O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2025 · Q66 Marks
The volume, V, of a sphere is increasing at the constant rate of 2π cm³ s⁻¹. Find the rate of change of the surface area, S, of this sphere when the volume of the sphere is 36π cm³.
Mark scheme
Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.
- [B1] r = 3 nfww
- [M1] dV/dr = 4πr² and dS/dr = 8πr (Uses relevant chain rule e.g. dV/dt = dV/dr * dr/dt to find the derivatives and values of dr/dt and dS/dr)
- [A1] dr/dt = 1/18
- [B1] Correct chain rule for dS/dt e.g. dS/dt = dS/dr * dr/dt
- [M1] Substitute values into dS/dt chain rule, e.g. 24π x 1/18 or 2/3 x 2π
- [A1] dS/dt = 4π/3 isw or 4.19 or 4.188 to 4.1888
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