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?What is the volume of the original tank?

   • A160π cubic meters160π cubic meters
   • B80π cubic meters80π cubic meters
   • C40π cubic meters40π cubic meters
   • D1,600π cubic meters1,600π cubic meters
   Show the answer and reasoningHide the reasoning

   160π cubic meters160π cubic meters

   Volume is π r² h = π x 16 × 10 = 160π cubic meters. Using r rather than r squared gives 40π, which is the commonest slip and the one the next question depends on not having made.Volume is π r^2 h = π x 16 × 10 = 160π cubic meters. Using r rather than r squared gives 40π, which is the commonest slip and the one the next question depends on not having made.

2. 18.2VolumeStandard

   Builds on: The 160π from question 1, which is what the new figure is compared against.The 160π 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?The radius is doubled and the height is unchanged. What happens to the volume?

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

   It quadruples.It quadruples.

   The radius appears squared, so doubling it multiplies the volume by 2² = 4: the new volume is π x 64 × 10 = 640π, four times 160π. Doubling a length does not double a volume unless every dimension doubles.The radius appears squared, so doubling it multiplies the volume by 2^2 = 4: the new volume is π x 64 × 10 = 640π, four times 160π. 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.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?Starting again from the original tank, the height is doubled and the radius is unchanged. What happens to the volume?

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

   It doubles.It doubles.

   The height appears to the first power, so doubling it doubles the volume: π x 16 × 20 = 320π. Set against question 2, this is the whole point — the same change to a different dimension has a different effect.The height appears to the first power, so doubling it doubles the volume: π x 16 × 20 = 320π. 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.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?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 halfOne half
   • BOne quarterOne quarter
   • COne eighthOne eighth
   • DOne sixteenthOne sixteenth
   Show the answer and reasoningHide the reasoning

   One eighthOne eighth

   Halving every dimension halves the radius twice over in the r² term and once in h: (12)^2 x (12) = 18. Check it directly: π x 4 × 5 = 20π, and 20π is one eighth of 160π.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: π x 4 × 5 = 20π, and 20π is one eighth of 160π.