Standard +0.3 This is a standard connected rates of change problem requiring the chain rule with the sphere volume formula V = (4/3)πr³. Students must differentiate, substitute given values, and solve for dr/dt. While it requires understanding of implicit differentiation and careful algebraic manipulation, it follows a well-practiced template with no conceptual surprises, making it slightly easier than average.
3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
Find the rate at which the radius of the balloon is increasing when the radius is 10 cm .
## Question 3:
| $(Derivative =) 4\pi r^2 \rightarrow 400\pi$ | B1 | SOI Award this mark for $\frac{dr}{dV}$ |
|---|---|---|
| $\frac{50}{\text{their derivative}}$ | M1 | Can be in terms of $r$ |
| $\frac{1}{8\pi}$ or $0.0398$ | A1 | AWRT |
---
3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of $50 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.
Find the rate at which the radius of the balloon is increasing when the radius is 10 cm .\\
\hfill \mbox{\textit{CAIE P1 2020 Q3 [3]}}