Polynomial Problem: Solving for Radius of Gas Tank Volume

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In summary, the conversation is about finding the radius of a gas tank with a volume of 50 cubic meters. The problem involves a right-cylinder and two hemispheres, resulting in a cubic equation that cannot be easily solved algebraically. Solutions using the "cubic formula" or numerical methods such as "Newton's method" are suggested. The conversation also touches on the usefulness of numerical methods in modern mathematics.
  • #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.
 
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  • #2
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
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
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
This is from a pre-calculus book in a chapter before one titled "Finding factors and zeros of polynomials." (i.e. polynomial division)
 
  • #6
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
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
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 want to 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.
 

1. How do I solve for the radius of a gas tank volume using polynomials?

To solve for the radius of a gas tank volume using polynomials, you will need to use the formula V = πr^2h, where V is the volume of the tank, π is pi (approximately 3.14), r is the radius, and h is the height of the tank.

2. What information do I need to find the radius of a gas tank volume?

In order to find the radius of a gas tank volume, you will need to know the volume of the tank and the height of the tank. You may also need to know the value of pi (π) depending on the specific problem.

3. Can I use polynomials to find the radius of any gas tank volume?

Yes, you can use polynomials to find the radius of any gas tank volume as long as you have the necessary information (volume and height) and use the correct formula (V = πr^2h).

4. Is there a specific method for solving polynomial problems related to gas tank volumes?

Yes, to solve for the radius of a gas tank volume using polynomials, you can use the quadratic formula or factor the polynomial into two binomials and solve for the radius that way.

5. Are there any common mistakes to avoid when solving for the radius of a gas tank volume using polynomials?

One common mistake to avoid is not converting the units of measurement to be consistent. For example, if the volume is given in cubic inches, the height should also be in inches before plugging into the formula. Another mistake is using the wrong formula or not properly rearranging the formula to solve for the radius.

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