Question 990465: The volume V (in cubic feet) of a rectangular room can be modeled by V(x)=2x^3−19x^2+24x, where x is the length (in feet) of the room. Factor the function. Use your factorization to determine the values of x for which the model makes sense.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The function V(x) = 2x^3 - 19x^2 + 24x can be factored as:
x(2x-3)(x-8)
Since volume must be greater than zero, the function only makes sense for positive values.
Taking the first factor, we must have x>0, so only positive values of x are allowed.
Also, the product of the other two factors must be greater than 0
The product is positive if both factors are positive, so we must have x-8 > 0 -> x > 8 AND 2x-3 > 0 -> x > 3/2.
This gives x > 8
The product is also positive if both factors are negative, so we must have x < 8 AND x < 3/2
This gives x < 3/2
So the values for which the function makes sense are
x>0 AND x<3/2 OR x>8
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