Measurement in Two and Three Dimensions Math A Honors

Module #7 Homework 2017 - 2018

Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project

San Dieguito Union High School District

7.1A Homework: Perimeter and Area Problem Solving* Name:

Period:

A rectangle has a perimeter of 24 cm. What is the smallest possible area the rectangle can have, using only whole number dimensions (length and width)? What is the greatest perimeter you can make for a rectangle with an area of 24 square units? Answer each of the questions below. 1. What is the least number of tiles you can add to the figure below to create a shape with a perimeter of 16 units? Note: When adding a tile, the new tile must share at least one side with the original shape; each tile is 1 unit by 1 unit.

2. Use the original figure given above to answer a-d. Draw your answers. An answer of just “yes” or “no” is not sufficient. Use pictures or words to justify your answer. a. Can you add a tile to this figure to increase the perimeter by 1? If so, draw the new figure.

b. Can you add a tile to this figure to increase the perimeter by 2? If so, draw the new figure.

c. Can you add a tile to this figure to increase the perimeter by 3? If so, draw the new figure.

d. Can you add a tile to this figure so that the perimeter doesn’t change? If so, draw the new figure.

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You have a supply of unit tiles. Show at least three ideas on the grid paper in each problem. 3. Can you make more than one shape with the 4. Can you make more than one shape with the same perimeter as #1, but with a different area same area (same number of tiles) as #1, but with (adding tiles)? Show your ideas with grid paper. a different perimeter? Show your ideas with grid paper.

5. If you pick any whole number between 12 and 24, and use that many unit tiles, can you make a shape where the area and perimeter are equal? Show your ideas.

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6. A triangle has an area of 56 𝑐𝑚 ! . If the base is 14 𝑐𝑚, what is the height? Start with a formula and show your algebra steps.

7. What is the formula for the base, b, of a triangle if the area is A and the height is h? Hint: start with the formula for the area of a triangle, and use algebra to isolate b.

!

8. A triangle’s height is reduced by a factor of . If the base remains the same, what is the height of the ! triangle, h, in terms of area, A? Hint: start with the formula for area of a triangle, change the value of h, and then use algebra to isolate h.

Spiral Review: 9. Solve: −8 < −3𝑚 + 10

11. Simplify each expression. a. 𝑥 + 3𝑥

10. Solve: −12 > 3𝑥

b. 𝑥(3𝑥)

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c. 4𝑥 2𝑥 + 3 − 3𝑥

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12. Find the area of each of the following shapes. Show the formula you are using, and substitutions. Label your answer. Leave your answers in terms of 𝜋 when applicable. a. b.

c.

d.

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7.1B Homework: Areas of Irregular Shapes* Name:

Period:

1. The following shapes have been drawn on square dot paper. The distance between each dot represents one unit. Use what you have learned about area to find the area of each shape (A-J). A

B C

E D

F

G H

I J

A:

B:

C:

D:

E:

F:

G:

H:

I:

J:

Solve the following area problems. Show all of your calculations. 2. A rectangular lap pool with a length of 40 ft. and a width of 15 ft. is surrounded by a 5-ft. wide deck. a. Find the area of the deck.

b. Draw a picture of the deck if the deck is extended 3 feet in every direction, then find the area of the new deck.

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3. A rectangular field with two semicircles at each of the shorter ends of the field measures 100 yards long and 40 yards wide. It is surrounded by a track that is 5 yards wide. Find the area of the field which includes the two semicircles on each of the shorter ends of the rectangle. Find the area of the track.

4. Laura is painting a sign for the new Post Office. She will paint the triangular portion blue and the lower rectangular portion red. Find the area of sign that she will paint blue. What percent of the sign will be in blue?

Spiral Review 5. Simplify the following expressions: a. −60 + 5𝑥 − 1 − 17𝑥

b. −9 5 − 𝑥 + 3𝑥 − 8

6. Anya and Bartholomew clean windows. Anya charges 50 cents per window plus $10 per job. Bartholomew charges 90 cents per window plus $6 per job. If on one job, they make the same amount of money, how many windows did they each clean?

7. Alfonso is tossing his baby brother’s cylinder toy. The following table shows the results of the tosses. Based on the observations, what is the probability the cylinder will land on its side? Land on side 9

Land on top or bottom 11

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7.1C Homework: Areas of Irregular Shapes and Expressions* Name:

Period:

Solve the following area problems. Write an expression showing how you got the area. 1. Find the perimeter and area of the region.

