Polynomial functions and Box volume: Consider a box with dimensions 3cm x 5xm x 11cm....

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nemo123

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Consider a box with dimensions 3cm x 5xm x 11cm. If all of its dimensions are increased by x cm, what values of x will give a box with a volume between 300cm3 and 900cm3?
I hope everyone can help me out with this problem. I have been taking the Advanced Function grade 12 math course so that I can go back to college. So far, I have come up with the following inequality:
300<=(x+3)(x+5)(x+11)<=900
I have tried split this in 2 simpler inequalities
a. 300<=(x+3)(x+5)(x+11)
b. (x+3)(x+5)(x+11) <= 900
When I expanded the terms, I got stuck. With (a), I have reached this far
0 <= x^3 + 19x^2 + 103x - 135
My plan is to find the 1st x - intercept and then conduct the synthetic division following by an interval chart/ table. Nevertheless, I still can't find an x-intercept.
Please help me out. Thank you.
 
Hi nemo. Your work looks good, so far. Those two cubic polynomials do not factor nicely. The exact solutions you seek come from the cubic formula, but that's complicated work to do by hand (each of your solutions involve a square root located inside of a cube root).

You'll need to approximate the solutions in decimal form. Are you allowed to use technology for that (eg: graphing calculator)? Otherwise, have you learned any root-finding algorithms (eg: Newton's Method )? :)
[imath]\;[/imath]
 
nemo123 said:
Consider a box with dimensions 3cm x 5xm x 11cm. If all of its dimensions are increased by x cm, what values of x will give a box with a volume between 300cm3 and 900cm3?
I hope everyone can help me out with this problem. I have been taking the Advanced Function grade 12 math course so that I can go back to college. So far, I have come up with the following inequality:
300<=(x+3)(x+5)(x+11)<=900
I have tried split this in 2 simpler inequalities
a. 300<=(x+3)(x+5)(x+11)
b. (x+3)(x+5)(x+11) <= 900
When I expanded the terms, I got stuck. With (a), I have reached this far
0 <= x^3 + 19x^2 + 103x - 135
My plan is to find the 1st x - intercept and then conduct the synthetic division following by an interval chart/ table. Nevertheless, I still can't find an x-intercept.
Please help me out. Thank you.
Click to expand...
You seem to be expecting integer roots. Did it specify that x is an integer?
Where did the problem come from e.g. what course?
I suspect that they're not asking you to find the exact interval that satisfies the inequalities but rather a "good enough" one.
 
Last edited:
Otis said:
Hi nemo. Your work looks good, so far. Those two cubic polynomials do not factor nicely. The exact solutions you seek come from the cubic formula, but that's complicated work to do by hand (each of your solutions involve a square root located inside of a cube root).

You'll need to approximate the solutions in decimal form. Are you allowed to use technology for that (eg: graphing calculator)? Otherwise, have you learned any root-finding algorithms (eg: Newton's Method )? :)
[imath]\;[/imath]
Click to expand...
Thanks so much Otis. I have tried the cubic formula and was not a very pleasant experience doing calculating. lol. Nevertheless, I have tried your another suggestion which is the Newton's method, and it works perfect! In a short time, I have come up with the 1st x-intercept which matched perfectly with my Casio function calculation. I am allowed to use Desmos, so I did utilize the graph calculator as you had suggested. Thanks so much for the quick response. You've made my day !
 
BigBeachBanana said:
You seem to be expecting integer roots. Did it specify that x is an integer?
Where did the problem come from e.g. what course?
I suspect that they're not asking you to find the exact interval that satisfies the inequalities but rather a "good enough" one.
Click to expand...
Thanks a lot BigBeachBanana.
I had no clue where to start honestly. I watched couple videos on Youtube and figured out the cubic function, then I got stuck with the inequality. I wanted to find the x-intercept so that I could make an interval chart to find out at what "values" of x will the given box volume falls between the given range. Like you have said, I thin they want a good enough answers rather than exact ones. The problem came from a course called Advanced Function MHF4U which is a prerequisite for a computer programming program I want to be in.
 
You can get exact answers here:

(x+3)(x+5)(x+11)=300 - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.
www.wolframalpha.com www.wolframalpha.com
 
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