Unit 3: Geometry in Engineering, Architecture, and Fine Arts, Part 2

1. A sphere has a radius of 9 inches. An artist creates a replica of the structure that has been scaled up by a factor of 1.5. What is the volume of the replica in in3?

10,305.99

6. A sphere has a radius of 9 inches. inches. An artist creates a replica of the structure that has been scaled up by a factor of 1.5. What is the volume of the replica in in2?

10,305.99 in3

7. Kiara is drawing a photo of a car using a scale factor of 3. The radius for each of the car's tires is 2 cm. What will the area of a tire in Kiara's drawing be?

113 cm^2

5. The length, width, and height of a wooden crate are 1 m, 3 m, and 2 m. What is the maximum number of such crates that can be transported in a cargo ship with a storage hold that has a volume of 72,000 m3?

12,000

2. The angle of elevation of a ladder is 19∘ and the ladder is 12 feet from the wall. What is the height of the ladder in feet?

12.7 ft

2. Examine this short piece of music for the guitar: CEFC DGCF CE Which notes would best continue the pattern?

FC

14. Examine the following image, which shows sound waves from two musical instruments.

Graph A shows a louder instrument because the amplitude is larger.

9. A ramp is built in the shape of a right triangle. The hypotenuse of the ramp is 17 feet and the length of one leg is 8 feet. What is the length of the other leg?

15 ft

8. The famous Louvre Pyramid is a metal and glass structure that serves as the main entrance to the Louvre Museum in Paris, France. The height from the floor to the apex, or the top vertex of the pyramid, is 21.6 meters If the hypotenuse is a side of the pyramid, and the pyramid has a height of 28 m, m, what is the angle of depression from the apex of the pyramid to the ground below?

50.5

2. An architect designs a castle tower for a new attraction at a popular amusement park. Blueprints of the tower, which is in the shape of a rectangular prism, show that it is 47 cm tall, 23 cm in length, and 25 cm in width. The blueprints have a scale factor of 1:96.

5218.1 m^2

8. A building that is 150 m tall, 100 m wide, and 80 m deep has glass on all sides except the roof and the base. What is the area of the glass needed for this building?

54,000 m^2

8. A 45-45-90 triangle has a hypotenuse with a length of 10 feet. What are the lengths of the other legs?

7.07 ft

Unit 3 Session ( 4 questions)

Code: 6669

3. An artist is asked to scale up a model of Pluto to 3 times its original size. If the original model of Pluto has a radius of 12 cm, what is the volume of the new model?

The volume of the new model of Pluto is 195,432.20 cm3.

3. Examine the image. Does the image display periodic behavior? Why or why not?

Yes, because the pattern of curving lines repeats to form spirals that flow into each other.

1. What are the basic characteristics of music that display periodic behavior?

pitch and rhythm

13. What geometric transformation is used in the following musical score?

reflection

6. The approximate volume of the Statue of Liberty is 2500 m3. A gift shop sells a replica with a scale factor of 1/18. What is the approximate volume of the replica?

0.43 m3

10. A right triangle has legs with lengths of 7 inches and 8.5 inches. Which expression, and which value, depicts the length of the hypotenuse?

- 11.01 ft - √7^2 + 8.5^2

7. A 30-60-90 triangle has a hypotenuse with a length of 10 feet. What are the lengths of the short leg and the medium leg?

- The medium leg has a length of 8.66 ft - The short leg has a length of 5 ft

1 Students measure the sides of 4 triangles in a test. The lengths of the sides of the different triangles follow: Triangle 1: 6, 12 and 13 Triangle 2: 5, 12 and 13 Triangle 3: 3, 4 and 5 Triangle 4: 2, 4 and 6

- triangle 3 - triangle 2

3. Roy is creating a miniature model of the kitchen in his parents' restaurant using a ratio of 1:5. There are two refrigerators in the kitchen that are 1.5 m tall and 2 m wide. Both have a depth of 1 m. How much space will the refrigerators occupy in Roy's model?

0.048 m^3

2. An artist wants to paint a scale drawing of a mural. The mural has a length of 160 feet and a width of 75 ft. if the length of the painting is 32 inches and the width is 15 inches, what is the scale factor

1/5

1. Jacob designs a building with a square base. He starts by building a model for the building, which has a base with an area of 36 feet2. The base of the real building will have an area of 6084 ft.2. What is the scale factor between the model and the real building?

13

1. A rectangular prism has a height of 6 inches, a width of 20 in, and a length of 25 in. What is the surface area of the rectangle prism in in2?

1540 in^2

4. The length of one leg of a right triangle is 6 units. What is the approximate length of the second leg if the hypotenuse is 3 times the length of the given leg?

17 units

3. The Army Corps of Engineers has been instructed to measure the effectiveness of a special wireless satellite tower's signal at various distances. The height of the satellite tower is 250 feet. If the tower's signal reaches its equipment truck, which forms an angle of elevation of 55∘, approximately how far is the truck from the base of the tower?

