Question

A square piece of cardboard of side 10 inches is taken and four equal square pieces are removed at the corners, such that the side of this square piece is also an integer value. The sides are then turned up to form an open box. Then the maximum volume such a box can have is:

• A. 72 cubic inches
• B. 24.074 cubic inches
• C. 2000/27 cubic inches
• D. 64 cubic inches

Hint:

Use trial values or calculus to maximize open box volume. Check integers around critical point.

Answer 1

The Correct Option is A

Let the square removed from each corner be of side $x$ (must be integer) Then box dimensions become: Length = Width = $10 - 2x$ Height = $x$ \[ \text{Volume} = x(10 - 2x)^2 \] Try integer values: - $x = 1$: $1 \cdot 8^2 = 64$ - $x = 2$: $2 \cdot 6^2 = 72$ - $x = 3$: $3 \cdot 4^2 = 48$ - $x = 4$: $4 \cdot 2^2 = 16$ - $x = 5$: $5 \cdot 0^2 = 0$ Max volume = $\boxed{72}$ cubic inches at $x = 2$