One cylinder, four changes

One cylinder, four changes

Deriving which dimension squares and which does not, instead of recalling a rule.

1. 18.1VolumeFoundation

   What is the volume of the original tank?

   • A160 pi cubic metres
   • B80 pi cubic metres
   • C40 pi cubic metres
   • D1,600 pi cubic metres
   Show the answer and reasoningHide the reasoning

   160 pi cubic metres

   Volume is pi r^2 h = pi x 16 x 10 = 160 pi cubic metres. Using r rather than r squared gives 40 pi, which is the commonest slip and the one the next question depends on not having made.

2. 18.2VolumeStandard

   Builds on: The 160 pi from question 1, which is what the new figure is compared against.

   The radius is doubled and the height is unchanged. What happens to the volume?

   • AIt doubles.
   • BIt quadruples.
   • CIt is multiplied by eight.
   • DIt is unchanged.
   Show the answer and reasoningHide the reasoning

   It quadruples.

   The radius appears squared, so doubling it multiplies the volume by 2^2 = 4: the new volume is pi x 64 x 10 = 640 pi, four times 160 pi. Doubling a length does not double a volume unless every dimension doubles.

3. 18.3VolumeStandard

   Builds on: Question 2. Answering both correctly requires noticing that r is squared in the formula and h is not, which a single question can be got right without noticing.

   Starting again from the original tank, the height is doubled and the radius is unchanged. What happens to the volume?

   • AIt quadruples.
   • BIt is multiplied by eight.
   • CIt doubles.
   • DIt is unchanged.
   Show the answer and reasoningHide the reasoning

   It doubles.

   The height appears to the first power, so doubling it doubles the volume: pi x 16 x 20 = 320 pi. Set against question 2, this is the whole point — the same change to a different dimension has a different effect.

4. 18.4VolumeChallenge

   Builds on: Questions 2 and 3 supply the two factors separately — a quarter from the radius, a half from the height — and this question is the first that needs both at once.

   A second tank is a scale model of the original at half size in every dimension. What fraction of the original volume does it hold?

   • AOne half
   • BOne quarter
   • COne eighth
   • DOne sixteenth
   Show the answer and reasoningHide the reasoning

   One eighth

   Halving every dimension halves the radius twice over in the r^2 term and once in h: (1/2)^2 x (1/2) = 1/8. Check it directly: pi x 4 x 5 = 20 pi, and 20 pi is one eighth of 160 pi.