2. Find the perimeter and area of the region. (use 3.14 for pi)

3. A stage design is illustrated in the figure to the right. a. Find the area of the stage.

b. What percent of the stage is the rectangular portion?

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4. Rachel is painting a sign for the new Health Center. a. Find the area of sign that she will need to paint.

b. Suppose Rachel decides to paint the rectangular portion (the part shown below the dotted line) of the sign blue. What percent of the sign would be blue?

5. Write a simplified expression to represent the area and perimeter of the following irregular shape. Remember to use ! proper labels. Height of trapezoid= 𝑥 inches; Base of ! trapezoid=7 inches; Diameter of semi-circle = 4 inches; each of ! leg of the trapezoid= (𝑥 + ) inches. Show all work clearly. ! Use 𝜋 ≈  3.14. Label the dimensions of the irregular shape.

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Write a simplified expression to represent the area of the following irregular shapes. 6. 8.

8n 17 2n 13 6n

7. Find the perimeter of this region:

9.

10. (Reminder: the arrows below mean lines are parallel.)

4x 2

2 8

x+6 6x

x+2

5

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11. Calculate the area of the shape below. Leave your answer in terms of π.

Write a simplified expression to represent the unshaded area of the following irregular shapes. 12. 13. 8 cm

x cm 15 cm

Leave your answer in terms of π. Round answer to the nearest hundredth.

If y = 6 feet, what is the area of the unshaded region?

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7.1D Homework: Problem Solving with Area* Name:

Period:

1. A rectangular garden has an area of 45 square feet. One of the sides is 6 feet. a. You want to put a fence around the garden. How long will the fence need to be?

b. You decide you want to increase the length of each side by a scale factor of 3.2. What is the area of your new garden?

2. Mrs. Garcia has a table shaped like an isosceles trapezoid in her third grade classroom. The two parallel sides have lengths of 6 feet and 8 feet. The distance between them is 4 feet. a. What is the area of the top of Mrs. Garcia’s table?

b. Suppose Mrs. Garcia has a 1.75 ft by 0.8 ft rectangular puzzle and a triangular piece of pizza with a base of 4 inches and a height of 6 inches on the table. How much area is now available on her table?

3. You’re making a 12 inch diameter pizza. You want the sauce to cover the pizza with a 1.5 inch ring left around the outside without sauce for the crust. (Use 3.14 as an approximation for pi.) a. What is the area that the sauce will cover?

b. What percent of the dough will be covered by sauce? Round to the nearest whole percent.

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c. If one 8 oz can of tomato sauce covers about 125 sq. inches of pizza dough, how many cans of sauce will you need to buy to make a dozen pizzas?

4. The three-point line in basketball is approximately a semi-circle with a radius of 19 feet and 9 inches. The entire court is a rectangle 50 feet wide by 94 feet long. What is the approximate area of the court that results in 3 points for a team rounded to the nearest whole number? Remember you shoot from anywhere but one of the semi-circles.

5. Your neighbor’s backyard lawn is shaped like a rectangle. The back fence is 38.2 feet long and the side fence is 32.6 feet long. He will pay you $0.04 per square foot for mowing and $0.11 per foot for trimming all the edges. How much will you get paid total for mowing and trimming? Remember to show all your work.

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6. You’re the manager of a county recreation center that has a 50 by 25 meter rectangular pool. Currently, there is an 1.5 meter cement walkway around the pool (see diagram). The community is concerned about the safety of the walkway and would like to cover it with a non-slip rubber substance that costs $78 a square meter to be installed. The county has budgeted $15,000 for the project. Is that enough money to cover the walkway? Explain you answer.

7. A hot tub is surrounded by a square deck as pictured to the right. What is the area of the deck to the nearest whole number?

8. Cristian is building a rectangular garden in his backyard. The width of the garden is set at 29 inches. He wants the fence to be 5 inches longer than the garden on each side. If he wants the area enclosed by the fence to be 2028 square inches, how long should the garden be?

Spiral Review: 9. Eugene’s math class has 20 boys and 10 girls. If the teacher draws a student’s name at random for a candy bar, what is the probability Eugene will be chosen? What is the probability that a girl will be chosen?

