"Sally has 6 chocolate bars. She wants to give each of her 2 friends 3 chocolate bars."
We recognize this as a situation that involves division. Sally will divide her 6 chocolate bars equally among her 2 friends, giving each friend 3 chocolate bars with none left over, or \(\small\mathtt{6 \div 2 = 3}\), or \(\small\mathtt{\frac{6}{2} = 3}\). Sharing and caring. And chocolate.
But the situation below also involves simple division. It's just that this fact is concealed by a nightmarish stew of words and symbols. Here we are trying to learn how to mathematically quantify the concept of "intensity":
"Over a sphere centered around the point source, the equation [for power, \(\small\mathtt{P}\)] becomes \(\small\mathtt{P = |I| \cdot A_{surf} = |I| \cdot 4\pi r^2}\), where \(\small\mathtt{I}\) is the intensity at the surface of the sphere, and \(\small\mathtt{r}\) is the radius of the sphere. Solving for \(\small\mathtt{I}\) gives \(\small \mathtt{|I| = \frac{P}{A_{surf}} = \frac{P}{4\pi r^2}}\)."
Sally Is the Power, and Chocolate Is the Point
Let's imagine Sally is a power source--giving off chocolate power! She's the red dot at the center of the sphere shown above. And the amount of chocolate power she has is 6, because she has 6 chocolate bars. Instead of sharing her chocolate power with 2 friends, she spreads it out to an infinite number of friends in all directions, represented by the other red points (and all the points I didn't draw). These are all the points on the surface of the sphere.
Because there are an infinite number of points on the surface of a sphere, we don't count them, we measure them in some way. And the measurement we use is an area measurement, \(\small\mathtt{4 \cdot \pi \cdot r^2}\). This tells us the area covered by all of the points on the surface of a sphere, so long as we know the radius, \(\small\mathtt{r}\), of the sphere. Intensity, then, is just Power divided by the surface area of a sphere: \[\small\mathtt{I = \frac{P}{4\pi r^2}}\]
The radius of the sphere matters because when it's larger the sphere is bigger and therefore has a larger measure of the number of points that cover its surface. So, as the power gets spread out across the surface of a larger sphere, the intensity of the power decreases. This is the same as Sally having more friends to share chocolate with.
Both of these situations can be represented similarly as functions, too.