The stallholder at a festive night market makes a candle holder from a right hollow cone, \(C_1\), and a cylindrical frame as shown in the diagram below. The candle holder has negligible thickness. The solid base of the cylinder is in contact with the inner surface of \(C_1\) and the open top of the cylinder is level with the base of \(C_1\). \(C_1\) has base radius 4 cm and height 6 cm. The cylinder has radius \(x\) cm and height \(h\) cm.
Given that \(V\) denotes the volume of the cylinder, show that \( V=\pi\left(16h-\frac{16}{3}h^2+\frac{4}{9}h^3\right). \)[2]
By using differentiation, find the exact maximum volume of the cylinder.[3]
The cylinder now has the dimensions found in part (a), where its volume is maximised. Another inverted hollow cone, \(C_2\), of radius 8 cm and height 3 cm is filled to the brim with a special liquid wax. The wax drains from \(C_2\) through a small hole at its vertex into the cylinder at a constant rate that would empty \(C_2\) in 120 seconds. Assuming the wax remains in liquid form throughout the process, find the rate of decrease of the depth of the wax in \(C_2\) at the instant the cylinder is filled to the brim.[7]
[The volume of a cone of base radius \(r\) and height \(h\) is given by \(V=\frac{1}{3}\pi r^2h\).]
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Answer:(a) \(V_{\max}=\dfrac{128\pi}{9}\text{ cm}^3\) (b) rate of decrease \(=\dfrac3{40(\sqrt[3]{21})^2}=0.00985\text{ cm s}^{-1}\)