10. Simplify: −1 −4 −7 − 28

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Section 7.1 Review* Name:

Period:

For #1-2, find the area of the following figures. 1.

2.

3. Find the perimeter and area.

4. Find the perimeter and area.

7 cm 3 cm

5 cm

4 cm

2.5 cm

4.1 cm 4 cm

11 cm

5. Phillip is mowing his backyard. His backyard is oddly shaped, and has an 8 ft by 8 ft square pond in his backyard. What is the total area Phillip will be mowing?

a. b. c. d.

64 square feet 106 square feet 336 square feet 400 square feet

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6. Find the perimeter and area of the shape below. Document all work.

13.9'

13'

7. Write simplified expressions for the perimeter and area of the shape below.

9

15' 2n

6

5n

10' 20

17'

8. Find the area and perimeter of the shaded region Round your answer to the nearest tenth.

9. Find the area of the shaded region. Round your answer to the nearest tenth. 2 in

12 mm

9 in

20 mm

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10. A soybean farmer uses an irrigation system in which a 150 ft. straight pipe shown at right travels in a circle around a well as shown in the figure below. a. How many square feet of the field are irrigated? Round to the nearest whole number.

b. The farmer usually plants wheat in the corners of the field, which are not irrigated. How many square feet of wheat is planted in this field? Round to the nearest whole number.

11. Jett created the ceramic tile design below. Each tile is a square with side lengths of 15 cm. He wants to paint the circles red, and the remaining tile white. When the tiles are laid edge to edge, they create a pattern of red circles with a diameter of 6 cm. What is the area of the red circles in one tile? What is the area of the white space in one tile? (use 3.14 for pi) What percent of Jett’s tile will be painted white? Round to the nearest whole percent.

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12. The rectangular field with two semi-circles at each of the shorter ends of the field (below) measures 100 meters long and 60 meters wide. It is surrounded by a track that is 7 meters wide.

a. Find the area of the field which includes the two semi-circles on each of the shorter ends of the rectangle. Round to the nearest whole number.

b. Find the area of the track. Round to the nearest whole number.

13. For the figure below, find area and perimeter of the shaded region. Simplify your expression and give units. (use 3.14 for pi) x

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7.2A Homework: 3D Objects* Name:

Period:

Make a cube and a rectangular prism. Orient them as in the figure below:

1. What is the shape of the base of the cube and the rectangular prism? 2. Will all cuts parallel to the base result in the same planar figure for the cube? Justify your answer.

3. Will all cuts parallel to the base result in the same planar figure for the rectangle prism? Justify your answer.

Now rotate each 90° as shown in the figure below:

4. What is the shape of the base of the cube and the rectangular prism now? 5. Will all cuts parallel to the base result in the same planar figure for the cube? Justify your answer.

6. Will all cuts parallel to the base result in the same planar figure for the rectangle prism? Justify your answer.

7. Name three things you discovered about plane sections of rectangular prisms, cubes and spheres.

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Spiral Review: 8. Find the area of each figure below: a.

b.

c. A square with perimeter 20 cm.

d.

9. Simone rolls a die 64 times. Approximately, how many times will she roll a 6?

10. Find the unit rate for BOTH units for the statement below: Izzy drove 357 miles on 10 gallons of gasoline.

11. The temperature at midnight was 8° C. By 8 am, it had risen 1.5°. By noon, it had risen another 2.7°. Then a storm blew in, causing it to drop 2.7° by 6 pm. What was the temperature at 6 pm?

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7.2B Homework: Area of Plane Sections* Name:

Period:

1. What will the cross section look like of any prism that is cut parallel to its base?

2. What will the cross section look like of any prism that is cut perpendicular to its base?

3. What will the cross section look like of any pyramid that is cut parallel to its base?

4. What will the cross section look like of any pyramid that is cut perpendicular to its base?

For each plane-section described below, state the shape of the plane-section and its area. Refer to previous class activities, as needed. 5. Imagine cutting a cube parallel to its base. a. What shape is the plane section?

b. If you cut a 5 × 5 × 5 inch cube parallel to any face, what will the area of the plane section be?

