Edexcel A-Level Mathematics A2 P4.5.2 Formation of Simple Differential Questions

Practise forming connected-rate equations for changing volumes, areas, radii and heights in physical geometry models.

Exam points

link rates with dV/dt=(dV/dr)(dr/dt) or equivalent chains for the model variables
derive a volume or area formula for cones, spheres, rods or tablets before differentiating
interpret signs and units when finding rates of increase or decrease at a given value

[Maximum number: 3]

Figure 1

A regular icosahedron of side length $x \mathrm{~cm}$ , shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length $x \mathrm{~cm}$ .

Find the rate of change of the volume of the icosahedron when x=2, giving your answer to 2 significant figures.

$\frac{dA}{dt}=0.025.$ $\frac{dV}{dt} =\frac{dV}{dA}\frac{dA}{dt} =\frac{(3+\sqrt5)x}{8\sqrt3}\cdot0.025.$
When x=2,
$\frac{dV}{dt} =\frac{2(3+\sqrt5)}{8\sqrt3}\cdot0.025 =0.019\ldots\ \mathrm{cm^3s^{-1}}.$

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Edexcel A-Level Mathematics A2 P4.5.2 Formation of Simple Differential Questions question 1