12 spheres of the same size are made from melting a solid cylinder of 1 cm diameter and 2 cm height. find the diameter of each sphere.

The answer is: The diameter is 4 centimeters.

The explanation  is shown below:

1. The volume of the cylinder is:

Where is the radius () and is the height ().

Then:

2. The total volume of the 12 spheres is:

3. The volume of the cylinder and the total volume of all the 12 spheres, are equal, therefore:

4. Now, you must solve for the radius:

5. The diameter is:

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for