If 40 square feet is sheet metal are to be used in the construction of an open tank with a - Maths - Application of Derivatives

Question

If 40 square feet is sheet metal are to be used in the construction
of an open tank with a square base find the dimension so that the
capacity is greatest possible

Manbar Singh · Accepted Answer

Let the length and breadth of the open tank be x units
Let the height of the tank be y units
Now, surface area of open tank = 40 sq feet⇒x2 + 4xy = 40⇒y = 40-x24x    
1Now, volume = x2y⇒V = x2 × 40-x24x⇒V = 40x - x34Now, dVdx = 40-3x24For maxima or minima, we have       dVdx = 0⇒ 40-3x24 = 0⇒x = 403Now, d2Vdx2 = -6x4 = -3x2Now, d2Vdx2x=40/3 = -32 ×  403 < 0So, volume is maximum at x = 403Now, y = 103So, length = breadth = 403 unitsheight = 103 units

If 40 square feet is sheet metal are to be used in the construction of an open tank with a square base find the dimension so that the capacity is greatest possible