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

Practise Edexcel IAL P4.5.2 connected rates: derive geometry formulae, link related derivatives, and report rates with units.

Exam points

Link rates with a chain such as dV/dt=(dV/dr)(dr/dt)
Derive the required volume, area or geometric formula before differentiating
Interpret signs, units and the requested rate at the stated value

[Maximum number: 4]

Figure 2

A cone, shown in Figure 2, has
- fixed height 5 cm
- base radius $r \mathrm{~cm}$
- slant height $l \mathrm{~cm}$

Given that the base radius is increasing at a constant rate of 3 cm per minute,

find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm. Give your answer in $\mathrm{cm}^{2}$ per minute to one decimal place.
[The total surface area, S, of a cone is given by the formula $S=\pi r^{2}+\pi r l$ ]

$S=\pi r^2+\pi r\sqrt{25+r^2}$ $\frac{dS}{dr}=2\pi r+\pi\sqrt{25+r^2}+\frac{\pi r^2}{\sqrt{25+r^2}}$ $\frac{dS}{dt}=\frac{dS}{dr}\frac{dr}{dt}$ $\frac{dS}{dt} =\left(2\pi(1.5)+\pi\sqrt{27.25}+\frac{\pi(1.5)^2}{\sqrt{27.25}}\right)3 =81.5\text{ cm}^2\text{/min}$

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