
2 = (50) 2 = 2,500
r
1 =
d
;
r
2 = 50 exact area (2,500)(3.14) Use 3.14 for .
= 7,850 approximate area The area of the region is about 7,850 mi 2 .
Quick Check
Area: Circles PRE-ALGEBRA LESSON 10-3
A pound of grass seed covers approximately 675 ft 2 . Find the area of the lawn below. Then find the number of bags of grass seed you need to buy to cover the lawn. Grass seed comes in 3-lb bags.
Area of region that is one fourth of a circle: area of circle =
r
2 area of quarter circle =
r
4
A
1 (3.14)(15) 2 4 = 176.625 ft 2 2 Replace with 3.14 and
r
with 15.
Area: Circles PRE-ALGEBRA LESSON 10-3 (continued)
Area of region that is a rectangle: area of rectangle =
bh A
= 45 • 25 = 1,125 ft 2 Replace
b
with 45 and
h
with 25.
The area of the lawn is about 177 ft 2 + 1,125 ft 2 = 1,302 ft 2 .
1,302 ÷ 675 1.93
Divide to find the number of pounds of seed.
You need to buy one 3-lb bag of grass seed.
Quick Check
Area: Circles PRE-ALGEBRA LESSON 10-3
Find the area.
1.
Find the exact area of a circle with diameter 32 in. 256 in.
2
2.
A 5-ft-diameter round table is in a 12 ft-by-15 ft room.
a.
What is the area covered by the table? Round to the nearest unit.
20 ft 2
b.
What is the area of the rest of the room? Round to the nearest unit.
160 ft 2
Space Figures PRE-ALGEBRA LESSON 10-4
Find the area of a square 15.7 mm on each side.
246.49 mm 2
Space Figures PRE-ALGEBRA LESSON 10-4
Judging by appearance, classify each polygon.
1.
2.
(For help, go to Lesson 9-3.) 3.
4.
Check Skills You’ll Need
Space Figures PRE-ALGEBRA LESSON 10-4 Solutions 1.
triangle
3.
rectangle
2.
square
4.
hexagon
Space Figures PRE-ALGEBRA LESSON 10-4 a.
For each figure describe the bases and name the figure.
b.
The bases are circles.
The figure is a cylinder.
The bases are rectangles.
The figure is a rectangular prism.
Quick Check
Space Figures PRE-ALGEBRA LESSON 10-4 a.
Name the space figure you can form from each net.
b.
With two hexagonal bases and rectangular sides, you can form a hexagonal prism.
With a rectangular base and triangular sides, you can form a rectangular pyramid.
Quick Check
Space Figures PRE-ALGEBRA LESSON 10-4
Describe the base(s) and name each solid.
1.
2.
circular base; cone triangular bases; triangular prism
3.
Name the solid you can form from this net. octagonal pyramid
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5
A prism has 7 faces and 10 vertices. How many edges does the prism have?
15
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 (For help, go to Lesson 9-6.)
Find the circumference of each circle with the given radius or diameter.
1. r = 5 in.
3. d = 8 ft 2. r 4. d = 4.2 cm = 6.8 in.
Check Skills You’ll Need
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Solutions
1. C = 2
r C C
= 2(3.14)(5 in.) = 31.4 in.
3. C =
d C
= (3.14)(8 ft)
C
= 25.1 ft 2. C = 2
r C C
= 2(3.14)(4.2 cm) = 26.4 cm 4. C =
d C
= (3.14)(6.8 in.)
C
= 21.3 in.
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5
Find the surface area of the rectangular prism using a net.
Draw and label a net.
Find the area of each rectangle in the net.
60 + 60 + 150 + 90 + 150 + 90 = 600 The surface area is 600 cm 2 .
Add the areas.
Quick Check
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5
Find the surface area of the rectangular prism.
Step 1
Find the lateral area.
L.A.
=
ph
Use the formula for lateral area.
= (5 + 6 + 5 + 6)20
p
= 5 + 6 + 5 + 6 and
h
= 20 = 440
Step 2
Find the surface area.
S.A. = L.A.
= 440 + 2
B
+ 2(5 • 6) = 440 + 60 = 500 L.A. = 440 and
B
= 5 • 6
Quick Check
The surface area of the rectangular prism is 500 in.
2 .
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5
Find the surface area of the cylindrical water tank.
