# Finding two possibilities of depth when volume is given?

1. Mar 7, 2015

### VLP1

1. The problem statement, all variables and given/known data

Problem - If the volume of the carry-on luggage is 2000 cubic inches, determine two possibilities for its depth. Where necessary, round to the nearest tenth of an inch

2. Relevant equations

V(x) = x(x+10)(30-2x)

Volume function is given above

3. The attempt at a solution

I like to think I learn Math logic well, and I love Math...but this problem has me stumped...something is just not clicking in my brain when I try to start solving this problem.

Now I set up the problem:

2000 = x(x+10)(30-2x)

I then said OK, I need to solve for x and by doing that I will find 1 solution, or maybe both, but I can't figure out how to combine the x's into one and then solve for the 1 remaining x.

I tried finding the zeros of the 3 factors, but then wondered to myself why did I do that, it doesn't seem I need to do that.

I was unable to ask my professor help on this problem because this was the last problem on my homework and I left the last couple of questions for today so I can bring in all my homework tomorrow (which is when it is due anyway) and I didn't think I would run into a mindwreck of a problem like this...not to mention this problem has a chance to be on the test tomorrow!!

Would really appreciate help in solving this problem.

2. Mar 7, 2015

### LCKurtz

Apparently you aren't telling us everything you know about this problem. Anyway, simplify your equation and write it as a cubic equation = 0. Then look for a rational root to reduce it to a quadratic.

3. Mar 7, 2015

### VLP1

The only thing I left out is the beginning format of the V(x) which was:

(x)(x+10)(40-(x+x+10))

The x being depth
The x + 10 being length
The (40 - (x+x+10)) being width(40) minus (depth + length)

OK I compact the problem into the below:

-2x3+30x2-20x2+300x

Now I need a constant to try the Rational Zero Theorem

So I simplify it as follows

x(-2x2+10x+300)

I then quickly jump to find a zero using -10 as a value for x to get zero

I then use Synthetic Division which just brings me right back to the original:

x(x+10)(-2x+30)

And the roots from that is 0, -10, 15

But I am not sure if finding the roots for this problem helps any...

If it is any help, the 2 depth values is 7.8 inches and 10 inches that shows in the back of my text book as the solution...I just don't know how to get these 2 values?

Should I be using another theorem or method to break this problem down?

4. Mar 7, 2015

### LCKurtz

You have to set the volume = 2000 and solve that cubic. You should get two positive values of $x$ which agree with your given answers.

5. Mar 7, 2015

### VLP1

Another thing to note is this problem is at the end of a Chapter dealing with:

The Rational Zero Theorm
The Fundamental Theorem of Algebra
The Linear Factorization Theorem
Descartes Rule

6. Mar 7, 2015

### VLP1

OMG unbelievable.

Firstly, I didn't think about setting it equal to 2000 and then moving the 2000 over to make it equal to 0.

Secondly, What was also messing me up was that I wasn't sure if looking for roots was the way to solve this, not realizing that was the way to solve it and that I had already found one of the answers (10 inches) when I found the zero for one of the simple factors from before.

Thirdly, because I saw the problem asking for 2 possible depths, I was under the assumption that this problem would only result in 2 possible solutions, when in reality it was a couple solutions, some positive numbers and some negative numbers.

Fourthly, it blew my mind how moving the 2000 over allowed me get a constant on the cubic equation...it totally transformed the entire problem as I started solving more and more of it, and I started to grin when I had to use the quadratic formula to get possible solutions with radicals, and one of those gave me the 7.8 inches (The second answer I was looking for)

Thanks for the help!!

7. Nov 13, 2015

### J. Doe

Can anyone break this down with a step by step guide? After getting the roots, I'm not sure how to proceed.

8. Nov 13, 2015

### HallsofIvy

Staff Emeritus
Well, what did you get for the roots? And what, exactly, does the problem ask you to find?