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Polynomial problem

  • Thread starter mouseman
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mouseman

I'm stumped! I'm on this question in my math book that reads something like this:

"A gas tank that is 10 meters in length (end to end) consists of a right-cylinder and is capped at either end by a hemisphere. What is the radius of the tank if the volume is 50 cubic meters?"

Ok i got as far as
4/3[pi]r^2(r + 15/2) = 50

but I can't seem to figure out how to isolate r. I know I'm overlooking something mundane, but can someone give me a hint?

Tanks! Ha ha ha! (get it? Tanks?)

Sorry.
 

HallsofIvy

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You have a cubic equation:
r3+ (15/2)r2- 75/(2[pi])=0.

I don't think there is a simple way to solve that. There is, of course, the "cubic formula" but that's not going to be easy here. You could also use a numerical method like "Newton's method".
 

mouseman

Originally posted by HallsofIvy
You have a cubic equation:
r3+ (15/2)r2- 75/(2[pi])=0.
Ok yeah I got that too.

Originally posted by HallsofIvy
I don't think there is a simple way to solve that. There is, of course, the "cubic formula" but that's not going to be easy here. You could also use a numerical method like "Newton's method".
By "cubic formula" you mean x3 + y3 = (x + y)(x2 - 2xy + y2) or whatever it is?
 
Last edited by a moderator:
65
1
You may need to do the substitution of r=y-a/2 where a is the coeffcient of the x^3 term. Once you do that you get a cubic in y and you can then try to solve by algebraic long division to get the roots.

One question though ..... what level of textbook is this problem from?
 

mouseman

This is from a pre-calculus book in a chapter before one titled "Finding factors and zeros of polynomials." (i.e. polynomial division)
 

petitbear

As far as I can see, you've made an error at the beginning.
Sphere volume: 4/3*Pi*r^3
Cylinder vol.: Pi*r^2*h
The equation should be: 4/3*Pi*r^3+Pi*r^2*(10-2*r)=50
... (errors possible)
r^3-15*r^2+75/Pi=0

After this... the "easy way" is with the cubic formula. The one you'll find useful.(googled)
 
18
0
hi buddy
the answer using calculator is r= 1.1731 meters to four places of decimals.you can do it without using calculator by pen-and-paper iteration(i used calc. for exactly the same thing).
1)start with r=1.
2)use r^2 = 2/15*(75/2pi - r^3) to get new r
3)repeat 2) until convergence occurs(in some cases it may not converge but here it does converges.see any text-book for more info on convergence)

numerical methods are much more useful nowadays than analytic ones due to their (almost) infinite range of application .
 

mouseman

Yeah I used a calculator to find it out too but I was hoping I could do it algebraically...
See I'm just trying to understand the math, I don't wanna end up memorizing it. I just figured with the information in all the previous chapters I could do it without any "advanced" math.
But maybe my search is in vain.
 

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