Question

The volume in cubic feet of a box can be expressed as V(x)=x^3-5x^2+6x or as the product of three linear factors with integer coefficients. The width of the box is 2 – x.
a. Factor the polynomial to find linear expressions for the height and the width.
b. Graph the function. Find the x-intercepts. What do they represent?
c. Describe a realistic domain for the function.
d. Find the maximum volume of the box.

Answer 1

Do you know how to factorize a polynomial?

Answer 2

x(2 + -1x)(3 + -1x)

Answer 3

for the height

Answer 4

?
2 + -1x for the width

Answer 5

Your addition signs are redundant. Just write:\[V(x)=x(2-x)(3-x)\]

Answer 6

ok for the height

Answer 7

and the width 2-1x

Answer 8

and the width 2-1x

Answer 9

You know 2-x is the width. Now the length is either x or 3-x. Which one makes sense? Can we decide this with the information we are given

Answer 10

x

Answer 11

dont I just need to find the height and width

Answer 12

I don't see a way to determine this. We can pick 3-x as the length, making x the height. Or we could call 3-x the height and then x is the length. It is arbitrary which one we assign to each dimension since the volume comes out the same regardless of the labels we choose.

Answer 13

I have to head work...good luck!

Answer 14

is this calculus or pre calc?

Answer 15

factors as
\[x(x-3)(x-2)\] no need for the minus sign. reasonable domain would be something like \(0<x<2\) because these are lengths and you can't have a negative length

Answer 16

to graph, i would use this
http://www.wolframalpha.com/input/?i=x^3-5x^2%2B6x+domain+0..2