### medicine tablet volume and surface area

Question Sample Titled 'medicine tablet volume and surface area'

A pharmaceutical manufacturer is designing the shape of a tablet. As shown in the figure, there are two designs. The first one is a cylinder with base radius ${6}$ $\text{mm}$ and height ${1.5}$ $\text{mm}$. The second one consists of three parts. The upper and lower parts are two identical hemispheres with radius ${2}$ $\text{mm}$ and the middle part is a cylinder with height ${h}$ $\text{mm}$.

${1.5}$ $\text{mm}$The first designThe second design${h}$

 (a) If the volumes of tablets in the two designs are equal, find the value of ${h}$ . (b) It takes longer time to dissolve a tablet with larger surface area. If shorter dissolution time is perferred, which design should be chosen? Explain your answer.  (4 marks)

 (a) $\pi{\left({6}\right)}^{{2}}{\left({1.5}\right)}$ $={2}{\left(\dfrac{{1}}{{2}}\times\dfrac{{4}}{{3}}\pi{2}^{{3}}\right)}+\pi{\left({2}\right)}^{{2}}{h}$ 1M Solving, we get ${h}$ $=\dfrac{{65}}{{6}}$ 1A  (b) The surface area of the first design $={2}\pi{\left({6}\right)}{\left({1.5}\right)}+{2}\pi{\left({6}\right)}^{{2}}$ 1A  $={90}\pi$ $\text{mm}^{{2}}$ The surface area of the second design $={4}\pi{\left({2}\right)}^{{2}}+{2}\pi{\left({2}\right)}{\left(\dfrac{{65}}{{6}}\right)}$  $=\dfrac{{178}}{{3}}\pi$ $\text{mm}^{{2}}$  $\therefore$The surface area of the second design is smaller. Thus, the second design should be chosen. 1A

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