Math problem set Grade 7 Engineering & Design
Stonecrop Ridge Cistern: Scale Drawing and Tank Volume Set
Eight problems where students convert a rainwater cistern blueprint's scale measurements to actual feet, find its volume in gallons, and redraw it at a new scale.
Objectives
- Students will convert scale-drawing measurements into actual dimensions using a stated scale.
- Students will calculate the volume of a cylindrical tank in cubic feet and convert it to gallons.
Vocabulary
Key words: scale drawing, cylinder, volume, conversion factor
Teacher overview
The fictional Stonecrop Ridge school district is planning a rainwater harvesting cistern, and its blueprint is drawn at a scale where 1 inch represents 4 feet of the actual tank. This problem set has students convert the blueprint's measurements into real dimensions, use those dimensions to find the cistern's volume, and then redraw the same tank at a more detailed scale. A common misconception in this math is calculating the tank's volume directly from the blueprint's inch measurements instead of converting to actual feet first, which produces an answer that is far too small to be a real tank's volume.
Problems
Stonecrop Ridge's cistern blueprint uses a scale of 1 inch = 4 feet. On the blueprint, the cistern (drawn as a cylinder) measures 2.5 inches across and 1.5 inches tall. Use these facts for all problems:
- Volume of a cylinder (cubic feet) = 0.785 × diameter² × depth.
- 1 cubic foot = 7.48 gallons.
Using the scale, find the cistern's actual diameter in feet.
Using the scale, find the cistern's actual depth in feet.
Using your answers from problems 1 and 2, find the cistern's volume in cubic feet.
Convert your answer from problem 3 to gallons.
The district wants to redraw the blueprint at a more detailed scale of 1 inch = 2 feet. At this new scale, what would the actual diameter from problem 1 measure on the blueprint?
At the same new scale (1 inch = 2 feet), what would the actual depth from problem 2 measure on the blueprint?
A second, smaller cistern design has half the diameter of the original but the same depth. Without recalculating from scratch, explain whether its volume is half as large, more than half, or less than half of the original, and why.
Challenge. The district needs the cistern to hold at least 3,000 gallons. Does the original design (problem 4) meet that requirement? By how many gallons does it exceed or fall short?
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: 10 feet. 2.5 in × 4 ft/in = 10 ft.
Answer: 6 feet. 1.5 in × 4 ft/in = 6 ft.
Answer: 471 cubic feet. Volume = 0.785 × 10² × 6 = 0.785 × 100 × 6 = 471 cu ft.
Answer: 3,523 gallons. 471 cu ft × 7.48 gal/cu ft = 3,523.08, which rounds to 3,523 gallons.
Answer: 5 inches. 10 ft ÷ 2 ft/in = 5 in.
Answer: 3 inches. 6 ft ÷ 2 ft/in = 3 in.
Answer: Less than half. Volume depends on the square of the radius (or diameter), so halving the diameter while keeping the depth the same cuts the volume to one-fourth, not one-half, of the original.
Answer: Yes, it exceeds the requirement by 523 gallons. 3,523 gallons (problem 4) − 3,000 gallons (requirement) = 523 gallons more than needed.
Differentiation
Support
Provide the scale conversion already modeled with one worked example (for instance, a 1-inch measurement converted to 4 feet), and give students a printed reference card listing "1 cu ft = 7.48 gal" so they do not need to recall the conversion from memory.
Extension
Ask students to find what blueprint scale (in the form "1 inch = ___ feet") would make the cistern's actual diameter measure exactly 8 inches on the drawing, working backward from the known 10-foot actual diameter.
ELL supports
Provide a labeled diagram of the cylinder showing diameter and depth as distinct measurements, marked in both blueprint inches and actual feet, before problem 1. Allow students to show their work using the diagram plus numbers rather than full sentences.