Step 1
Find the lateral area.
L.A.
= 2
rh
2(3.14)(8)(15) 754 Use the formula for lateral area.
Step 2
Find the surface area.
S.A. = L.A.
= L.A.
754 + 2
B
+ 2(
r
2 ) + 2(3.14)(8) 2 Use the formula for surface area.
1,156
Quick Check
The surface area of the water tank is about 1,156 ft 2 .
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5
Find the surface area of each figure rounded to the nearest whole unit.
1.
triangular prism with base perimeter 24 cm, base area 24 cm 2 , and height 15 cm 408 cm 2
2.
rectangular prism with base perimeter 30 cm, base area 50 cm 2 , and height 150 cm 4,600 cm 2
3.
cylindrical candle with radius 2 cm and height 16 cm about 226 cm 2
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6
Give the number of faces, edges, and vertices of a rectangular prism.
6 faces, 12 edges, 8 vertices
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 (For help, go to Lesson 5-4.)
Use the Order of Operations to simplify each expression.
1.
2 3 1 (9 ) + (8 ) 2
2.
3 4 2 (12 ) + (15 ) 5
3.
1 6 (24 ) + (3 ) 3
4.
5 8 (32 ) + (14 ) 7
Check Skills You’ll Need
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Solutions 1.
2 3 1 • 9 3 = 6 + 4 = 10 + • 8 4 2 1
3.
1 6 1 • 24 4 = 4 + = 5 + • 3 1 3 1
2.
3 4 1 12 3 5 1
4.
8 1 = 9 + 6 = 15 • 32 = 22 4 + • = 20 + 2 7 1 15 3 14 2
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6
Find the surface area of the square pyramid.
Step 1
L.A.
1 =
p
2 = • 20 • 8 = 80 2 Use the formula for lateral area.
p
= 4(5) and = 8.
Step 2
S.A. = L.A. +
B
= 80 + 5 2 = 80 + 25 = 105 Lateral area = 80 and The surface area of the pyramid is 105 m 2 .
B
= 5 2 .
Quick Check
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6
Find the surface area of the cone.
Step 1
L.A.
=
r
3.14(3)(7) = 65.94
r
Use the formula for lateral area.
= 3 and = 7.
Step 2
S.A. = L.A. +
B
65.94
+ 3.14(3) 2 = 65.94 + 28.26
= 94.2
Use the formula for surface area.
L.A. 65.94 and The surface area of the cone is about 94 m 2 .
B
= (3) 2 .
Quick Check
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6
Earth has an average radius of 3,963 mi. What is Earth’s approximate surface area to the nearest 1,000 mi 2 ? Assume that Earth is a sphere.
S.A. = 4
r
2 4(3.14)(3,963) 2 = 197,259,434.64
r
Use the formula for surface area.
3,963 Multiply.
197,259,000 Round to nearest 1,000.
The surface area of Earth is about 197,259,000 mi 2 .
Quick Check
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6
Find the surface area of each space figure. Round to the nearest whole unit.
1.
a square pyramid with base edge 80 m and slant height 100 m 22,400 m 2
2.
a cone with slant height 22 cm and radius 7 cm about 637 cm 2
3.
a sphere with radius 12 cm about 1,809 cm 2
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7
Use graph paper. Design and draw a diagram to determine which has the greater area —a square with sides 10 cm or a circle with a diameter 10 cm?
Draw a circle within the square to prove that the square has the greater area.
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7
Find the area of each circle.
1.
2.
(For help, go to Lesson 10-3.) 3.
Check Skills You’ll Need
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Solutions
1. A =
r
2 = (8 2 )( ) = 64(3.14) 201 cm 2 2. A =
r
2 = (12 2 )( ) = 144(3.14) 452.2 cm 2
3.
r
=
d
2 1 = (20)
A
= 10 =
r
2 = (10 2 )( ) = 100(3.14) 314 cm 2
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7
Find the volume of the triangular prism.
V
=
Bh
= 63 • 20 = 1,260 Use the formula for volume.
B
= • 9 • 14 = 63 cm 2 2 Simplify.
The volume is 1,260 cm 3 .
Quick Check
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7
Find the volume of the juice can, to the nearest cubic centimeter.
V
=
Bh V
=
r
2
h
3.14 • 3.4
2 • 16 Use the formula for volume.
B
=
r
2 Replace
r =
580.7744
Simplify.
