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I think I got it but I just want to double check. In case I got it wrong, if you could explain your steps that would be awesome :)
I think I got it but I just want to double check. In case I got it wrong, if you could explain your steps that would be awesome :)
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Semi vertical angle of cone = arc tan (2/3)
Let the radius of cylinder be r and height h
From 2-D perspective of the figure, we see a triangle with a rectangle in it.
By some trigonometry, we get,
r / (9-h) = 6 / 9 (Equating tan of semi-vertical angle)
3r = 18 - 2h
Volume of cylinder = πr²h
Replacing h = (18-3r) / 2
V = πr²(18-3r) / 2
Differentiate Volume with respect to radius
dV / dr = (36πr - 9πr²) / 2
For maxima/minima, equate it to zero
(36πr - 9πr²) / 2 = 0
This gives us r = 4cm
Differentiate again
d²V/dr² = 36π - 18πr
Putting r = 4
d²V/dr² = -36π < 0
Hence, r = 4 corresponds to maxima.
r = 4 cm
h = 3 cm
Happy to help...
Let the radius of cylinder be r and height h
From 2-D perspective of the figure, we see a triangle with a rectangle in it.
By some trigonometry, we get,
r / (9-h) = 6 / 9 (Equating tan of semi-vertical angle)
3r = 18 - 2h
Volume of cylinder = πr²h
Replacing h = (18-3r) / 2
V = πr²(18-3r) / 2
Differentiate Volume with respect to radius
dV / dr = (36πr - 9πr²) / 2
For maxima/minima, equate it to zero
(36πr - 9πr²) / 2 = 0
This gives us r = 4cm
Differentiate again
d²V/dr² = 36π - 18πr
Putting r = 4
d²V/dr² = -36π < 0
Hence, r = 4 corresponds to maxima.
r = 4 cm
h = 3 cm
Happy to help...