2018 MJC P2 Q3

A semicircle has radius \(r\) cm, perimeter \(P\) cm and area \(A\) cm\(^{2}\). Show that

\(\frac{\mathrm{d}P}{\mathrm{d}A}=\frac{2+\pi }{\pi r}\).

Determine the exact value of the radius when the area of the semicircle is increasing at a constant rate of \(3\) cm\(^{2}\)/s and the perimeter is increasing at a constant rate of \(\frac{3}{5}\)cm/s.[4]

An architectural firm wants to make a model of a greenhouse as shown.

The model is to be made up of three parts.

• The roof is modelled by a pyramid with a square base \(2r\) cm by \(2r\)cm and whose apex is \(r\) cm directly above the center of its base.

• The walls are modelled by four rectangles measuring \(2r\)cm by \(h\) cm.

• The floor is modelled by a square measuring \(2r\)cm by \(2r\)cm.

The three parts are joined together as shown in the diagram. The model is made of material of negligible thickness.

It is given that the external surface area of the model has a fixed value of \(160\)cm\(^{2}\). Show that \(V=80r-\left( \frac{2}{3}+2\sqrt{2} \right){{r}^{3}}\).

Hence, using differentiation, find the value of \(r\) which gives the maximum value of \(V\). (You do not need to verify that the volume is a maximum for this value of \(r\).)[7]

[Volume of Pyramid\(=\frac{1}{3}\times \)base area\(\times \)height]