### Volume of container with frustum bottom

Question Sample Titled 'Volume of container with frustum bottom'

A factory is going to produce a type of cylindrical container with outer height ${64}$ $\text{cm}$ and outer base radius ${28}$ $\text{cm}$. Inside the container, the upper part of is a cynlinder and the lower part is a frustum with height ${8}$ $\text{cm}$, upper and lower radii of ${28}$ $\text{cm}$ and ${8}$ $\text{cm}$ respectively as shown in Figure (a). (Ignore the thinkness of the container.)

${8}$ $\text{cm}$${64}$ $\text{cm}$${28}$ $\text{cm}$${8}$ $\text{cm}$Figure (a)Figure (b)

 (a) (i) Find the capacity of the container in terms of $\pi$. (ii) Find the percentage of the capacity of the whole container.  (5 marks) (b) The designer of the factory claimed that changing the lower part of the container from a frustum into a hemisphere as shown in Figure (b) can reduce the capacity by more than ${10}\%$. Do you agree? Explain your answer. (4 marks)

 (ai) Let ${h}$ $\text{cm}$ be the height of the cone in which the lower part is identical to the frustum$.$ $\dfrac{{8}}{{28}}$ $=\dfrac{{{h}-{8}}}{{h}}$ 1M ${h}$ $=\dfrac{{56}}{{5}}$ $\text{cm}$ 1A Required capacity $=\pi{\left({28}\right)}^{{2}}{\left({64}-{8}\right)}+\dfrac{{1}}{{3}}\pi{\left({28}\right)}^{{2}}{\left(\dfrac{{56}}{{5}}\right)}\cdot{\left({1}-{\left(\dfrac{{8}}{{28}}\right)}^{{3}}\right)}$  $=\dfrac{{140288}}{{3}}\pi$ $\text{cm}^{{3}}$ 1A  (aii) Required percentage $=\dfrac{{\dfrac{{140288}}{{3}}\pi}}{{{28}^{{2}}\times{64}\pi}}$ 1M  $={0}$ 1A  (b) Note that the radius of the hemisphere is ${28}$ $\text{cm}$ . New capacity of the container $=\pi{\left({28}\right)}^{{2}}{\left({64}-{28}\right)}+\dfrac{{1}}{{2}}\times\dfrac{{4}}{{3}}\times\pi{\left({28}\right)}^{{3}}$ 1M  $=\dfrac{{128576}}{{3}}\pi$ $\text{cm}^{{3}}$ 1A  % change of capacity $=\dfrac{{\dfrac{{128576}}{{3}}\pi-\dfrac{{140288}}{{3}}\pi}}{{\dfrac{{140288}}{{3}}\pi}}\times{100}\%$ 1M  $\approx-{8.35}\%$ $\therefore$The new design does not reduce more than ${10}\%$ of capacity. Thus, the claim is disagreed. 1M

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