6. Imagine a rectangular prism with edge lengths

3 3 1 × × 4 inches. 4 4 2

a. If you make a cut parallel to the square base, what will the plane section be?

b. What will the area of the plane section described in “a” be?

c. What will the plane section be if the cut is made parallel to the lateral face? d. What will the area of the plane section in “c” be?

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7. Imagine cutting a sphere with diameter 10 cm through the center. a. What shape will any plane section be? b. What will the area of the plane section be?

8. Imagine cutting a cylinder of diameter 12.62 cm and height 8 cm, parallel to the base. a. What shape is the plane section and what is its area? Give your answer in both exact form and round to the hundredths place.

b. What shape is the plane section if the cut is perpendicular to the base? What do you know about the figure?

9. Imagine a triangular prism: a. What shape is the plane section parallel to the base? b. What shape is a plane section perpendicular to the base? c. If the area of the plane section in “b” is 6.25 cm2 and the height of the prism 5 cm. What was the length of the cut?

10. Imagine cutting a square based right pyramid parallel to the base. a. What shape is the plane section? b. If the dimensions of the length and the width of the plane section are

3 3 in. and in., what is the area 2 2

of the plane section?

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Given the following shapes, what planar figure would result with a cut parallel to the base and perpendicular to the base? 11.

12.

13.

a. Planar figure parallel to base?

a. Planar figure parallel to base?

a. Planar figure parallel to base?

b. Planar figure perpendicular to base?

b. Planar figure perpendicular to base?

b. Planar figure perpendicular to base?

14.

15.

16.

a. Planar figure parallel to base?

a. Planar figure parallel to base?

a. Planar figure parallel to base?

b. Planar figure perpendicular to base?

b. Planar figure perpendicular to base?

b. Planar figure perpendicular to base?

17. Summarize the 2D planar results of any parallel and perpendicular slices of prisms and any other figure that may include points and segments. Give an example for each.

18. Summarize the 2D planar results of any parallel and perpendicular slices of pyramids and any other figure that may include points and segments. Give an example for each.

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Spiral Review: 19. Solve and graph the following inequality:   −97 ≥ 4 19 − 2𝑥 + 3

20. Find the perimeter and area of each figure below: a.

b.

21. Solve: 17𝑥 − 3 5𝑥 + 7 = − 𝑥 − 3  

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7.2C Homework: Nets, Surface Area, and Volume of 3D Objects* Name:

Period:

1. Which of the following nets make a cube? a. b.

d.

e.

c.

f.

2. Suppose there is a rectangular prism that has dimensions 8 × 8 × 12 inches. Find the surface area and volume of the prism. Show all your work.

!

3. Still using the prism above, find the new dimensions of the prism if you scaled it by . Then find the new ! surface area and volume.

!

4. Still using the original prism above, find the new dimensions of the prism if you scale it by . Then find the ! new surface area and volume.

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5. Find the surface area and volume of the right triangular prism below.

6. Explain your procedure for finding surface area of a trapezoidal prism.

7. Explain your procedure for finding the height of a rectangular prism given its volume and the other needed dimensions.

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8. If the base area of a rectangular prism is 28 cm2 and the volume is 177.8 cm3, what is the height of the prism?

9. If the height of a square based rectangular prism is 13 in and its volume is 637 in3, what is the length of each side of the base?

10. Give three possible edge lengths for a rectangular prism of volume 96 m3.

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Find the Surface Area and Volume of the following shapes. 11. 12.

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Spiral Review: 13. Simplify: a. 3 − 9 − 12

b. 9 − 6𝑥 + 7𝑥

14. On a map of New York City, with a scale of 1 inch= ½ mile, Central Park is 1 inch wide and 5 inches long. What is the area of the park?

15. Find the additive inverse and multiplicative inverse of each of the following numbers: Number

Additive Inverse

Multiplicative Inverse

2 17 7 −2.15

16. Given the following table, find the indicated unit rate:

push-ups per day

Days Total Push-ups 2

30

4

60

29

435

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7.2D Homework: Growing and Shrinking Objects* Name:

Period:

Are the two figures below similar? If so, state the scale factor. 1. 2.

Each pair of figures is similar. Use the information given to find the scale factor of the figure on the left to the figure on the right. 3. 4.

Each pair of figures is similar. Find the scale factor of the figure on the left to the figure on the right. Then find the scale factor of surface areas and the scale factor of volumes.

