Polynomial problem

  • Thread starter mouseman
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  • #1
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.
 

Answers and Replies

  • #2
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".
 
  • #3
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?
 
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  • #4
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?
 
  • #5
mouseman
This is from a pre-calculus book in a chapter before one titled "Finding factors and zeros of polynomials." (i.e. polynomial division)
 
  • #6
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)
 
  • #7
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 .
 
  • #8
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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