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Question
Find the maximum volume of a right circular cylinder if the sum of the radius and the height of the given cylinder is 6?
Options:
A .  28 π 
B .  32 π 
C .  64 π 
D .  40 π 
Answer: Option B
:
B
Conventional method
Volume of a cylinder, V = πr2h ( r = radius and h = height)
Given that r+h=6. Hence, h = 6-r
Hence, V = πr2(6-r)
V = 6πr2 - πr3
For maximizing volume, we need to differentiate V with respect to r and equate it to 0.
dvdr = 0
dvdr = 12pr - 3πr2= 0
Hence, r = 4
R + h is given as 6, hence h = 2
V = πr2h = π×42×2 = 32π
Alternate Method:
Volume of a cylinder = πr2h ( r =radius and h= height)
Given that r+h=6
To maximize πr2h, we need to maximize r2h which happens when r2 = h1,
r = 4 and h =2
Maximum volume = π×42×2 = 32π
Points to remember
1. If a+b=constant, ab will be maximum when a=b
2. If ab=constant, the minimum value of a+b will be obtained at a=b
3. If a+b+c is a constant, then am. bn .cp is maximum when am = bn = cp (the above question is an example of this)

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