Scaling a water tank

Scaling a water tank

Applying the length, area and volume scaling rules to one object, where each step uses a different power.

1. 5.1VolumeStandard

   What is the volume of the original tank, in terms of π?What is the volume of the original tank, in terms of π?

   • A12π cubic meters12π cubic meters
   • B6π cubic meters6π cubic meters
   • C24π cubic meters24π cubic meters
   • D18π cubic meters18π cubic meters
   Show the answer and reasoningHide the reasoning

   12π cubic meters12π cubic meters

   A cylinder holds π r² h. With r = 2 and h = 3, that is π x 4 × 3 = 12π cubic meters.A cylinder holds π r^2 h. With r = 2 and h = 3, that is π x 4 × 3 = 12π cubic meters.

2. 5.2ScalingChallenge

   Builds on: The 12π volume from question 1.The 12π volume from question 1.

   How much water does the larger tank hold?How much water does the larger tank hold?

   • A96π cubic meters96π cubic meters
   • B24π cubic meters24π cubic meters
   • C48π cubic meters48π cubic meters
   • D144π cubic meters144π cubic meters
   Show the answer and reasoningHide the reasoning

   96π cubic meters96π cubic meters

   Doubling every linear dimension multiplies volume by 2³ = 8, so 12π x 8 = 96π. Checking directly agrees: radius 4 and height 6 give π x 16 × 6 = 96π. Doubling the volume instead of cubing the factor is the standard error here.Doubling every linear dimension multiplies volume by 2^3 = 8, so 12π x 8 = 96π. Checking directly agrees: radius 4 and height 6 give π x 16 × 6 = 96π. Doubling the volume instead of cubing the factor is the standard error here.

3. 5.3ScalingChallenge

   Builds on: The volume factor of 8 from question 2, which this question asks you not to reuse.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?The tanks are painted on the outside. How does the painted area of the larger tank compare with the original?

   • AFour times as largeFour times as large
   • BTwice as largeTwice as large
   • CEight times as largeEight times as large
   • DSixteen times as largeSixteen times as large
   Show the answer and reasoningHide the reasoning

   Four times as largeFour times as large

   Area scales with the square of the linear factor, so doubling every dimension multiplies surface area by 2² = 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.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.