A cylindrical tank without a top lid is being manufactured to hold a fixed volume of 125 cubic cm. The minimum surface area required to construct this tank is ............square cm

Step-by-step solution

Let r be the radius and h be the height of the cylindrical tank. Given the volume V = r^2 h = 125 h = 125 r^2 The surface area of the tank without a top lid is S = 2 r h + r^2 Substituting the value of h , we get: S = 2 r ( 125 r^2 ) + r^2 = 250 r + r^2 To find the minimum surface area, we differentiate S with respect to r and equate it to zero: dS dr = - 250 r^2 + 2 r = 0 2 r = 250 r^2 r^3 = 125 r = 5 Also, d^2S dr^2 = 500 r^3 + 2 > 0 for r = 5 , which confirms a minimum. The minimum surface area is: S = 250 5 + (