A rectangular sheet of cardboard 64 cm by 93 cm is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain the largest possible volume?
D) 4

Question

Quizplus · Accepted Answer

Let x be the size of the squares cut from each corner. Then the length of the base of the box will be 93 - 2x, and the width of the base will be 64 - 2x. The height of the box will be x. Therefore, the volume of the box will be:
V = x(93 - 2x)(64 - 2x)
Expanding this expression and simplifying, we get:
V = -4x^3 + 314x^2 - 5952x
To find the value of x that maximizes the volume, we need to find the critical points of V. Taking the derivative of V with respect to x, we get:
V' = -12x^2 + 628x - 5952
Setting this expression equal to zero and solving for x, we get:
x = 13/3 or x = 48
Since x must be less than half the width and length of the cardboard sheet, we have x ≤ min{32, 46.5/3}. Therefore, the only critical point in the feasible region is x = 13/3.

To confirm that this critical point is a maximum, we need to check that V''(13/3) < 0. Differentiating V' with respect to x, we get:
V'' = -24x + 628
Substituting x = 13/3, we get:
V''(13/3) = 4/3 > 0
Therefore, x = 13/3 is a minimum point, and x = 48 is a maximum point but not feasible. Hence, the optimal size for the squares that should be cut is approximately 4.33333. Therefore, the closest choice is (B), 11ead0bc_8cf0_ed5b_99a0_df04747bad31_TB6988_11.

A Rectangular Sheet of Cardboard 64 Cm by 93 Cm