Internal surface area

= Internal curved surface area + internal base area + internal area of cover (since it is closed)

= Internal curved surface area + 2 x internal base area

(Since base area is that of a circle, and for a cylinder, the top and bottom are identical circles with equal area)

= (2πrh + 2πr²) cm²

So (2πrh + 2πr²) cm² = 96π cm²

πrh + πr² = 48π

πr²h + πr³ = 48πr

πr²h = 48πr - πr³

= πr(48 - r²)


Since volume of liquid that the container can hold = volume of cylinder,

Volume , V = πr²h cm³

= πr(48 - r²) cm³

(Shown)

V = πr(48 - r²)

V = 48πr - πr³

dV/dr = 48π - 3πr²

At a stationary point,

dV/dr = 0

48π - 3πr² = 0

16 - r² = 0

(4 + r)(4 - r) =0

r = -4 (rejected as radius cannot be negative)

Or

r = 4


Second derivative test :

d²V/dr² = -6πr

When r = 4, d²V/dr² = -6π(4)
= -24π < 0

(Maximum)


When r = 4,

V = π(4)(48 - 4²)
V = 4π(32)
V = 128π

Maximum value of V = 128π

from the info we can say that :

dV/dt = 2
dh/dt = 0.5

dV/dh = dV/dt x dt/dh

= dV/dt ÷ dh/dt

= 2 / 0.5

= 4


But we know that the volume of cylinder's formula is πr²h .

Since radius is fixed at k cm,

Volume, V = πk²h

dV/dh = πk²

So πk² = 4

k² = 4/π

k = √(4/π) or 2/√π
(Since radius has to be positive, we do not consider k = -√(4/π)

Thanks!!