Differentiation?

Question

I only need help with the first part cos i can't figure it out

A right circular cylinder is to be made such that the sum of its radius and height is 6m. Find the max volume of the cylinder.

So all i know is: a rt. circular cylinder is a cylinder whose base is perpendicular to its sides.
r+h =6m
V=pie r sq. h, having 3 variables, i have to find a variable using r+h=6m. i need help finding this variable, i know how to work out the rest.

Answer 1

r + h = 6
V = r^2 pi h
V = r^2 pi (6 - r), 0 < r < 6
Consider voume as a function of radius
dV/dr = ....

Answer 2

volume of cylinder, V = pi r^2 h

given that r + h = 6

h = 6-r

substituting this in V

V = pi r^2(6-r)

= 6 pir^2- pi r^3

differentiate with respect to r

dV/dr = 12 pi r - 3 pi r^2

volume will be maximum when dV/dr = 0

12 pi r - 3 pi r^2 = 0

divide by 3 pi r

4 - r = 0

r = 4 m

h = 6-4 = 2 m

Volume = pi r^2 h

pi (16)2

= 32 pi

= 100.53 cubic meters

Answer 3

V = pi*r^2*h with r+h=6
so V = pir^2(6-r)
dV/dr = pi[ 2r(6-r) -r^2]= pi(-3r^2+12r) = 0 so r= 0 and r= 4
solution r= 4m and h = 2m

Answer 4

h = 6 - r
V = πr²h = πr²(6 - r) = π(6r² - r³)
find 0 < r < 6 such that dV/dr = 0

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