Suppose a single drop of rain falls from a cloud base of height of *H* km. Suppose that at an above-ground height of *h* metres, it has diameter *d* mm. Suppose it falls alongside a tall glass skyscraper. Suppose the rain-drop-to-skyscraper distance remains a constant *k* metres. Suppose the sun is upon the rain drop unobstructed, at angle a to the ground.

Consider the reflective properties of the drop, the glass, and the human eye. Consider the energies acting upon the rain drop. Consider other relevant factors and/or a few irrelevant ones.

Q1. How does the shape and width of the rain drop change over time?

Q2. From what locations and angles would the rain drop (or any reflection of it) be visible to the human eye?

Q3. What determines the temperature of a particular point within the rain drop? What is the formula?

Q4. Does the raindrop reach the ground?

Q5. At what height does it cease to be a rain drop?

Q6. What is altered by the raindrop’s mathematical/existential journey?

**No question is mandatory - no answer is wrong**

### SAMPLE ANSWERS

#### Sample A

- It stays mostly round, and gets smaller.
- Locations of open eyes looking within the optical range of the observer’s eye, symmetrical to the angle of the sun to the ground, and the glass building wrt to 2k
- The sun. f(k,H,h,…) = g(f(k),f(H),f(h), f(…)) = 42
- Yes.
- 0 + d/2 – x; 0<x<d/2
- Dirt.

#### Sample B

- It vibrates and gets smaller.
- Near windows.
- The average temperature of neighbouring points. sum(temp(points))/sum(points);inclusive
- No.
- 6ft 8 inches.
- An umbrella.

#### Sample C

- It gets smaller over time t.
- Anywhere where view of the rain drop is unobstructed.
- Atomic theory. f(fhHkd) = shit happens!
- No, the ground reaches it.
- 0.
- The ground and a little weed.