### medicine tablet: surface area and volumes

Question Sample Titled 'medicine tablet: surface area and volumes'

A pharmaceutical manufacturer is designing a new shape of a tablet. As shown in the figure, the old design is a cylinder with base radius ${5}$ $\text{mm}$ and height ${1.5}$ $\text{mm}$; while the new design consists of three parts. The upper and lower parts are two identical hemispheres with radius ${2.5}$ $\text{mm}$ and the middle part is a cylinder with height ${h}$ $\text{mm}$.

${1.5}$ $\text{mm}$The old designThe new design${h}$

 (a) If the volumes of tablets in the two designs are equal, find the value of ${h}$ . (b) If the new design is adopted, find the percentage decrease of the total surface area of each tablet.  (5 marks)

 (a) $\pi{\left({5}\right)}^{{2}}{\left({1.5}\right)}$ $={2}{\left(\dfrac{{1}}{{2}}\times\dfrac{{4}}{{3}}\pi{2.5}^{{3}}\right)}+\pi{\left({2.5}\right)}^{{2}}{h}$ 1M Solving, we get ${h}$ $=\dfrac{{8}}{{3}}$ 1A  (b) The surface area of the old design $={2}\pi{\left({5}\right)}{\left({1.5}\right)}+{2}\pi{\left({5}\right)}^{{2}}$ 1A  $={65}\pi$ $\text{mm}^{{2}}$ The surface area of the new design $={4}\pi{\left({2.5}\right)}^{{2}}+{2}\pi{\left({2.5}\right)}{\left(\dfrac{{8}}{{3}}\right)}$  $=\dfrac{{115}}{{3}}\pi$ $\text{mm}^{{2}}$  $\therefore$Required % decrease $=\dfrac{{{65}\pi-\dfrac{{115}}{{3}}\pi}}{{{65}\pi}}\times{100}\%$ 1M  $={41.0}\%$ (cor. to 3 sig. fig.) 1A

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