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JAMB Mathematics 2023 Paper

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Question : 7

Total: 80

The area A of a circle is increasing at a constant rate of 1.5 cm

{}^{2}s^{-1}

. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm

{}^{2}

.

0.200 cms ${}^{-1}$
0.798 cms ${}^{-1}$
0.300 cms ${}^{-1}$
0.299 cms ${}^{-1}$

Solution:

Area of a circle (A) = $\pi r^{2}$

Given

\frac{dA}{dt} = 1.5\,\text{cm}^{2}\,\text{s}^{-1}

\frac{dr}{dt}

= ?

A = 2cm

{}^{2}

Now

2 =

\pi r^{2}

= r

{}^{2} = \frac{2}{\pi}

r =

\sqrt{\frac{2}{\pi}}

cm = 0.798cm

\frac{dr}{dt} = \frac{dA}{dt} \times \frac{dr}{dt}

\frac{dA}{dr} = 2\pi r

(differentiating A =

\pi r^{2})

\frac{dr}{dA} = \frac{1}{2\pi r}

\frac{dr}{dt} = 1.5 \times \frac{1}{2 \times \pi \times 0.798} = 1.5 \times 0.199

\frac{dr}{dt} = 0.299\,\text{cm}\,\text{s}^{-1}

(to 3 s.f)

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