A box of rectangular base and an open top has a surface area 600 cm2.

Question

If the height of the box is equal to its width, find the dimensions that
give a maximum volume.

Answer 1

Let x be the width and height of the box, and y be the length of the box. Then the surface area of the box is given by xy + 2x^2 = 600.

We want to maximize the volume of the box V = x^2y. We will do this by expressing y in terms of x using the surface area formula, and then finding the maximum of the resulting expression.

From the surface area formula, we have y = (600 - 2x^2) / x. Substituting this expression for y in the volume formula, we get:

V(x) = x^2 * (600 - 2x^2) / x
V(x) = 600x - 2x^3

To find the maximum of V(x), we take its derivative and set it equal to zero:

V'(x) = 600 - 6x^2
0 = 600 - 6x^2
x^2 = 100
x = 10

So the width and height of the box are both 10 cm. Now we can find the length of the box using the surface area formula:

10y + 2(100) = 600
y = 40

The dimensions of the box that give a maximum volume are 10 cm by 10 cm by 40 cm.