SOLUTION: A manufacturer wants to create a box that has a height of 6 inches, a length of x inches, and a width that is 4 inches less than twice the length. The box must have a volume great

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Question 1098101: A manufacturer wants to create a box that has a height of 6 inches, a length of x inches, and a width that is 4 inches less than twice the length. The box must have a volume greater than 1000 cubic inches.
what is an inequality that could be used to find all possible lengths (x) of the box.

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Given Info:
height = 6 inches
length = x inches
width = 2*(length) - 4 = (2*x - 4) inches
where x is some positive number

The volume of the box V is found by
Volume of box = (length)*(width)*(height)
V = (x)*(6)*(2x-4)
V = 6x*(2x-4)
V = 6x*(2x)+6x*(-4)
V = 12x^2-24x

We want the volume V to be "greater than 1000 cubic inches" indicating that V > 1000. We can replace V with 12x^2-24x
V > 1000
12x^2-24x > 1000

Now subtract 1000 from both sides
12x^2-24x > 1000
12x^2-24x-1000 > 1000-1000
12x^2-24x-1000 > 0

Use the quadratic formula to find the roots of the equation 12x^2-24x-1000 = 0. The approximate roots are x = -8.183 or x = 10.183

The function f(x) = 12x^2-24x-1000 will be negative on the interval -8.183 < x < 10.183 as this graph shows below
This region is between the two roots (points shown in red)

So the region to the left of x = -8.183 represents when f(x) > 0. Also the region to the right of x = 10.183 is when f(x) > 0.

Keep in mind that x > 0, so it makes no sense to have x = -8.183. We only need to focus on the region to the right of x = 10.183

Therefore, if x > 10.183, then the volume of the box is going to be larger than 1000 cubic inches. If x must be a whole number, then we can say x > 10 or ; just add on a note "x is a whole number" or "x is an integer"