A box in the shape of a rectangular prism has a volume of 90 cubic feet. The box has a length of x feet, a width of (x-1) feet and a height of (x+10) feet. Find the dimensions of the box.

Answer 1

x(x-1)(x+10) = 90

x=-9 is one root but x must be > 0, one of the other 2 roots is >0

-9(-10)(1)=90

x(x^2+ 9x -10) = 90

x^3 +9x^2 -10x -90= 0

divide by x+9 to get x^2-10 =0, x^2 =10, x= sqr10 = about 3.16

x intercepts = 3.16, 0, and -3.16

only 3.16>0 for the value of one side of the prism

one side of the prison = x = 3.16 feet

another side = x-1 = 2.16 feet

last side = x+10 = 13.16 feet

or if you prefer exact dimensions, they're sqr10 by sqr10 -1 by sqr10 +10

3.16(2.16)(13.16) = 89.825.

sqr10 is slightly larger than 3.16

sqr10 =3.16227766...

how did you find the root x=-9?

trial and error, try simple integers. when you try 3 and 4 the sign switches, so you know one root is between them, same with -3 and -4. That's two roots which are sqr10 and -sqr10

but that first root came from noticing the graph is an upside down U shape followed by a U shape

the 3+ root is a local minimum, the -3^- root is a local max. to follow the upside down U shape there had to be another root less than -3. so try -5, -6, -7, -8 and then -9 works