$\dfrac{12 - h}{r} = \dfrac{12}{6}$
$12 - h = 2r$
$ h = 12 - 2r$
For maximum volume of cylinder:
$V = \pi r^2h$
$V = \pi r^2(12 - 2r)$
$V = 2\pi(6r^2 - r^3)$
$\dfrac{dV}{dr} = 2\pi(12r -3r^2) = 0$
$12r -3r^2 = 0$
$r = 4 \, \text{ m}$ Part 1: [ B ]
$h = 12 - 2(4)$
$h = 4 \, \text{ m}$
$V_{max} = \pi(4^2)(4)$
$V_{max} = 201.062 \, \text{ m}^3$ Part 2: [ D ]
For maximum lateral surface area:
$A_L = 2\pi rh$
$A_L = 2\pi r(12 – 2r)$
$A_L = 4\pi(6r – r^2)$
$\dfrac{dA_L}{dr} = 4\pi(6 - 2r) = 0$
$6 - 2r = 0$
$r = 3 \, \text{ m}$
$h = 12 - 2(3)$
$h = 6 \, \text{ m}$ Part 3: [ C ]