Question

Air is escaping from a spherical balloon at the rate of 2𝑐𝑚3 per minute. How fast is the radius shrinking when the volume is 36𝜋 𝑐𝑚3 ?

Find the rate of change of the area 𝐴, of the circle with respect to its circumference C, 𝑖. 𝑒 𝑑𝐴 𝑑�

Answer 1

$Volume={4 \over 3} \pi r^3$

Since volume is 36𝜋 𝑐𝑚³

Then, $36\pi= {4 \over 3}\pi r^3$

$r^3=27$

r=3cm

Since, $V={4 \over 3} \pi r^3$ and ${dv \over dt}=$ 2cm³ per mins

${dv \over dt}=4\pi r^2{dr \over dt}$

$2=4\pi(3^2) {dr \over dt}$

${dr \over dt}={1 \over 18\pi}cm/mins$

$Area(A)=\pi r^2$

$Circumference (C)= 2\pi r$

$r={C \over 2\pi}$

$\therefore A= \pi({C \over 2\pi})^2$

$A= {C^2 \over 4\pi}$

${dA \over dC}={C \over 2\pi}units$