A decorative artifact, as shown in the diagram above, is in the shape of a cone with radius $r$ an

Question

A decorative artifact, as shown in the diagram above, is in the shape of a cone with radius $r$ and height $h$ with a sphere of fixed radius $a$ inscribed in it. Show that the volume of the cone, $V$, is given by $V=\frac{\pi {{a}^{2}}{{h}^{2}}}{3\left( h-2a \right)}$. [Volume of cone, $V=\frac{1}{3}\pi {{r}^{2}}h$][3] Use differentiation to find, in terms of $a$, the minimum value of $V$. Leave your answer in exact form.[5] In a triangle $PQR$, $PQ=3$cm and $PR=2$cm. If angle $QPR$ is increasing at a constant rate of $0.1$ radians per second, find the rate of increase of the length $QR$ at the instant when angle $QPR$ is $\frac{\pi }{3}$ radians.[4]

Accepted Answer

(a)(i) $V=\frac{\pi a^2h^2}{3(h-2a)}$ (a)(ii) $\frac{8\pi a^3}{3}\text{ units}^3$ (b) $\frac{3\sqrt{21}}{70}\text{ cm/s}=0.196\text{ cm/s}$ (3 s.f.)

2019 ASRJC Promo Q11

2019 ASRJC Promo Q11

Video Solution:

Solution:

Solution locked