A sheet of area 40m^2 is used to make an open tank of square base then find the dimensions of the square if its volume is at maximum
Answers
Explanation:
Square base of side = 2 × (10)^½ meters
Height = (80/3) / (8 × (10/3)^½) meters
Explanation:
We should note that the surface area of the sheet is equal to the total surface area of the open tank formed.
Let the tank be of base sides x and height h, the surface area of the open tank is given by :
SA = x² + 4(x × h)
= x² + 4xh
We have :
x² + 4xh = 40
We make h the subject of the equation:
4xh = 40 - x²
h = (40 - x²) / 4x
Volume of the tank = base Area × h
Substituting we have :
Volume = x² × (40 - x²)/4x
Canceling the x we have : x(40 - x²)/4
V = (40x - x³)/4
At maximum volume :
dV/dx = 0
Given that the volume is a quotient we will apply quotient rule in differentiation.
Given f/g :
d/x (f/g) = {f'g - fg'} / g²
We apply this in getting the derivative of the volume :
= {4(40 - 3x²) - (40 - x³) × 0}/4²
(160 - 12x²)/16 = 0
160 - 12x² = 0
12x² = 160
x² = 160/12
x² = 40/3
x = 2 × (10/3)^½ meters
h = (40 - 40/3) / (4 × 2 × (10/3)^½
h = (80/3) / 8 × (10/3)^½ meters