Question

What is the maximum volume of a box?

There is a wooden bog with an open top. The base is square, X cm by X cm, and the height is Y cm. It is to be constructed using 192 cm^2 of wood. Find the maximum volume of such a box.

Answer 1

See explanation.

Explanation:

First we have to calculatte the area of the box:

`A=4xy+x^2`

The area is `192cm^2`, so we have an equation which combines `x` and `y`:

`4xy+x^2=192`

`4xy=192-x^2`

`y=(192-x^2)/(4x)`

Now we can calculate the volume:

`V=x^2y`

`V=x^2*(192-x^2)/(4x)`

`V=(192x-x^3)/4=48x-x^3/4`

To maximize the volume we have to calculate the derivative:

`V'(x)=48-(3x^2)/4`

The maximum volume is reached for `x` where `V'(x)=0`

`48-(3x^2)/4=0`

`192-3x^2=0`

`64-x^2=0`

graph{64-x^2 [-150.2, 150, -75.1, 75.1]}

From the graph of `V'(x)` we see that the maximum volume is for `x=8`

The maximum volume is: `V=48*8-8^3/4=384-128=256`

Answer: The maximum volume of the box is `V_{max}=256cm^3`