Polynomial Equation-Application

[notsosmartman]

First of all this is from a book I own. It is not for any class or assignment. I am just curios on how you would go about solving such a problem.

The radii of four different-sized ball bearings differ by 1.0mm in radius from one size to the next. If the volume of the largest equals of the other three combined, find the radii.

Any help is greatly appreciated!

 

[Yuqing]

Well, the first thing to do when approaching these problems is to define an attribute as your variable and this is the part which many people find the most difficult. For this question, it might make sense to call the radius of the largest sphere "x".

Now we know that the radii differ by 1mm for each successive sphere, and we also know that the volume of the smaller three sum up to the larger sphere.

Therefore we know that

$$\frac{4}{3}\pi((x-1)^{3} + (x-2)^{3} + (x-3)^{3}) = \frac{4}{3}\pi x^{3}$$

which we can simplify into

$$x^{3} - 9x^{2} + 21x - 18 = 0$$

now there are numerous ways to solve this equation which I will not go into.