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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