Simple function question (Function to represent radius)

[nukeman]

Homework Statement

Ok here is the question, and I will give the answer below. I am not really clear on how the instructor got this answer. So, if anyone could explain the answer a little, I would REALLY appreciate it!

Question: A spherical balloon with radius r centimeters has volume given by the formula V(r) = 4/3 pi r^3

Find a function that represents the amount of air required to inflate the balloon from a radius of r centimeters to a radius of r + 1 centimeters.

ANSWER: f(r) = 4/3 pi (r+1)^3 -4/3 pi r^3 = 4/3 pi (3r^2 + 3r + 1)

I am not sure why and how this is the answer. ?

Homework Equations

The Attempt at a Solution

 

[Tomer]

Alright...
You know that the amount of air (volume units) in a balloon with radius r is given by the formula for the volume you've written, that is, V₁ = $\frac{4}{3}\pi r^{3}$.
Now, the volume (=the volume the air would take) of a balloon with radius r+1 is given by the same formula, only r+1 instead of r: V₂ = $\frac{4}{3}\pi (r+1)^{3}$.

What is then the difference between the two quantities? Well, for that you have to calculate the difference surprisingly!

Diff_V = V₂ - V₁

And that's what you've written.