We included HMH Into Math Grade 7 Answer Key PDF Module 10 Lesson 3 Describe and Analyze Cross Sections of Circular Solids to make students experts in learning maths.
HMH Into Math Grade 7 Module 10 Lesson 3 Answer Key Describe and Analyze Cross Sections of Circular Solids
I Can identify the shapes of cross sections of circular solids and solve problems involving the areas of cross sections.
Spark Your Learning
Cassie and Amanda are making a cylindrical layer cake with four different-flavored layers. Amanda wants a circular piece of cake with only one flavor. Cassie wants a piece of cake with a rectangular face and all four flavors. Show how each girl could make a single cut to the cake to get the piece each one wants. Can both girls get the piece of cake they want from the same cake? Explain.
Answer:
Yes, both girls get the piece of cake they want from the same cake.
Explanation:
Vertical cross section cut of cake piece can have a rectangular face and all four flavors.
Horizontal cross section cut of cake piece can have a circular face and one flavor.
Turn and Talk For each type of cut, does the location where the cake is cut change the shape of the piece of cake?
Answer:
Yes, depends upon the location of cake cut there is a change of cake piece shape.
Explanation:
Vertical cross section cut of cake piece can have a rectangular shape.
Horizontal cross section cut of cake piece can have a circular shape.
Build Understanding
Question 1.
An art class is cutting foam figures for a project.
Connect to Vocabulary
A plane is a flat surface that has no thickness and extends forever. When a plane intersects a solid, the two-dimensional figure formed is called a cross-section. Some cross-sections of cones and cylinders are circles.
A. Jerome has two identical cylinders with the dimensions shown. He cuts one horizontally and one vertically through the centers of the bases. What are the figures and dimensions of the cross-sections? Make sketches of the cross-sections.
Answer:
Explanation:
circle of diameter = 2 cm
rectangle of length = 5 cm and width 2 cm
B. Jana has two identical cones like the one shown. She cuts one horizontally and one vertically through the center of the base. What are the figures and the given dimensions of the cross sections? Make sketches of the cross sections.
Answer:
Explanation:
Vertical cross section cut of cone can have a triangular shape.
Horizontal cross section cut of cone can have a circular shape.
C. Javier has two identical spheres like the one shown. He cuts one horizontally through the center and one vertically through the center. What are the figures and dimensions of the cross-sections? Make sketches of the cross-sections.
Answer:
Explanation:
When cross section takes horizontally through the center and vertically through the center,
out put of both is in circle in shape.
Turn and Talk Does the location of a horizontal cross-section of the cylinder change the dimensions of the cross-section? Explain.
Answer:
Explanation:
cross section of cylinder as shown in the above figure A, B and C
A is horizontal cross section
C is vertical cross section
B is cross section inclined position
as location changes the shape is also changes as shown in the above figures
Step It Out
Question 2.
A cylindrical barrel that is used to collect rainwater is shown.
A. Sketch the horizontal cross-section. Label the radius. Find the circumference and the area, to the nearest hundredth, of the horizontal cross-section of the barrel. Use 3.14 for π.
Answer:
Area = 99.35 sq in ,
Circumference = 35.325 in
Explanation:
the circumference and the area
C = 2πr area A = πr2
d = 11.25 in
r = 11.25/2
r = 5.625 in
circumference
C = 2πr
C = 2 x 3.14 x 5.625 = 35.325 in
Area A = πr2
A = 3.14 x 5.625 x 5.625
A = 99.35 sq in
B. Sketch the vertical cross-section through a diameter of the base and label the length and width. Find the area of this vertical cross-section of the barrel.
Answer:
376.875 sq in
Explanation:
the base and label the length and width
length = 33.5 in and width = 11.25
Area of a rectangle = length x width
A = 11.25 x 33.5
A = 376.875 sq in
Turn and Talk What do you notice about the width of a vertical cross-section through the centers of the bases and the diameter of the horizontal cross-section? Is this true for all cylinders? Explain.
Answer:
yes its true, for all cylinders.
