Worksheet Grade 10 Math
Cobre Vista's New Water Tank: Cylinder vs. Sphere Design Worksheet
A grade 10 geometry worksheet where students model cylindrical and spherical elevated-tank designs for a fictional town, comparing volume and surface-area efficiency.
Objectives
- Model a storage tank as a cylinder or a sphere and calculate its volume in cubic feet and gallons.
- Compare two tank designs of equal diameter to determine which holds more water.
- Calculate and compare surface area per 1,000 gallons of capacity to evaluate material efficiency.
Vocabulary
- Water tower
- A water tower is a big tank raised high above town that stores drinking water. Its height is what pushes water through the pipes, keeping the pressure steady even when the power goes out.
- Ground storage tank
- A ground storage tank is a large water storage tank that sits on or near the ground rather than up on a tower. Utilities often pump water from a ground storage tank up into an elevated tank or directly into the pipes as needed.
- Standpipe
- A standpipe is a tall, cylindrical water storage tank that sits directly on the ground instead of standing on legs like a water tower. It stores water and helps maintain steady pressure in the pipes nearby.
Teacher overview
The town of Cobre Vista, invented for this worksheet, is replacing an aging elevated water tank. Site rules limit any new tank to a 60-foot diameter, so the engineering team is comparing two shapes that both fit within that footprint: a cylindrical tank and a spherical tank. Students calculate each design's volume, check it against the town's required minimum capacity, and then compare surface area per gallon stored to judge which design uses material more efficiently, since cost tracks surface area rather than volume.
Student Worksheet
The town of Cobre Vista requires at least 470,000 gallons of new elevated storage. Two designs are under review, both with an overall diameter of 60 feet.
Option A: a cylindrical tank, 60 feet in diameter and 24 feet deep.
Option B: a spherical tank, 60 feet in diameter at its widest point.
Use these facts for all problems:
- Cylinder volume (cu ft) = 0.785 × Diameter² × depth
- Sphere volume (cu ft) = (4/3) × π × radius³, using π ≈ 3.14159
- Sphere surface area (sq ft) = 4 × π × radius²
- Cylinder lateral surface area (sq ft) = π × Diameter × depth; each flat end adds 0.785 × Diameter²
- 1 cubic foot holds 7.48 gallons
Find Option A's volume in cubic feet, using a diameter of 60 feet and a depth of 24 feet.
Convert Option A's volume from item 1 to gallons.
Does Option A meet Cobre Vista's minimum required capacity of 470,000 gallons? By about how many gallons does it exceed or fall short of that minimum?
Option B is a sphere with the same 60-foot diameter, so its radius is 30 feet. Find Option B's volume in cubic feet. Round to the nearest cubic foot.
Convert Option B's volume from item 4 to gallons.
Compare Option A and Option B. Which design holds more water for the same 60-foot diameter, and by about how many gallons?
Find Option B's (the sphere's) total surface area in square feet, using a radius of 30 feet.
Find Option A's (the cylinder's) total surface area in square feet, including the lateral surface and both flat ends.
Divide each design's surface area by its volume in thousands of gallons to find its surface area per 1,000 gallons of capacity. Show both calculations.
Based on item 9, which design is more material-efficient per gallon of water stored? Does this match the general geometric principle that, of all shapes, a sphere has the smallest surface area for a given volume?
Despite Option B's material efficiency, spherical elevated tanks are far less common in the United States than cylindrical ones. Propose one practical, non-mathematical reason an engineer might still choose the cylindrical design.
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: 67,824 cu ft. Diameter² = 60² = 3,600. Volume = 0.785 × 3,600 × 24 = 2,826 × 24 = 67,824 cubic feet.
Answer: 507,324 gallons. 67,824 cu ft × 7.48 gal/cu ft = 507,323.52, which rounds to about 507,324 gallons.
Answer: 37,324 gallons. Yes, Option A meets the minimum, exceeding it by about 37,324 gallons: 507,324 − 470,000 = 37,324.
Answer: 113,097 cu ft. Radius = 30 ft, so radius³ = 27,000. Volume = (4/3) × 27,000 × 3.14159 = 36,000 × 3.14159 ≈ 113,097.24, which rounds to 113,097 cubic feet.
Answer: 845,966 gallons. 113,097 cu ft × 7.48 gal/cu ft = 845,965.56, which rounds to about 845,966 gallons.
Answer: 338,642 gallons. Option B (the sphere) holds more water than Option A: 845,966 − 507,324 = 338,642 gallons more, even though both share the same 60-foot diameter.
Answer: 11,310 sq ft. Surface area = 4 × 3.14159 × 30² = 12.56636 × 900 ≈ 11,309.72, which rounds to about 11,310 square feet.
Answer: 10,176 sq ft. Lateral surface = 3.14159 × 60 × 24 ≈ 4,523.89, or about 4,524 sq ft. Each flat end = 0.785 × 3,600 = 2,826 sq ft, so both ends total 5,652 sq ft. Total surface area = 4,524 + 5,652 = 10,176 square feet.
Answer: 20.1 sq ft per 1,000 gallons (Option A); 13.4 sq ft per 1,000 gallons (Option B). Option A: 10,176 ÷ 507.324 ≈ 20.1 sq ft per 1,000 gallons. Option B: 11,310 ÷ 845.966 ≈ 13.4 sq ft per 1,000 gallons.
Answer: Option B (the sphere) is more material-efficient, at about 13.4 square feet of surface per 1,000 gallons versus about 20.1 for the cylinder. This matches the geometric principle that a sphere minimizes surface area for a given volume, even though the sphere's raw surface area number in item 7 is larger than the cylinder's, because the sphere also holds far more water.
Answer: Accept any reasonable practical explanation, such as: cylinders are simpler and cheaper to fabricate from flat steel plate, easier to support structurally on a tower, and easier to fit with standard ladders, pipes, and access hatches than a curved spherical shell.
Differentiation
Support (below grade level)
Provide the radius (30 ft) already calculated for the sphere problems, and give students a printed reference card listing both volume formulas and the 7.48 gallons-per-cubic-foot conversion so they do not need to recall them from memory. Work through item 1 together as a class before students attempt items 2 through 11 independently.
Extension (above grade level)
Ask students to find the diameter a cylindrical tank of the same 24-foot depth would need to match Option B's volume exactly, and to explain whether that diameter would still fit within Cobre Vista's 60-foot site limit.
ELL supports
Provide a labeled diagram showing a cylinder's diameter, radius, and depth alongside a sphere's diameter and radius before item 1. Allow students to show their work using the diagram and numbers, with a written explanation only required for items 10 and 11.