Oh, the poor cylindrical metal met a melting pot of fate and transformed into a spherical shape! How dramatic! Now, let's find out the radius of the new sphere in terms of x.
To do that, we need to remember that the volume of the cylinder is equal to the volume of the sphere. Volume, volume, what a magical word!
The volume of the cylinder is given by the formula V_cylinder = πr_cylinder^2h_cylinder, where r_cylinder is the base radius and h_cylinder is the height.
So, for our cylinder, V_cylinder = π(2x)^2(9x) = 36πx^3.
Now, let's move on to the sphere. The volume of a sphere is V_sphere = (4/3)Ï€r_sphere^3. Fun fact: when you roll a sphere down a hill, it just keeps on going and going!
Since the metal from the cylinder is used to make the sphere, the volumes are equal: 36Ï€x^3 = (4/3)Ï€r_sphere^3.
Now, we just need to solve for r_sphere. Let's punch some numbers.
36Ï€x^3 = (4/3)Ï€r_sphere^3.
Divide both sides by π to get rid of the pi's. We don't want pizza here!
36x^3 = (4/3)r_sphere^3.
To solve for r_sphere, we simply take the cube root of both sides:
r_sphere = (3(36x^3)/4)^(1/3).
Now, let's simplify this expression, because I know math is already complicated enough.
r_sphere = (108x^3/4)^(1/3).
And that, my friend, is your expression for the radius of the spherical creation out of our melted metal cylinder! Keep on rolling with that math!