175 feet

2. A statue that is 80 m tall stands on a 625 m2 base. An artist creates a replica using a ratio of 1:15. What is the area of the replica's base?

2.78 m^2

1. The angle of depression from a descending airplane to the control tower below is 60∘. If the plane is 35,000 feet above the ground, approximately what is the plane's horizontal distance from the control tower?

20,207 ft

3. A snail farmer's business is growing. He wants to build a new pen similar to the old one, but with a scale factor of 1:3. This will accommodate the expanding business. The old pen is square. If the diagonal of the old pen measures 10 meters, what is the length of one side of the new pen? Use the Pythagorean Theorem.

21.21 m

3. A park ranger at the top of a 700-meter station sees a fair that is 1800 m from the base of the tower. What is the angle of elevation?

21.3 degrees

11. The base of a ladder is 10 feet away from a vertical building. The ladder top rests at the top of the building. The ladder's angle of elevation from the bottom of the ladder to the top of the building is 70 degrees. What is the height of the building?

27.47 ft

12. A cable is connected to the top of a building that is 20 feet high. The cable is attached to the ground 35 feet away from the building. What is the angle of depression from the top of the building to the ground?

29.74 degrees

1. Mike has a plaster replica of the Eiffel Tower in his room. The volume of the plaster in the replica is 0.25 m3. He wishes to make a bigger replica using a scale factor of 5. What is the volume of plaster that he needs for the new replica?

31.25 m^3

4. A square pyramid has a height of 25 feet and each side of the base has a length of 15 feet. If a model of the square pyramid is scaled down by a factor of 1/5 , what is the surface area of the model in ft2?

40.32

3. A square pyramid has a height of 25 feet and each side of the base has a length of 15 feet. If a model of the square pyramid is scaled down by a factor of 1/5, what is the surface area of the model?

40.32 ft2

1. A computer science engineer is developing a model of a motherboard to use in product demonstrations. The original motherboard measures 5 1/4 inches in length. The model needs to be scaled up at a factor of 8:1. What would the length of the model be?

42 inches

7. Kevin rode his bike 2 miles east and 6 miles north from his home to the movie theater, forming a right angle. How far is the movie theater from Kevin's home?

6.32 miles

2. An architectural firm wants to replicate a nearby estate with some changes. The new estate will be larger than the original by a scale factor of 1:10. A section of the roof in the original estate is constructed using 45-45-90 triangles. The hypotenuse of the triangle is 10 meters long. What will be the length of the triangle legs for the larger estate?

70.71 m

3. A cylinder has a height of 5 feet and a radius of 2 feet. If a scale model of the cylinder is scaled up by a factor of 3, what is the surface area of the model in ft3?

791.68

4. A cylinder has a height of 5 feet and a radius of 2 feet. If a scale model of the cylinder is scaled up by a factor of 3 what is the surface area of the model?

791.68 ft2

3. Andrea made a replica of the Empire State Building. The surface area of the model is 45 feet2. What is the surface area of the actual Empire State Building in ft.2 if the scale factor is 44?

87,120 ft^2

6. Tommy is constructing a roof for a tree house. He decides to use 45 - 45 - 90 triangles with legs that are 8 feet in length to create the roof's frame. If the hypotenuse is the width of the house, which answers represent its length?

8√2 ft 11.31 ft √8^2 + 8^2 ft

2. Kamal made a replica of the Kaaba located in Mecca, Saudi Arabia. The dimensions of his replica are 24 cm long by 21 cm wide by 30 cm high. What is the surface area of the actual Kaaba in meters if the scale factor is 50?

927 m^2

2. A circular cone structure has a height of 50 feet and a radius of 15 feet. An artist creates a replica of the structure that has been scaled down by a factor of ⅕. What is the volume of the replica in ft3?

94.25

5. A circular cone structure has a height of 50 feet and a radius of 15 feet. An artist creates a replica of the structure that has been scaled down by a factor of 1/5. What is the volume of the replica in ft3?

94.25 ft3

4. A pyramid art installation has a surface area of 24 m2. An artist creates replicas with scale factors of 1/8, 1/10, and 1/12. What is the surface area of each replica?

???

1. What are the correct trigonometric ratios for angle H in a right triangle if GI=5, HI=12, and GH=13?

sin H = 5/13, cos H = 12/13, tan H = 5/12

5. Catalina extends a ladder 15 feet and places it diagonally on a tree to get her cat that is perched on the lowest branch. Which trigonometric ratio would she use to determine how high her cat was from the ground if the ladder formed a 65 degree angle with the ground?

sine

2. Consider triangle DEF. The measure of angle F is 90∘, DE=85 cm, EF=84 cm, and DF=13 cm. Which of the following expressions are true about this triangle?

tan D = 84/13 cos E = 84/85 sin E = 13/85