Explanation:
The width of a vertical cross-section through the centers of the bases and the diameter of the horizontal cross-section is same as shown in the below picture with blue line.
Question 3.
Kia has a funnel shaped like a cone. The cone has a diameter of 2\(\frac{5}{8}\) inches and is 6 inches tall.
A. Sketch the vertical cross-section through the center of the base of the cone and label its height and base.
Answer:
Explanation:
Draw a vertical cross-section through the center of the base of cone,
and mark the base and height of the cone as shown above.
B. Find the area of the vertical cross-section of the cone. Explain how you arrived at your answer.
Answer:
Area = 7.875 sq in
Explanation:
A = (1/2)bh
A = 0.5 x 2.625 x 6
A = 7.875 sq in
Check Understanding
Question 1.
Ralph has a cylindrical container of parmesan cheese. The diameter of the base of the container is 2.75 inches, and the height is 6 inches. What is the area of a horizontal cross-section of the cylinder to the nearest tenth of a square inch? Use 3.14 for π.
Answer:
Area = 5.94 sq in
Explanation:
d = 2.75 in
r = 2.75 / 2 = 1.375 in
Area A = πr2
A = 3.14 x 1.375 x 1.375
A = 5.94 sq in
Question 2.
Tori has a paper cup that is shaped like a cone. The radius of the base is 2.5 centimeters and the height is 11 centimeters. What are the shape and dimensions of the vertical cross-section through the center of the base?
Answer:
Explanation:
Given,
radius = 2.5 cm
height = 11 cm
Diameter = 2r = 2 x 2.5 = 5in
The Triangular shape and dimensions of the vertical cross-section through the center of the base is shown above.
Question 3.
Tito builds a model of the solar system. In his model, the diameter of the sphere representing Jupiter is 7 inches. What is the circumference of a cross-section through the center of the model of Jupiter? Use 3.14 for π.
Answer:
Circumference = 24.5 in
Explanation:
circumference = 2πr
d = 7 in
r = d/2
r = 7/2 = 3.5 in
C = 2 x 3.14 x 3.5
C = 24.5 in
On Your Own
Question 4.
Use Repeated Reasoning Brock has a cylindrical metal tin where he keeps his coins. The radius of the base is 5.5 inches, and the height is 4 inches.
A. What is the circumference of a horizontal cross-section of the cylinder? Use 3.14 for π.
Answer:
Circumference = 34.55 in
Explanation:
C = 2πr
r = 5.5 in
C = 2 x 3.14 x 5.5
C = 34.55 in
B. What is the area of a horizontal cross-section of the cylinder to the nearest hundredth? Use 3.14 for π.
Answer:
Area = 94.985 sq in
Explanation:
radius = 5.5 in
Area A = πr2
A = 3.14 x 5.5 x 5.5
A = 94.985 sq in
C. What is the area of a vertical cross-section of the cylinder through the center of the base?
Answer:
Area = 44 sq in
Explanation:
Area = length x width
Area = 11 x 4
A = 44 sq in
Question 5.
Find the circumference of a horizontal cross-section of the cylinder. Use 3.14 for π.
Answer:
Circumference = 31.4 ft
Explanation:circumference = 2πr
r = 5 ft
C = 2 x 3.14 x 5
C = 31.4 ft
Question 6.
Find the area of a vertical cross-section through the center of the base of the cone.
Answer:
Area = 16 sq in
Explanation:
The area of a vertical cross-section through the center of the base of the cone
A = (1/2) base x height
A = 0.5 x 4 x 8
A = 16 sq in
Question 7.
Find the area of a vertical cross-section through the center of the base of the cylinder.
Answer:
Area = 84 sq cm
Explanation:
Area A = length x width
A = 12 x 7
A = 84 sq cm
Question 8.
Find the area of a cross-section through the center of the sphere. Use 3.14 for π.
Answer:
Area = 3.14 sq m
Explanation:
radius = 1m
Area = πr2
Area = 3.14 x 1 x 1
Area = 3.14 sq m
Question 9.
