An open rectangular box with square base is to be made from 48 square feet of material. what dimensions will result in a box with the largest possible volume?

Sidelengths of the box will be:

x = 4 ft, h = 2 ft.

Step-by-step explanation:

As the open rectangular box is to be made from a sheet that is 48 square feet, the surface area of this box will be at most 48 square feet.

mathematically, we can write the above information as:

the surface area of any rectangular box can be found by adding the areas of each of its face. In our case:

now we can combine our two surface area equation

Now we're asked to find the dimension so that the volume is largest!

we first need to find the volume:

since to find the maximum volume we'll have to differentiate the above equation, but before that we need to have only ONE variable at each side of the equation.

We need to replace 'h'. To do that we'll use our derived equation for surface area.

Now we can substitute this 'h' in 'V'

Now we can find the dimension at which the volume is maximum by using:

this the dimension of sidelength 'x' of the rectangular box such that the volume is maximum.

we can use this value of x to find the height as well.

at these dimensions the maximum possible volume of the box will be!