## An open box is to be made from a 3 ft by 8 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners an

Question

An open box is to be made from a 3 ft by 8 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.

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4 days 2021-10-10T20:23:30+00:00 1 Answer 0

Maximum volume of the box will be 7.41 cubic feet.

Step-by-step explanation:

Open box has been made from a metal sheet measuring 3 ft and 8 ft.

Let four square pieces were removed from the four corners with one side measurement x ft.

Volume of the open box = Length × width × height

Length of the box = (3 – 2x) ft

Width of the box = (8 – 2x) ft

Height of the box = x ft

Volume of the box = (3 – 2x)(8 – 2x)x

V = (24 – 6x – 16x + 4x²)x

V = 24x – 22x² + 4x³

Now we take the derivative of V with respect to x and equate the derivative to zero, V’ = 24 – 44x + 12x²

V’ = 0

12x² – 44x + 24 = 0

3x² – 11x + 6 = 0

3x² – 9x – 2x + 6 = 0

3x(x – 3) – 2(x – 3) = 0

(3x – 2)(x – 3) = 0

(3x – 2) = 0

and (x – 3) = 0

Therefore, x = 3, For x = 0.67

Length of the box = (3 – 2x) = 3 – 1.34

= 1.66 ft

Width of the box = (8 – 2x) = 8 – 1.34

= 6.66 ft

Volume of the box = 0.67 × 1.66 × 6.66

V = 7.41 cubic feet.

Similarly, for x = 3,

Length of the box = (3 – 2\times 3) = -3

which is negative but the length of the box can not be negative.

Therefore, maximum volume of the box will be 7.41 cubic feet.