The volume is about 581 cm 3 .
with 3.4, and
h
with 16.
Quick Check
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7
Find the volume of each space figure.
1.
rectangular prism with base 12 m by 14 m and height 50 m 8,400 m 3
2.
cylindrical pool with diameter 24 ft and height 4 ft about 1,808.64 ft 3
3.
right triangular prism with base legs 8 cm and 10 cm and height 20 cm 800 cm 3
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8
A roll of wallpaper is 24 in. wide. You used all but 4 ft of one roll.
1 Another roll has 7 ft of wallpaper on it. What is the total area you can cover with the remaining wallpaper?
1 2 23 ft 2 3
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 (For help, go to Lesson 9-3.)
Draw each figure described below.
1.
a rectangle with small squares drawn in each corner
2.
a rectangle divided into eight congruent rectangles
3.
two parallelograms that have different shapes but the same perimeter
Check Skills You’ll Need
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 Solutions
Answers may vary. Samples:
1.
2.
3.
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8
A can company rolls rectangular pieces of metal that measure 8 in. by 10 in. to make the sides of cans. Which height, 8 in. or 10 in., will make the can with the greater volume?
Build two cans using 8 in.-by-10 in. pieces of paper. You do not need to make the bases, just the sides.
not to scale
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 (continued) not to scale
Measure your models to find approximate radii.
Radius of 10-in. high can 1.3 in.
Radius of 8-in. high can 1.6 in.
Find the volumes.
V
=
r
2
h
(3.14)(1.3
2 )(10) 53.1 in.
3
V
=
r
2
h
(3.14)(1.6
64.3 in.
The volume of the can with height 8 in. is greater.
3 2 )(8)
Quick Check
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8
Solve.
1.
You cut square corners from a piece of cardboard that has dimensions 32 cm by 40 cm. You then fold the cardboard to create a box with no lid. To the nearest centimeter, what are the dimensions of the box that will have the greatest volume? 20 cm by 28 cm by 6 cm
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Multiply. Write each answer in simplest form.
a.
3 3 4 1 2 1 7 8
b.
1 1 8 2 1 6 7 2 16
c.
1 7 2 5 2 35
10-9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Multiply.
1.
3.
1 (3.14)(2) 2 (5) 3 4 (3.14)(2) 3 3
2.
4.
1 (4) 2 (6) 3 4 3 (3.14)(0.5) 3
(For help, go to Lesson 5-4.) 10-9 Check Skills You’ll Need
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9 Solutions 1.
1 3 (3.14)(2 2 1 )(5) = (3.14)(4)(5) 3 = (62.8) 3 = 20.93
3.
4 3 (3.14)(2 3 ) = (3.14)(8) 3 = (25.12) 3 = 33.493
2.
1 3 (4 2 1 )(6) = (16)(6) = (96) 3 = 32
4.
1 (3.14)(0.5) 3 3 4 = (3.14)(0.125) 3 = (0.3925) 3 = 0.523
10-9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Find the volume of the cone.
V
1 =
Bh
3
V
= 1 3
r
2
h
1 3 (3.14)(2) 2 (12) = 50.24
Use the formula for volume.
B
=
r
Replace Simplify.
r
2 The volume of the cone is about 50 in.
3 .
with 2 and
h
with 12.
10-9 Quick Check
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Find the volume of the square pyramid.
V V
1 =
Bh
3 = 1 3
s
2
h
1 = (8) 2 (12) 3 Use the formula for volume.
B
=
s
2 Replace = 256 Simplify.
The volume of the pyramid is 256 in.
3 .
s
with 8 and
h
with 12.
10-9 Quick Check
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Earth has an average radius of 3,963 mi. What is Earth’s approximate volume to the nearest 1,000,000 mi 3 ? Assume that Earth is a sphere.
V
4 = 3
r
3 4 3 (3.14)(3,963) 3 260,579,713,159 Use the volume formula.
Replace Simplify.
r
with 3,963.
The volume of the Earth is about 260,580,000,000 mi 3 .
Quick Check 10-9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-9
Find the volume of each space figure to the nearest unit.
1.
a cone with diameter 9 cm and height 12 cm
2.
about 254 cm 2 a square pyramid with base edges 12 m and height 18 m 864 m 3
3.
a basketball with diameter 10 in. about 523 in.
3
10-9