[ The volume of a cone of base radius $r$ and height $h$ is given by \(V=\frac{1}{3}\pi {{r}^{2}

Question

[ The volume of a cone of base radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi {{r}^{2}}h$.] A disposable cup for water dispenser, in the form of a cone, is made from a circular piece of paper of radius $8$ cm with negligible thickness. A sector is cut off from the piece of paper as shown in Fig. 1. The rest of the paper is then folded to form a cone with slant height $8$ cm, base radius r cm and height $h$ cm as shown in Fig. 2. Fig. 1 Fig. 2 Find the maximum volume of the cone, giving your answer in exact form.[6] Show that $r=\sqrt{2}h$ when the volume of the cone is a maximum.[2] The cone with the volume found in (i) is then inverted and is held with its axis vertical and vertex downwards. Water from the dispenser flows into the cone at a rate of $3\pi $cm$^{3}$ per second. At time $~t$ seconds after the start, the radius of the water surface is $x$ cm as shown in Fig. 3. Find the rate of change of the height of the water in the cone after $6$seconds.[4] Fig. 3

Accepted Answer

(i) $\frac{1024}{27}\sqrt{3}\pi $cm$^3$ (iii) $\frac{1}{6}$cm$^3$/s

2023 JPJC P1 Q11

2023 JPJC P1 Q11

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