SOLUTION: (a) The volume of a box is represented by the function, V(x)=x^3+5x^2+2x-8. If the height of the box is represented by (x+4), determine the possible dimensions(binomials) of the b

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: (a) The volume of a box is represented by the function, V(x)=x^3+5x^2+2x-8. If the height of the box is represented by (x+4), determine the possible dimensions(binomials) of the b      Log On


   

Question 1141800: (a) The volume of a box is represented by the function, V(x)=x^3+5x^2+2x-8.
If the height of the box is represented by (x+4), determine the possible dimensions(binomials) of the box. Is there any restriction on the value of x?
(b) If the volume of the box is 70cm^3 determine possible whole number value(s) for x.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


(a) The volume of the box is length times width times height; you are given that the height is x+4.

(1) Use synthetic division to remove the factor (x+4).

  -4 |  1   5   2  -8
     |     -4  -4   8
     +-----------------
        1   1  -2   0

So

x%5E3%2B5x%5E2%2B2x-8+=+%28x%2B4%29%28x%5E2%2Bx-2%29

(2) Factor the quadratic.

x%5E3%2B5x%5E2%2B2x-8+=+%28x%2B4%29%28x%2B2%29%28x-1%29

ANSWER: The possible dimensions of the box are (x-1), (x+2), and (x+4). Since dimensions of a box must be positive, the restriction is that x has to be greater than 1.

(b) You could use any number of methods to solve the equation

x%5E3%2B5x%5E2%2B2x-8+=+70

However, since we know x > 1 and we are looking for whole number values for x, we can quickly find the solution by logical trial and error.

x=2: (x-1)(x+2)(x+4) = (1)(4)(6) = 24 no...

x=3: (x-1)(x+2)(x+4) = (2)(5)(7) = 70 YES!

And clearly larger values of x will produce volumes greater than 70.

ANSWER: x=3 makes the volume 70