Hint
Evaluate all quantities at the same instant. Use \(\displaystyle \frac{2rdr}{dt}\) after dividing the rate by π.
Worked solution
(a)
\(\displaystyle A=\pi r^{2}\), so \(\displaystyle \frac{dA}{dt}= \frac{2\pi rdr}{dt}=\)\(16 \pi\) \(\displaystyle \frac{\mathrm{cm}^{2}}{s}\).
Answer: \(16 \pi\) \(\displaystyle \frac{\mathrm{cm}^{2}}{s}\)
(b)
Differentiate \(\displaystyle V= \frac{4\pi r^{3}}{3}\) with respect to time and use the same radius and radius rate.
Answer: \(256 \pi\) \(\displaystyle \frac{\mathrm{cm}^{3}}{s}\)
(c)
Use r=r₀+qt.
\(5\)
Answer: \(5\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Differentiate \(\displaystyle \pi r^{2}\) in time and substitute the radius and its rate.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not confuse \(\displaystyle \frac{dA}{dr}\) with \(\displaystyle \frac{dA}{dt}\).
(b) · 2 marks
- 1 mark: Differentiate \(\displaystyle \frac{4\pi r^3}{3}\) in time and substitute the same instant.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The chain rule includes \(\displaystyle \frac{dr}{dt}\).
(c) · 2 marks
- 1 mark: Use r=r₀+qt.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not confuse \(\displaystyle \frac{dA}{dr}\) with \(\displaystyle \frac{dA}{dt}\).
Total: 6 marks.