The figure shows two sectors in which $PQ$ and $MN$ are arcs of concentric circles with centre \

Question

The figure shows two sectors in which $PQ$ and $MN$ are arcs of concentric circles with centre $O$. $OP=2r$ cm, $MP=r$ cm and the perimeter of the sector $OMN=90$ cm. Show that the area of the shaded region, $A$ cm$^2$, is given by $A=75r-5r^2$. It is given that $A$ and $r$ vary with time. If the area of the shaded region is increasing at the rate of $10$ cm$^2$ per second, find the rate of change of $r$ when $r=5$ cm.

Accepted Answer

(a) $A=75r-5r^2$ (b) $\frac{\mathrm dr}{\mathrm dt}=\frac{2}{5}\text{ cm s}^{-1}$

Differentiation

Differentiation

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