Find the dimensions of the largest rectangular box with a square base and open top that can be made from 300cm^2 of metal.

Any help would be great

if the box has dimensions x,x,h

x^2 + 4hx = 300
so, h = (300-x^2)/4x

by "largest" I assume you mean "having the largest volume".

v = x^2 * h = 4x(300-x^2) = 1200x - 4x^3
dv/dx = 1200 - 12x^2

max volume where 1200-x^2 = 0, or x = 10

So, the box is 10x10x5

To find the dimensions of the largest rectangular box with a square base and an open top that can be made from 300cm^2 of metal, we need to understand the constraints and the geometry of the box.

Let's assume the side length of the square base is 'x' centimeters. The height of the box will also be 'x' centimeters since it is a cube. The base area is then x * x = x^2 cm^2.

Given that the total amount of metal available is 300cm^2, the sum of the base area and the area of the four sides (which are rectangles with a length of 'x' and a height of 'h', where 'h' is the height of the box) must be equal to 300cm^2.

So, the equation becomes:
x^2 + 4xh = 300

To find the largest box, we need to maximize the volume. The volume of a rectangular box is given by V = length * width * height, which in this case is x^2 * h.

We can solve the equation x^2 + 4xh = 300 for 'h' and then substitute it back into the volume formula to maximize the volume.

To solve for 'h', we isolate it in the equation x^2 + 4xh = 300:
4xh = 300 - x^2
h = (300 - x^2) / 4x

Now that we have 'h' in terms of 'x', we substitute it back into the volume formula:
V = x^2 * ((300 - x^2) / 4x)
Simplifying this equation further, we get:
V = 75x - (x^3 / 4)

To find the maximum volume, we can differentiate this equation with respect to 'x' and set it equal to zero:

dV/dx = 75 - (3x^2 / 4) = 0

Solving for 'x', we find:
75 = (3x^2 / 4)
100 = 3x^2
x^2 = 100/3
x = √(100/3) ≈ 5.77 cm

Now we have the value of 'x', which is the side length of the square base. Since the height of the box is also 'x' in this case, we have:
Length = Width = x = 5.77 cm
Height = x = 5.77 cm

Therefore, the dimensions of the largest rectangular box with a square base and open top that can be made from 300cm^2 of metal are approximately 5.77 cm x 5.77 cm x 5.77 cm.