Reason Why can’t the dimensions of a horizontal cross-section of a cone be determined from just the dimensions of the cone?
Answer:
A horizontal plane passing through the vertex will cut the cone into two parts with a cross section of a triangle.
Explanation:
As, the vertical shape of the cone is of triangle.
The horizontal cross section of a cone is triangle but varies the radius from base to top edge.
Question 10.
A cone-shaped paperweight is 5 inches tall, and the base has a circumference of about 12.56 inches. What is the area of a vertical cross-section through the center of the base of the paperweight? Use 3.14 for π.
Answer:
Area = 10 sq in
Explanation:
Circumference = 12.56 in
C = 2πr
h = 5 in
12.56 = 2 x3.14 x r
r = 12.56/6.28
r = 2 in
d = 2r = 2 x 2 = 4 in
Area of a triangle shape cone
A = (1/2) base x height
A = 0.5 x 4 x 5
A = 10sq in
Question 11.
Rosalie has a can of soup that is shaped like a cylinder. The measurements of the can are shown. Find the areas of a vertical cross-section through the centers of the bases and a horizontal cross-section of the can. Use 3.14 for π.
Answer:
Area = 6,370 sq mm
Explanation:
radius = 32.5 mm
diameter = 2r = 2x x 32.5 2 = 65 mm
length = 98 mm
Area of rectangle = length x width
A = 98 x 65
A = 6,370 sq mm
Question 12.
Find the area of a vertical cross-section through the center of the base of a cone with a height of 5 feet and a circumference of about 28.26 feet. Use 3.14 for π.
Answer:
Area = 22.5 sq ft
Explanation:
height = 5 ft
Circumference = 28.26ft
radius = 28.26/2 x 3.14
r = 28.26/6.28 = 4.5 ft
A = (1/2) x 9 x 5
A = 0.5 x 45
A = 22.5 sq ft
Question 13.
Find the area of a vertical cross-section through the center of a sphere with a diameter of 16 centimeters. Use 3.14 for π.
Answer:
Area = 200.96 sq cm
Explanation:
diameter = 16 cm
radius = d/2
r = 16/2 cm
r = 8 cm
A = πr2
A = 3.14 x 8 x 8
A = 3.14 x 64
A = 200.96 sq cm
Question 14.
Find the area of a vertical cross-section through the centers of the bases of a cylinder with a height of 24 inches and a circumference of about 43.96 inches. Use 3.14 for π.
Answer:
Area = 336 sq in
Explanation:
Circumference = 2 π r
43.96 = 2 π r
43.96 / 2 x 3.14 = r
radius = 43.96/6.28
r = 7 in
d = 2r = 2 x 7 = 14 in
A = length x width
A = 14 x 24
A = 336 sq in
Question 15.
Find the area of a horizontal cross-section of a cylinder with a height of 34 centimeters and a circumference of about 131.88 centimeters. Use 3.14 for π.
Answer:
Area = 882 sq cm
Explanation:
Circumference = 2 π r
131.88 = 2 π r
131.88 / 2 x 3.14 = r
radius = 131.88/6.28
r = 21 cm
d = 2r = 2 x 21 = 42 cm
A = length x width
A = 42 x 21
A = 882 sq cm
I’m in a Learning Mindset!
What strategies do I use to stay on task when working on my own?
Answer:
One should be through with the concept and formulas of circle and their cross sections.
Lesson 10.3 More Practice/Homework
Question 1.
Clyde has a cone-shaped party hat. The height of the hat is 10 inches, and the radius of the base of the hat is 4 inches. What is the area of a vertical cross-section through the center of the base of the party hat?
Answer:
Area = 40 sq in
Explanation:
height = 10 in
radius = 4 in
diameter = 2r = 2 x 4 = 8 in
Area = (1/2) base x width
A = 0.5 x 10 x 8
A = 40 sq in
Question 2.
A cylindrical swimming pool has a height of 4 feet and a circumference of about 75.36 feet. What is the area of a vertical cross-section through the center of the pool? Use 3.14 for π.
