Question

find the dimension of rectangular box of maximum capacity whose surface area is given when
A box is open at the top
B box is closed

Answer 1

Box open : Volume=  (S/6)√(S/3)  Side = √(S/3)   & height = √(S/3) /2

Box closed : Volume = (S/6)√(S/6)  Side = height = √(S/6)

Step-by-step explanation:

Surface Area = S

Rectangular Box has to ve Square on Base

Let say x * x * h  is box

Case 1 - Box Closed

Surface area = 2x² + 4xh = S

=> h = (S - 2x²)/4x

V = x²h

V = x² (S - 2x²)/4x

V = x (S - 2x²)/4

V = xS/4  - x³/2

dV/dx = S/4 - 3x²/2

S/4 - 3x²/2 = 0

=> S/4 = 3x²/2

=> S = 6x²

h =  (6x² - 2x²)/4x  = x

Side = √(S/6)

Volume = x³ = (√(S/6))³ = (S/6)√(S/6)

Case 2 - Box open

Surface area = x² + 4xh = S

=> h = (S - x²)/4x

V = x²h

V = x² (S - x²)/4x

V = x (S - x²)/4

V = xS/4  - x³/4

dV/dx = S/4 - 3x²/4

S/4 - 3x²/4 = 0

=> S/4 = 3x²/4

=> S = 3x²

h =  (3x² - x²)/4x  = x/2

Side = √(S/3)

height - √(S/3) /2

Volume = x²h  = (S/3)√(S/3) /2

Volume=  (S/6)√(S/3)

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