Given that the solid obtained by rotating a rectangle about one of its side is a cylinder. If the perimeter of a rectangle is 48 ~cm and the volume of the cylinder formed by rotating it is maximum, then the dimensions of that rectangle is

Step-by-step solution

Let l and b be length and width of the rectangle. The perimeter is 48 ~cm . 2(l+b)=48 l+b=24 ...(i) Volume of the cylinder form by rotation is v= l^2 b v= l^2(24-l)=24 l^2- l^3 d v d l =48 l-3 l^2 ...(ii) For maxima or minima : d v d l =0 48 l-3 l^2=0 3 (16 l-l^2 )=0 l=0,16 Now, d^2 v d l^2 =48 -6 l At l=0 d^2 v d l^2 =48 >0 So l=0 is point of minima : At l=16 d v^2 d l^2 =48 -96 =-48 < 0 l=16 is point of manima. So, volume is maximum when l=16 . From eq. (i) 16+b=24 b=24-16=8 So, dimension of rectangle is 8,16 .