You are planning to make an open-top rectangular box from a 10 by 18 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides.
What are the dimensions of the box of largest volume you can make?
Draw a figure.
Originally we have a rectangle with the dimensions 10 cm x 18 cm.
Whenever you make the box, you are taking two congruent squares off each side. This causes two equal lengths, let's call them x, to be subtracted from the original length and width.
So now we have the lengths: 18 - 2x
And the widths: 10 - 2x
The height would be x, as you will see from the figure.
So the volume is l x w x h => (18 - 2x)(10 - 2x)(x)
To find the largest dimensions, you find the x maximum of the cubic equation, and fill that in for x.
Yes, there is a way to solve this using derivatives and yes, you have the correct answer. (Though if you use Calculus you should be able to show that the answer for x is $\displaystyle \frac{14 - \sqrt{61}}{3}$.)
You can also estimate this by graphing the cubic function and using a graphing calculator (or program) to find the value of x at the relative maximum point on the function.
-Dan