Answer:
Area = 96 sq ft
Explanation:
Circumference = 75.36 ft
C = 2πr
75.36/2×3.14 = r
radius = 12 ft
diameter = 2r = 2 x 12 = 24 ft
height = 4 ft
Area = length x width
Area= 24 x 4
Area = 96 sq ft
Question 3.
Open-Ended Alexandra is working on a float for a parade. She is making a sphere out of papier-mâché. She knows the circumference of a cross-section through the center of a sphere. How can Alexandra find the radius of the sphere?
Answer:
Radius = C / 2π
Explanation:
Circumference = 2πr
The radius of the sphere r
r = C / 2π
Question 4.
Find the area of a vertical cross-section through the center of the base of the cone.
Answer:
Area = 192 sq cm
Explanation:
base = 12 cm
height = 32 cm
Area of a triangle = (1/2) base x height
A = (1/2) x 12 x 32
A = 192 sq cm
Question 5.
Find the area of a cross-section through the center of the sphere. Use 3.14 for π.
Answer:
Area = 78.5 sq in
Explanation:
diameter = 10 in
radius = 10/2 = 5 in
A = πr2
A = 3.14 x 5 x 5
A = 78.5 sq in
Question 6.
Find the area of a vertical cross-section through the centers of the bases of a cylinder with a height of 27 inches and a circumference of about 47.1 inches. Use 3.14 for π.
Answer:
Area = 702 sq in
Explanation:
height = 27 in
Circumfrence = 47.1 in
C = 2πr = 81.64 m
81.64 = 2πr
81.64/ 2 x 3.14 = r
81.64/6.28 = r
radius = 13 in
diameter = 2r = 2 x 13 = 26 in
A = length x width
A = 27 x 26
A = 702 sq in
Question 7.
Find the area of a horizontal cross-section of a cylinder with a height of 11 meters and a circumference of about 81.64 meters. Use 3.14 for π.
Answer:
Area = 530.66 sq m
Explanation:
Circumference = 2πr = 81.64 m
C = 2 x 3.14 x r
81.64 = 6.28 r
r = 81.64 / 6.28
r = 13
Area of the circle
A = π.r.r
A = 3.14 x 13 x 13
A = 530.66 sq m
Test Prep
Question 8.
Two chefs are working on cylindrical cakes. David wants to make a stripe of frosting in his cake, and he makes a cut that shows a rectangle. Terri wants to put a layer of frosting in the middle of her cake, so she makes a cut that shows a circle. How was Terri’s cut different from David’s?
Answer:
David’s cut
It is rectangular in shape.
It has 4 corners, 6 sides and 8 faces.
Terri’s cut
It is circular in shape.
It has no corners, no edges, no sides.
Explanation:
Rectangle is a simple four sided polygon with internal angles equal to 90 degrees.
The two sides at each corner meet at right angles and parallel to each other.
A circle is a round shaped figure that has no corners or edges.
Question 9.
Name the figure that represents the two-dimensional cross-section of the cylinder cut by the plane.
Answer:
Vertical cross section of the cylinder, we get Rectangle figure.
Explanation:
Vertical cross section of the cylinder, we get rectangle with 2 sides parallel to each other.
Question 10.
The cone is sliced horizontally, parallel to its base, as shown. Choose the figure that represents the two-dimensional cross-section of the cone.
Answer:
Option (B)
Explanation:
The two-dimensional cross-section of the cone is circle.
Spiral Review
Question 11.
A straight path from the edge of a circular garden to the center of the garden is 7 meters long. What is the area of the garden? Use 3.14 for π.
Answer:
Area = 153.86 sq m
Explanation;
radius = 7 m
Area A = πr2
A = 3.14 x 7 x 7
A = 3.14 x 49
A = 153.86 sq m
Question 12.
How many unique triangles can be made with the angle measures 48°, 64° and 68°: none, one, or infinitely many?
Answer:
One
Explanation:
One unique triangles can be made with the angle measures 48°, 64° and 68°
as shown below
as the sum of the internal angle of a triangle is 180 degrees
