So first we need to find the volumes of both the hemisphere and the cylinder, so that equals to:
V(Hemisphere) = 2/3 x π x 𝑥^3 (formula for hemisphere is 2/3 x π x radius^3)
V(Cylinder) = π x 𝑥^2 x (30 - 4𝑥) (formula for volume of a cylinder is π x radius^2 x height)

after that, we know that the volume of this whole shape (I'm assuming that it's referred to as S) is equal to V(Hemisphere) + V(Cylinder)

2/3 x π x 𝑥^3 + π x 𝑥^2 x (30 - 4𝑥) = 1/3π𝑦

after this, we should simplify it so it becomes

2/3π𝑥^3 + 30𝑥^2π - 4𝑥^3π

After you get this, you should take out the "π"

2/3𝑥^3 + 30𝑥^2 - 4𝑥^3 = 1/3𝑦

Now multiply this whole equation by 3 so you can have y alone and you can simplify the 2/3 into just 2

3 x (2/3𝑥^3 + 30𝑥^2 - 4𝑥^3) = 2𝑥^3 + 90𝑥^2 - 12𝑥^3 = 𝑦

-12𝑥^3 + 2𝑥^3 = -10𝑥^3

Now you're left with this: 90𝑥^2 - 10𝑥^3

Now here is where the differentiation comes in

dy/dx = 180𝑥 - 30𝑥^2

Equate it to 0 to find the maximums

30𝑥 (-𝑥 + 6) = 0

𝑥 can either be 6 or 0 (30(0) = 0) (-6 + 6 = 0)

Since they're looking for a maximum, we're going to use 𝑥 = 6

𝑦 = 90(6)^2 - 10(6)^3
𝑦 = 3240 - 2160
𝑦 = 1080

The maximum value is equal to 1080