Polynomial inequality - word problem

Mar 2017
358
3
Massachusetts
Hi,

I hope someone can help. I'm trying to solve for the following question:

Screen Shot 2017-04-13 at 4.29.12 PM.png
<= "less than or equal to"
>= "greater than or equal to"

These are my steps:
1. I know that the volume of a rectangle is l x w x h - therefore the inequality to the equation must be x(50 - 2x)(30 - 2x) <= 4000, where x represents the height, 50 - 2x represents the width, and 30 - 2x represents the length.
2. Find the roots to the inequality: x = 0, x = 25, and x = 15
3. Create intervals for the corresponding roots: x <= 0, 0 <= x <= 15, 15 <= x <= 25, x >= 25
4. Identify values within those intervals, then sub those intervals within the inequality equation. If the outcome is less than or equal to 4000, then consider this a range of the side lengths. These ranges would represent my solution(s) to this question.

The textbook solution 5 < x < 7.19. My solution is wrong so I would appreciate some guidance on where I went wrong in my steps.

Sincerely,
Olivia
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
2. Find the roots to the inequality: x = 0, x = 25, and x = 15
the correct inequality is $x(50 - 2x)(30 - 2x) > 4000$

roots of the inequality are where $x(50-2x)(30-2x) - 4000 = 0$

the values of $5 < x < 7.19$ yield a volume $\ge 4000 \, cm^3$
 

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