for a spherical balloon;
volume, V=34πr3...(i)
area, A=4πr2...(ii)
circumference,C=2πr...(iii)
a) given dtdv=−2cm3/min the negative sign to denote reduction in volume with time and V=36πcm3 we have;
dtdv=4πr3dtdrfrom(i)
∴dtdr=4πr21dtdv...(iv)
when V=36π⟹34πr3=36π
r3=43∗36=27
⟹r=3cm
∴dtdr=4π(3)21cm/min=18π−1cm/min
the radius is shrinking at the rate of 18π1cm/min
b) from (iii) r=2πc...(v)
now (v) in (ii) gives;
A=4π(2πc)2=4π24πc2=π1c2...(vi)
∴dCdA=π2C
when V=36πcm3,r=3cm [from part (a)]
and thus C=6πcm
⟹dCdA=π2(6π)(whenc=6π)
=12
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