Connected rates of change

A question is this type if and only if it requires using the chain rule to connect two rates of change (e.g. dr/dt and dV/dt, or dx/dt and dy/dt) given one rate and a geometric formula or curve equation.

3 questions · Moderate -0.3

1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
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CAIE P1 2012 November Q3
4 marks Moderate -0.8
3 An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday.
Edexcel C4 2014 June Q5
6 marks Moderate -0.3
5. At time \(t\) seconds the radius of a sphere is \(r \mathrm {~cm}\), its volume is \(V \mathrm {~cm} ^ { 3 }\) and its surface area is \(S \mathrm {~cm} ^ { 2 }\). [You are given that \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and that \(S = 4 \pi r ^ { 2 }\) ] The volume of the sphere is increasing uniformly at a constant rate of \(3 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
  1. Find \(\frac { \mathrm { d } r } { \mathrm {~d} t }\) when the radius of the sphere is 4 cm , giving your answer to 3 significant figures.
  2. Find the rate at which the surface area of the sphere is increasing when the radius is 4 cm .
OCR C3 Q4
6 marks Standard +0.3
Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = (h^6 + 16)^{\frac{1}{2}} - 4.$$
  1. Find the value of \(\frac{dV}{dh}\) when \(h = 2\). [3]
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures. [3]