The volume in cubic feet of a box can be expressed V(x) = x^3 - 6x^2 + 8x, 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.

Question

The volume in cubic feet of a box can be expressed V(x) = x^3 - 6x^2 + 8x, 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.

anonymous · Answer

What's the problem? Factoring, graphing, finding the x-intercepts, finding a domain, or finding maximum volume? ("All of the above" is not an answer, heh)

anonymous · Answer

$$V(x) = x^3 - 6x^2 + 8x$$
$$V(x) = x(x^2-6x+8)$$
$$=x(x-2)(x-4)$$,

anonymous · Answer

i guess if one of the factors is supposed to be $2-x$ you would have to write 
$$V(x)=-x(x-4)(2-x)$$

anonymous · Answer

here is a nice graph finding the max is a calculus problem http://www.wolframalpha.com/input/?i=x^3+-+6x^2+%2B+8x++++domain+0..2

anonymous · Answer

in fact i would say the domain is $(0,1)$ because the sides of the box must be positive

anonymous · Answer

@satellite73: Please please please don't just DO the problem for people asking questions. It gives no incentive to learn.

valpey · Answer

Or more likely:?
$$V(x)=x(4−x)(2−x)$$

anonymous · Answer

i guess that would work too right?

valpey · Answer

Usually in spatial problems we have non-negative dimensions, so it is useful to describe them as such. It makes sense to me that x would be greater than 0 and less than 2.