Question

A company wishes to manufacture a box with a volume of 24 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material.

A) 5.2 ft
B) 3.2 ft
C) 2.6 ft
D) 6.4 ft

Answer 1

The width of the box that can be produced using the minimum amount of material, given it is twice as long as it is wide and has a volume of 24 cubic feet, is found to be 2.6 feet.

Step-by-step explanation:

To find the width of the box that can be produced using the minimum amount of material, we must first define the dimensions of the box in terms of width. Let w be the width of the box, then the length would be 2w because the box is twice as long as it is wide, and let h be the height of the box.The volume of the box is given as 24 cubic feet, so:

V = l × w × h = 2w × w × h = 24 ft³

This implies that 2w²h = 24. The surface area of the box, which we want to minimize since the box is open on the top, is given by:

SA = lw + 2lh + 2wh = 2w² + 4wh

We can express h in terms of w using the volume equation, h = ³√(24/2w^2). Now, we can substitute h into the surface area equation to find the function SA(w).

After solving the equations and differentiating to find the value of w that gives the minimum surface area, it is found that the width w corresponding to the minimum surface area is 2.6 feet.

The correct answer is C) 2.6 ft.