Rotating a parabolic cake mould
A pastry shop in Santiago is designing a new cake mould. The interior cross-section, in decimetres, is the region bounded by y=x^2 and y=2x for 0 ≤ x ≤ 2. The mould is obtained by rotating this region about the x-axis. The shop's standard recipe yields 10 litres of batter, and the batter expands by 25% during baking. Shop rules require that the risen batter occupy at most 85% of the mould volume, leaving headspace. The manager first wants the cross-sectional area to estimate release agent at 0.05 litres per square decimetre. She then wants the mould volume and a yes-or-no decision on the standard recipe. Finally, a smaller recipe yields 9 litres; the manager asks whether that version would satisfy the 85% rule. All lengths are in decimetres, so 1 dm^3 equals 1 litre.
1. Find the area of the cross-section and the release agent needed at 0.05 litres per square decimetre.
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The curves meet where x^2=2x, i.e. x=0 or x=2. On [0,2], 2x ≥ x^2. Area = ∫0^2 (2x-x^2) dx = [x^2 - x^3/3]_0^2 = 4 - 8/3 = 4/3 dm^2. Release agent = 0.05*(4/3)=1/15 L ≈0.0667 L, about 66.7 mL.
2. Compute the mould volume and decide whether the 10-litre recipe satisfies the 85% rule.
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Using washers about the x-axis, V=π∫0^2 [(2x)^2-(x^2)^2] dx = π∫0^2 (4x^2-x^4) dx = π[4x^3/3 - x^5/5]_0^2 = π(32/3 - 32/5)=π(160-96)/15=64π/15≈13.40 dm^3, i.e. about 13.40 L. The 10 L of batter expands to 12.5 L. The fill fraction is 12.5/13.40≈0.933, or 93.3%, which exceeds 85%. The maximum risen volume allowed is 0.85*13.40≈11.39 L, so the maximum initial batter is 11.39/1.25≈9.11 L. The 10 L recipe is not safe.
3. Would the 9-litre recipe satisfy the rule? How much headspace would it leave?
Read the worked answer
9 L expands to 9*1.25=11.25 L. Fill fraction = 11.25/13.40≈0.840, or 84.0%, which is below 85%. It is acceptable, but only just. The risen batter occupies 11.25 L, so headspace is 13.40-11.25≈2.15 L, about 16.0% of the mould. The manager should note that the margin is small: a slightly larger expansion than 25% could break the rule.