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?

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- Oct 22nd 2007, 03:40 PMMr_Greenvolume of a box
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? - Oct 22nd 2007, 03:51 PMJonboy
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. - Oct 22nd 2007, 04:28 PMMr_Green
is there a way to solve this using derivatives?

i am getting x= 2.06325, so the dimensions would be:

13.8735, 5.8735, and 2.06325

is this correct?? - Oct 22nd 2007, 06:57 PMtopsquark
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