The scale factor between two similar figures is given. The surface area and volume of the smaller figure are given. Find the surface area and volume of the larger figure.

5.

6. Scale Factor is 2 Surface Area = 90 𝑦𝑑 ! Volume = 216 𝑦𝑑 !

SA = 675

SA = 432

Scale Factor is ______.

Surface Area of larger figure is ____________. Volume of larger figure is __________.

Surface Area Scale Factor is ______. Volume Scale Factor is ______. SDUHSD Math A Honors Module #7 – HOMEWORK 2017-2018

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For each of the following problems, find the necessary measurements to answer all the questions. 7. A mini cereal box has the following dimensions: 4.5 in by 6 in by 2 in. a. If all the dimensions are doubled, will it require double the amount of cardboard to make the box? Why or why not?

b. If all the dimensions are doubled, will it hold double the amount of cereal? Why or why not?

c. If one of the dimensions is doubled, will it require double the amount of cardboard to make the box? Why or why not?

d. If one of the dimensions is doubled, will it hold double the amount of cereal? Why or why not?

e. If all the dimensions are increased by 4 inches how does this change the amount of cardboard needed to make the box?

f.

If all the dimensions are increased by a value of 4 inches, as described above, how will this change the amount of cereal the box can hold?

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8. A container of hot chocolate mix has the following dimensions: a square base with sides of 6 in. and height 9 in. a. If all the dimensions are reduced by a scale factor of 13 , will it require a third the amount of materials to make the container? Why or why not?

b. If all the dimensions are reduced by a scale factor of mix? Why or why not?

1 3

, will it hold a third the amount of chocolate milk

c. If the length of each side of the base is reduced by a scale factor of volume will it now have?

1 3

, how much surface area and

d. What is the percent change of surface area and volume for “c”?

Spiral Review: 9. Mrs. Zamora will not tell you how the class in general did on the last test. You really want to know how you compare, so you survey 10 random students out of 35. The results are shown below. Estimate the average score for your class and describe how far off the estimate might be. 77 100

89 80

79 81

100 88

94 87

10. Draw and describe the plane section that results from the following cut: a cylinder cut parallel to the base.

11. Kurt puts 70% of his earnings into his savings. Write and solve an equation to find how much money he earned if he had $165 to spend.

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Section 7.2 Review* Name:

Period:

Find the area and perimeter or each figure. Use 𝜋 ≈ 3.14 when appropriate. Round to the nearest hundredth. 1. 2. 3. 9 cm 4 cm

3 cm

6 in

2 cm 12 cm

Find the area of the shaded region. Leave in terms of 𝜋 when appropriate. 4. 5. 6.

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Find the Surface Area and Volume of the following shapes. 7. 8.

9.

10. If the prism in #7 was increased by a scale factor of 4, how would the surface area change?

11. If the volume of the prism in #8 changed to 137,280 mm3, how did the lengths of the sides change?

12. If a triangle has an area of 36  𝑖𝑛! and a base of 4  𝑖𝑛, what is the measurement of its height?

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Read each question carefully. Show all of your work clearly. State your answer in a complete sentence. 13. Haley and Madison are arguing over a math problem. Haley insists that a square with a area of 196 square units is bigger than Madison’s square that has a perimeter of 196 units. Who is correct and why?

14. A trapezoid has an area of 168 square inches. a. If the sum of the bases is 28 inches, what is the height of the trapezoid?

b. A second trapezoid has an area that is 75% smaller than the area of the original 168 in2. Name one possible height of the smaller trapezoid assuming that the sum of the bases stays the same.

c. A third trapezoid has an area that is 75% smaller than the area of the original 168 in2. Name one set of possible bases of the smaller trapezoid assuming that the height of the trapezoid stays the same.

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15. A rectangular pig pen is 40 feet wide by 80 feet long. One circular fountain will be placed in the pen that has a diameter of 20 feet. A circular bathing pond with a radius of 7 feet, will be placed inside the pen as well. The rest of the garden will be covered by grass. What is the approximate square feet of grass needed? Draw and label a picture. Use 𝜋 ≈  3.14.

16. A rectangular garden is 2.5 times as long as it is wide. If it has a perimeter of 168 feet, what is the area of the garden?

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