Math problem set Grade 8 Math
Netherby Sludge Thickener: Cone, Cylinder, and Sphere Volumes
A grade 8 problem set finding the volume of a fictional plant's cone-bottomed tank and a spherical digester, converting cubic feet to gallons and comparing sizes.
Objectives
- Students will find the volumes of a cylinder, a cone, and a sphere using the standard formulas.
- Students will convert tank volumes from cubic feet to gallons and compare them.
Vocabulary
Key words: cylinder, cone, sphere, volume
Teacher overview
A sludge thickener is often a cylinder with a cone-shaped bottom, which makes it a natural place to use both the cylinder and cone volume formulas on one tank. This set walks students through that combined tank, then a separate spherical digester, and asks them to compare the two in gallons. A common mistake to watch for is using the diameter in place of the radius, and forgetting that the cone formula includes the factor of one-third while the sphere formula uses the radius cubed.
Problems
Netherby Water Reclamation Facility is a fictional plant. Its sludge thickener is a tank shaped like a cylinder with a cone-shaped bottom. Use 3.14 for pi, and use the fact that one cubic foot holds 7.48 gallons.
- The cylinder part is 20 feet across (its diameter) and 12 feet tall, so its radius is 10 feet. Find its volume in cubic feet using Volume = 3.14 x radius squared x height.
- The cone bottom has the same radius, 10 feet, and is 6 feet deep. Find its volume using Volume = (1/3) x 3.14 x radius squared x depth.
- Find the total volume of the thickener by adding the cylinder volume and the cone volume.
- Convert the total volume from problem 3 to gallons by multiplying by 7.48.
- The thickener fills at 2,000 gallons per hour. How many hours does it take to fill from empty? Round to the nearest tenth.
- A separate spherical digester at the plant is 30 feet across, so its radius is 15 feet. Find its volume using Volume = (4/3) x 3.14 x radius cubed.
- Convert the digester's volume from problem 6 to gallons by multiplying by 7.48.
- How many times as large is the digester's volume as the thickener's total volume, using the cubic-foot values? Round to the nearest tenth.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: 3,768 cubic feet.
Radius = 10 ft (half of the 20 ft diameter). Volume = 3.14 x 10 squared x 12 = 3.14 x 100 x 12 = 3,768 cubic feet. - Answer: 628 cubic feet.
Volume = (1/3) x 3.14 x 10 squared x 6 = (1/3) x 3.14 x 100 x 6 = (1/3) x 1,884 = 628 cubic feet. - Answer: 4,396 cubic feet.
3,768 cubic feet + 628 cubic feet = 4,396 cubic feet. - Answer: 32,882 gallons.
4,396 cubic feet x 7.48 = 32,882.08, which rounds to 32,882 gallons. - Answer: 16.4 hours.
32,882.08 gallons / 2,000 gallons per hour = 16.44, which rounds to 16.4 hours. - Answer: 14,130 cubic feet.
Volume = (4/3) x 3.14 x 15 cubed = (4/3) x 3.14 x 3,375 = (4/3) x 10,597.5 = 14,130 cubic feet. - Answer: 105,692 gallons.
14,130 cubic feet x 7.48 = 105,692.4, which rounds to 105,692 gallons. - Answer: 3.2 times.
14,130 cubic feet / 4,396 cubic feet = 3.21, which rounds to 3.2 times as large.
Differentiation
Support
Provide the three formulas on a card with the radius already found for each shape (10 ft and 15 ft), so students focus on substitution and arithmetic. Let them use the cubic-foot answers from problems 1 and 2 when they work problem 3 rather than recomputing.
Extension
Ask students to find the depth a plain cylinder (no cone) of the same 10-foot radius would need to hold the same total volume as the cone-bottomed thickener, and explain why the cone bottom changes the needed height.
ELL supports
Provide a labeled diagram of each shape showing diameter, radius, height, and depth as separate arrows before problem 1. Pair each formula with a small sketch of the matching shape so students connect the words to the picture.