The volumes of two spheres differ by 336pi cubic meters and the lengths of their radii differ by 6 meters. Find the length of the radius of the smaller sphere. (Round your answer to the nearest hundredth)
To find the length of the radius of the smaller sphere, we can set up an equation using the formula for the volume of a sphere.
The formula for the volume of a sphere is given by V = (4/3) * pi * r^3, where V is the volume and r is the radius.
Let's assume the radius of the smaller sphere is r1, and the radius of the larger sphere is r2 = r1 + 6 (since the lengths of their radii differ by 6 meters).
We can then set up the equation using the volume difference:
(4/3) * pi * r2^3 - (4/3) * pi * r1^3 = 336pi
Simplifying the equation, we get:
4/3 * pi * (r1 + 6)^3 - 4/3 * pi * r1^3 = 336pi
To solve for r1, we'll isolate it:
4/3 * pi * (r1 + 6)^3 - 4/3 * pi * r1^3 = 336pi
(4/3 * pi * (r1 + 6)^3) = 4/3 * pi * r1^3 + 336pi
(r1 + 6)^3 = r1^3 + 336
Expanding (r1 + 6)^3, we get:
r1^3 + 3 * r1^2 * 6 + 3 * r1 * 6^2 + 6^3 = r1^3 + 336
Simplifying and canceling out the common terms, we have:
3 * r1^2 * 6 + 3 * r1 * 6^2 + 6^3 = 336
Now, we can solve this equation to find the value of r1.
4/3 pi (r+6)^3 = 4/3 pi r^3 + 336pi
4(r+6)^3 = 4r^3 + 1008
now just solve for r