Expert: Richard J. Raridon - 8/23/2009

Question
Hi!  I have a question concerning solving inequalities.

"An open box is formed my cutting squares from the corners of a regular piece of cardboard and folding up the flaps.  The figure shows the flat piece of cardboard with the dimensions of 15 inches by 12 inches.

a. What size corner squares should be cut to yield a box with a volume of 125 inches^3?

b. What size corner squares should be cut to yield a box with a volume more than 125 inches^3?

c. What size corner squares should be cut to yield a box with a volume of at most 125 inches^3?"

To begin, I made the dimensions of the corner square to be cut out to be x by x inches.  I then multiplied (15-2x)(12-2x)(x) and set it equal to 125.  But after I multiplied it through, I became stuck.  I understand that for part b, you must set the equation to be more than 125 and for part c, you must set the equation to be less that or equal to 125.

Thanks for your time and help!
Jenny

Answer
I presume that when you multiplied it out and combined terms you got 4x^3-54x^2+180x-125=0.  
Since you can't factor that, one way to solve it is by trial and error.  
Set f(x) = 4x^3-54x^2+180x-125
If you put in a value of x and it is a solution, then f(x) = 0.  
For example, f(1) = 3 which means that x=1 is just slightly too big.  
By trying x = 0.99, 0.98, etc., you can narrow it down.  
b. Obviously, x has to be bigger than the value just found.  However, it can't be bigger than 6 since that would lead to 12-2x to be negative.  
c. any value of x that is less than the value found in a.