Finding two possibilities of depth when volume is given?

[VLP1]

Homework Statement

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

Homework Equations

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

Volume function is given above

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.

 

[LCKurtz]

Homework Statement

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

Homework Equations

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

Volume function is given above

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.

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.

 

[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 textbook 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?

 

[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.

 

[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

 

[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!

 

[J. Doe]

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

 

[HallsofIvy]

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