differentiation

1.
R m: radius
V m³: volume

dR/dt ∝ R
Hence dR/dt = kR where k is a constant

When R = 2, dR/dt = 0.2 :
0.2 = k(2)
k = 0.1
Hence, dR/dt = 0.1R

When R = 3 :
dR/dt = 0.1(3)
dR/dt = 0.3

V = (4/3)πR³
dV/dt = (d/dt)(4/3)πR³
dV/dt = 4πR²(dR/dt)
dV/dt = 4π(3)²(0.3)
dV/dt = 10.8π
dV/dt = 10.8(3.14)
dV/dt = 33.9

Rate at which the volume of the balloon is increasing = 33.9m³/s

dR/dt = λR

Sub. R = 2, dR/dt = 0.2 => λ = 0.2/2 = 0.1

When R = 3, dR/dt = 0.1 * 3 = 0.3 m/s

The radius of a spherical ballon is expanding at 0.3 m/s when the radius is 3m