Finding radius of cylindrical tank with hemispherical caps

[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]

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?

 

[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)

 

[teddy]

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 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.