Find the mmaximum volume of cylinder, generated by rotating a rectangle of perimeter 48 about one of its side - Maths - Application of Derivatives

Question

Find the mmaximum volume of cylinder, generated by rotating a
rectangle of perimeter 48 about one of its side

Manbar Singh · Accepted Answer

Let the length of the rectangle = x unitsLet the breadth of the rectangle = y unitsNow, perimeter of rectangle = 48 units⇒2length + breadth = 48⇒x + y = 24⇒y = 24 - x   
1Suppose the rectangle is rotated about the breadth
Now, radius of base of cylinder = r = x2Height of cylinder = h = yNow, volume of cylinder = πr2h⇒V = π×x2424-x⇒V = π424x2 - x3⇒dVdx = π448x - 3x2For maxima or minima    dVdx = 0⇒π448x - 3x2 = 0⇒48x - 3x2 = 0⇒3x16-x = 0⇒x = 16  or  x = 0  rejectedd2Vdx2 = π448-6x d2Vdx2x=16 = π448-6×16 = -12π < 0So, volume is maximum at x = 16Maximum volume = π424162 - 163 = π4×2048 = 512π cubic units

Find the mmaximum volume of cylinder, generated by rotating a rectangle of perimeter 48 about one of its side.