A cylinder is such that sum of its height and circumference of its base is 10 m Find - Maths - Application of Derivatives

Question

A cylinder is such that sum of its height and circumference of
its base is 10 m Find the greatest volume of the cylinder

Manbar Singh · Accepted Answer

Let r be the radius of the base of cylinder and h be its height
Now, height of cylinder + circumference of base = 10 m   given⇒h + 2πr = 10⇒h = 10 - 2πr    
1Now, volume of cylinder = πr2h⇒V =  πr2h⇒V =  πr210 - 2πr    
2⇒V = 10 πr2 - 2 π2r3⇒dVdr = 20πr - 6 π2r2Now, for maxima or minima,    dVdr = 0⇒ 20πr - 6 π2r2 = 0⇒2πr10 - 3πr = 0⇒r = 0 rejected   or   10 - 3πr = 0⇒r = 103πNow, d2Vdr2 = 20π - 12π2r    ⇒d2Vdr2r=103π =  20π - 12π2 ×103π = -20π <0So, Volume is maximum at r = 103πPutting the value of r in 2, we getNow, maximum volume = π×103π2 - 10 - 2π × 103π = 100027π m3

A cylinder is such that sum of its height and circumference of its base is 10 m . Find the greatest volume of the cylinder.