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Volume & Capacity
Transformations
Tank Calculations
Pythagorean & Distance
100

A company produces small cubic sugar cubes, each with side length 2 cm. These cubes are packed into cylindrical containers for shipping. Each cylindrical container has a radius of 5 cm and a height of 12 cm. What is the maximum number of sugar cubes that can be placed in one full cylindrical container? Round your answer to the nearest whole number. (Assume the cubes can be "melted" and poured, or consider volume packing for simplification).

117 cubes

100

Triangle DEF has vertices D(2,−1), E(5,3), F(1,4). If triangle DEF is reflected over the y-axis to form triangle D′E′F′, what are the vertices of triangle D′E′F′?

A. D′(2,1),E′(5,−3),F′(1,−4)

B. D′(−2,−1),E′(−5,3),F′(−1,4)

C. D′(−2,1),E′(−5,−3),F′(−1,−4)

D. D′(1,−2),E′(−3,−5),F′(−4,−1)

B. D'(-2,-1) E'(-5,3) F'(-1,4)

100

Question 2: Water Tank Radius 

A cylindrical water storage tank has a capacity of 250 gallons. Its height is 60 inches. If one cubic foot of water is equivalent to 7.48 gallons, find the maximum possible radius for the base of the tank. Round your answer to the nearest thousandth of an inch.

  1. Convert gallons to cubic feet: 250 gallons÷7.48 gallons/ft3≈33.42246 ft3
  2. Convert cubic feet to cubic inches: 33.42246 ft3×1728 in3/ft3≈57731.81 in3
  3. Calculate Radius: Volume (V) = π×radius2(r2)×height(h). So, r=V/(π×h) r=57731.81 in3/(π×60 in)=57731.81/188.4955=306.286≈17.5009 inches
  • Answer: Approximately 17.501 inches
100

Flagpole Height

 From a point on the ground 75 meters away from the base of a flagpole, the angle of elevation to the top of the flagpole is 45∘. What is the height of the flagpole, to the nearest meter?

Tan x = opp/adj

75 meter

200

A spherical water balloon has a radius of 4 inches. It is used to fill a rectangular children's pool that measures 30 inches in length, 20 inches in width, and 10 inches in height. What is the least number of full water balloons needed to completely fill the pool? Round your answer to the nearest whole number.

23 balloons 

200

Point Q(4,−2) is rotated 90° clockwise about the origin. What are the coordinates of Q′?

A. (−2,−4)

B. (−4,2)

C. (2,4)

D. (2,−4)

  • A. (−2,−4)

    • Explanation: The rule for a 90° clockwise rotation about the origin is (x,y)→(y,−x).
      • Q(4,−2)→Q′(−2,−(4))=Q′(−2,−4).


200

Question 3: Grow Tank Width

A rectangular hydroponic grow tank has a capacity of 75 gallons. Its length is 40 inches and its height is 16 inches. If one cubic foot of water is equivalent to 7.48 gallons, what is the maximum possible width for this grow tank? Round your answer to the nearest thousandth of an inch

  1. Convert gallons to cubic feet: 75 gallons÷7.48 gallons/ft3≈10.02674 ft3
  2. Convert cubic feet to cubic inches: 10.02674 ft3×1728 in3/ft3≈17326.16 in3
  3. Calculate Width: Volume (V) = Length (L) × Width (W) × Height (H). So, W=V/(L×H) W=17326.16 in3/(40 in×16 in)=17326.16 in3/640 in2≈27.0721 inches
  • Answer: Approximately 27.072 inches
200

Ramp Angle

 A wheelchair ramp is 12 feet long and has a vertical rise of 2 feet. What is the angle of elevation of the ramp with the ground, to the nearest degree?

sin x = opp/hyp

Answer: Approximately 10 degrees

300

A cone-shaped funnel has a radius of 3 cm and a height of 10 cm. This funnel is used to transfer sand into a rectangular planter box that has dimensions of 25 cm in length, 15 cm in width, and 12 cm in height. What is the least number of full funnels of sand needed to completely fill the planter box? Round your answer to the nearest whole number.

48 funnels 

300

Quadrilateral GHIJ has vertices G(−2,4), H(6,2), I(4,−4), J(−4,−6). If quadrilateral GHIJ is dilated by a scale factor of 21 centered at the origin to form quadrilateral G′H′I′J′, what are the vertices of G′H′I′J′?

A. G′(−1,2),H′(3,1),I′(2,−2),J′(−2,−3)

B. G′(−4,8),H′(12,4),I′(8,−8),J′(−8,−12)

C. G′(2,−4),H′(−6,−2),I′(−4,4),J′(4,6)

D. G′(−1,−2),H′(3,−1),I′(2,2),J′(−2,3)

A. G′(−1,2),H′(3,1),I′(2,−2),J′(−2,−3)

300

Question 4: Drum Height

 A large cylindrical drum used for storing liquid has a capacity of 300 gallons. The diameter of its base is 30 inches. If one cubic foot of liquid is equivalent to 7.48 gallons, what is the maximum possible height for this drum? Round your answer to the nearest thousandth of an inch.

  1. Convert gallons to cubic feet: 300 gallons÷7.48 gallons/ft3≈40.10695 ft3
  2. Convert cubic feet to cubic inches: 40.10695 ft3×1728 in3/ft3≈69299.79 in3
  3. Calculate Radius: Diameter is 30 inches, so radius (r) = 30 in/2=15 in.
  4. Calculate Height: Volume (V) = π×radius2(r2)×height(h). So, h=V/(π×r2) h=69299.79 in3/(π×(15 in)2)=69299.79/(π×225)=69299.79/706.858≈98.0381 inches
  • Answer: Approximately 98.038 inches
300

Airplane Approach

 An airplane is at an altitude of 5,000 feet and begins its descent towards an airport. The angle of depression from the airplane to the airport is 15∘. What is the horizontal distance, in feet, from the airplane to the airport? Round your answer to the nearest foot.

Tan x = opp/ adj

Answer: Approximately 18660 feet

400

A cylinder with a dimension of diameter 6 inches and height 8 inches is used to fill the rectangular prims fish tank with water. What is the least number of full cylindrical containers needed to complete fill the fish tank?

31

400

The ABC has vertices at A(1,2) B(4,6) C(4,2) in the coordinates plane. The triangle will be reflected over the x axis and the rotated 180 degrees about the origin. What are the vertices for ABC

A. A'(1,2) B'(4,6) C'(4,2)

B. A'(1,-2) B'(4,-6) C'(4,-2)

C. A'(-1,2) B'(-4,6) C'(-4,2)

D. A'(-1,-2) B'(-4,-6) C'(-4,-2)

C. A'(-1,2) B'(-4,6) C'(-4,2)

400

The Pisces Place is company that makes fish tanks. Corey has sent the company a special order for a fish tank that is a rectangular prism with an open top. the dimension length 54 inches height = ? width = 18 inches and capacity = 100 gallons. one cubic foot water is equivalent to 7.48 gallons. if the capacity of the fish tank is 100 gallons find the máximum possible height for the fish tank. Round you answer to nearest thousand of an inch 

V=LWH

13.369=4.5x1.5xH

H=1.981

400

In a right-angled triangle ABC, the hypotenuse AC measures 15 cm. If angle A is 30∘, what is the length of side BC (the side opposite to angle A)?

25.98