Question

If 6400 cm² of material is available to make a box with a square base and an open top, find the dimensions of the box that give the largest possible volume. What is the maximum value of the volume?

Answer 1

The dimensions of the box that give the largest possible volume are a square base with side length 40 cm and a height of 40 cm. The maximum volume of the box is 64,000 cm³.

Step-by-step explanation:

To find the dimensions of the box that give the largest possible volume, we need to maximize the volume function. Let's say the side length of the square base is x cm. The height of the box would then be (6400 - x^2)/4x. We can then find the maximum volume by taking the derivative and setting it equal to zero. Solving for x gives us x = 40. The dimensions of the box that give the largest possible volume are a square base with side length 40 cm and a height of 40 cm. The maximum volume of the box is 64,000 cm³.