Scaling a water tank
Applying the length, area and volume scaling rules to one object, where each step uses a different power.
Scenario
A market garden stores rainwater in a cylindrical tank of radius 2 metres and height 3 metres. Demand has outgrown it, so a second tank is built to the same shape, with every linear dimension multiplied by 2.
Both tanks stand upright and are painted on the outside, including the curved wall and the flat top, but not the base, which sits on concrete.
Answers in order, explanations withheld, and a report on where the chain broke.
- 5.1VolumeStandard
What is the volume of the original tank, in terms of pi?
- A12pi cubic metres
- B6pi cubic metres
- C24pi cubic metres
- D18pi cubic metres
Show the answer and reasoningHide the reasoning
12pi cubic metres
A cylinder holds pi r^2 h. With r = 2 and h = 3, that is pi x 4 x 3 = 12pi cubic metres.
- 5.2ScalingChallenge
Builds on: The 12pi volume from question 1.
How much water does the larger tank hold?
- A96pi cubic metres
- B24pi cubic metres
- C48pi cubic metres
- D144pi cubic metres
Show the answer and reasoningHide the reasoning
96pi cubic metres
Doubling every linear dimension multiplies volume by 2^3 = 8, so 12pi x 8 = 96pi. Checking directly agrees: radius 4 and height 6 give pi x 16 x 6 = 96pi. Doubling the volume instead of cubing the factor is the standard error here.
- 5.3ScalingChallenge
Builds on: The volume factor of 8 from question 2, which this question asks you not to reuse.
The tanks are painted on the outside. How does the painted area of the larger tank compare with the original?
- AFour times as large
- BTwice as large
- CEight times as large
- DSixteen times as large
Show the answer and reasoningHide the reasoning
Four times as large
Area scales with the square of the linear factor, so doubling every dimension multiplies surface area by 2^2 = 4 — not by 8, which is what happened to the volume. The same scale factor produces different multipliers for area and volume, and the question is which one applies.