A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2021 · Q96 Marks

The volume V m³ of a large circular mound of iron ore of radius r m is modelled by the equation V = (9/2)(r - 1/2)³ – 1 for r ≥ 2. Iron ore is added to the mound at a constant rate of 1.5 m³ per second.

(a)Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m.[3]
(b)Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] dV/dr = 9/2 * 3(r - 1/2)² OE. Accept unsimplified.
[M1] Correct use of chain rule with 1.5, their differentiated expression for dV/dr and using r = 5.5 .
[A1] 0.0133 or 3/225 or 1/75 [metres per second]
[B1 FT] Correct statement involving dV/dr or their dV/dr, 1.5 and 0.1.
[B1] (r - 1/2)² = 15 => r = 0.5 + √(15)
[B1] [Volume =] 8.13 AWRT

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2021 Paper 1 Variant 2, question 9.

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