How do i find how the box is?

Question

The volume of a rectangle box is (X^3 +6x^2 +11x +6) cm^3. The box is (×+3) cm long and (×+2) cm wide. How high is the box?

Answer 1

x³+6x²+11x+6 this is the Volume of the rectangle, which is product of its length, (x+3), whidth (x+2) and height, h that should be found.

You can factorize Volume, that is x³+6x²+11x+6=(x+1)(x+2)(x+3)

h(x+3)(x+2)=(x+1)(x+2)(x+3) (divide both sides by (x+2)(x+3) )

h=x+1.

 

 

Answer 2

Let L(x) = length of box (cm) = x + 3

     W(x) = width of box (cm) = x + 2

     H(x) = height of box (cm) 

 

     V = volume of box (cm³) = L(x)*W(x)*H(x) = x³ + 6*x² + 11*x + 6      

 

    Substitute for L(x) and W(x):

   

    (x + 3)*(x + 2)*H = x³ + 6*x² + 11*x + 6     

 

(x² + 5*x + 6)*H = x³ + 6*x² + 11*x + 6

 

H(x) = height of box = [x³ + 6*x² + 11*x + 6]/((x² + 5*x + 6)     for x ≥ 0

 

     

 

 

Of course I suppose you wish to simplify this ( ha ha) therefore; finding the rational roots (from rational root theorem) of cubic: H = (x + 1)*(x +2)*(x + 3)/[(x + 2)*(x + 3)] = x + 1 (cancel like terms)

Answer 3

Since the volume equals the product of length times width times height, all you have to do is to divide the volume expression by length, and then divide the quotient again by width.  You could use synthetic division (twice) in this instance.