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Math problem set Grade 8 Math

Blackstock Treatment Plant Tanks: Cylinder, Cone, Sphere Volume

A grade 8 problem set on the volume formulas for a cylinder, a cone, and a sphere, applied to three tanks at a fictional drinking-water treatment plant, with fill time.

Objectives

  1. Students will find the volume of a cylinder, a cone, and a sphere from given dimensions.
  2. Students will convert cubic feet to gallons and use volume to solve a fill-time problem.

Vocabulary

Key words: cylinder, cone, sphere, radius

Teacher overview

A drinking-water plant stores and moves water in tanks of different shapes, and each shape has its own volume formula. This set uses the fictional Blackstock Water Treatment Plant, which has a cylindrical clearwell, a cone-bottom hopper, and a spherical elevated tank. Students find each volume, convert to gallons, and then answer a fill-time and a comparison question. A common error is putting the diameter into the formula where the radius belongs, which makes the volume far too large; another is dropping the one-third factor for a cone or the four-thirds factor for a sphere. Use 3.14 for pi and 7.48 gallons per cubic foot throughout.

Blackstock Treatment Plant Tanks: Cylinder, Cone, Sphere Volume — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

All tanks below belong to the fictional Blackstock plant.

  1. The clearwell is a cylinder 30 feet across and 16 feet deep. Find its volume in cubic feet using V = pi × r^2 × depth, with r as the radius.
  2. Convert your answer from problem 1 to gallons, using 7.48 gallons per cubic foot.
  3. The sludge hopper is a cone 12 feet across at the top and 9 feet deep. Find its volume in cubic feet using V = one-third × pi × r^2 × depth.
  4. Convert your answer from problem 3 to gallons.
  5. The elevated tank is a sphere 20 feet across. Find its volume in cubic feet using V = four-thirds × pi × r^3.
  6. Convert your answer from problem 5 to gallons.
  7. A pump fills the empty clearwell at 250 gallons each minute. About how many minutes does it take to fill, using your gallons from problem 2? Round to the nearest minute.
  8. About how many times as much water does the clearwell hold as the spherical tank? Use your gallon answers from problems 2 and 6, and round to the nearest tenth.
  9. A student found the sphere's volume by using 20 in place of the radius. Explain whether that makes the volume too big or too small, and about how many times off it is.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 11,304 cubic feet. The radius is 15 feet (half of 30). V = 3.14 × 15^2 × 16 = 3.14 × 225 × 16 = 3.14 × 3,600 = 11,304 cubic feet.
  2. Answer: 84,554 gallons. 11,304 × 7.48 = 84,553.9, which rounds to 84,554 gallons.
  3. Answer: 339 cubic feet. The radius is 6 feet. V = (1/3) × 3.14 × 6^2 × 9 = (1/3) × 3.14 × 36 × 9 = (1/3) × 1,017.36 = 339.12, which rounds to 339 cubic feet.
  4. Answer: 2,537 gallons. 339.12 × 7.48 = 2,536.6, which rounds to 2,537 gallons.
  5. Answer: 4,187 cubic feet. The radius is 10 feet. V = (4/3) × 3.14 × 10^3 = (4/3) × 3.14 × 1,000 = (4/3) × 3,140 = 4,186.7, which rounds to 4,187 cubic feet.
  6. Answer: 31,316 gallons. 4,186.7 × 7.48 = 31,316.3, which rounds to 31,316 gallons.
  7. Answer: 338 minutes. 84,554 ÷ 250 = 338.2, which rounds to 338 minutes.
  8. Answer: about 2.7 times. 84,554 ÷ 31,316 = 2.70, so the clearwell holds about 2.7 times as much as the spherical tank.
  9. Answer: Too big, about 8 times too big. The formula cubes the radius, and 20 is twice the correct radius of 10; since 2^3 = 8, using the diameter makes the volume about 8 times larger than it should be.

Differentiation

Support

Give students the radius already found for each tank (15, 6, and 10 feet) and a reference card listing the three formulas and the 7.48 conversion. Let them use a calculator for every step and check each squared or cubed value before multiplying.

Extension

Ask students to design a fourth tank, a cylinder, that holds close to the same number of gallons as the spherical tank, stating its diameter and depth and showing the volume calculation that supports their choice.

ELL supports

Provide a labeled picture of each shape marking the diameter, the radius, and the depth as separate measurements. Pair each formula with the words "one-third for a cone" and "four-thirds for a sphere" so the fraction is tied to the shape name.

Extensions

Ask your local water utility how many gallons its largest tank holds and what shape it is. Students can compare that real number with the Blackstock tanks and decide which formula the utility's engineers would have used.