2025 Crescent Girls' School Prelims A Math Paper 1
Cement is poured into an empty right circular cone mould at a constant rate of \(36\pi\ \text{cm}^3/\text{s}\). The mould stands on its vertex. The cement forms a similar cone whose depth is twice the radius of its horizontal surface.
Show that the radius of the cement surface after 1.5 minutes is approximately \(16.94\ \text{cm}\). [The volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2h\).][2]
Find the rate of change of the radius of the cement surface after 1.5 minutes.[3]
The intensity of light, \(I\) candelas, at a certain distance, \(x\) metres, from a light source is given by the formula \(I=\frac{k}{x^2}\), where \(k\) is a constant. Given that the intensity of light is 12 candelas at a distance of 4 metres from the source, find the rate at which the intensity of light is changing with respect to the distance when \(x=8\).[3]
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Answer:(a)(i) \(r=16.94\text{ cm}\) (a)(ii) \(\frac{dr}{dt}=0.0627\text{ cm s}^{-1}\) (b) \(\frac{dI}{dx}=-0.750\text{ cd m}^{-1}\)