An open box with a volume of 1500cm^3 is to be constructed by taking a piece of cardboard of 20 cm by 40 cm, cutting squares of sides length x cm from each corner and folding up the sides. Show that this can be done in two different ways, and find exact dimensions of the box in each case.

Question

I have already found the solution using the quadratic equation, can someone help me find the solution using synthetic division?

Answer 1

L = 40 - 2x
w = 20 -2x

V = 1500 = L w x
1500= (40-2x)(20-2x)x = (800 -120 x + 4 x^2)x
so
4x^3 - 120 x^2 + 800 x -1500 = 0
make a reasonable guess
x = 5 for example
then the box would be 30*10 * 5 = 1500
good, height of five is a solution
so divide by (x-5)

x^3 - 120 x^2 + 800 x -1500
-----------------------------
(x-5)

= 4 x^2-100 x +300

factor that
4 (x^2- 25 x + 75) = 0
I get about 21 which is too big to take off the width and 2.89
check x = 2.89
20 - 5.78 = 14.22
40 - 5.78 = 34.22
v = 14.22*34.22*2.89 = 1406
well, better do it a bit more accurately

Answer 2

Could you explain how to do this with the quadratic equation as well?