## Developing a Working Formula for Energy

Equation 5 gives us what seems like a reasonable formula. However, we are not interested in what happens when one crow tries to break one whelk open; we are interested in what happens in general. We can generalize equation 5 by using the data Zach gathered in Table 1, which gave the __average__ number of drops it took to break whelks from particular heights.

To start, let's graph equation 5 for all the different heights that Zach tried, using the steps in exercise 4.

Exercise 4: Answer the following questions:

a) What shape would the graph of *E = HN* be, where *E* is a function of *N*, and *H *is a constant (Hint: consider the shape of *y *= 5*x*, for example)?

b) On a large grid, sketch the graph of energy expended versus number of drops, for a height of 6 metres.

c) Consider where the curves *E* = 10*N* and *E* = 2*N *should be, relative to the curve *E *= 6*N* you drew in part (b). Here's a hint: if a crow is flying higher up for each drop, it will expend more energy for each drop. Sketch these on the same grid.

d) For *H *= 3,4,5,7,8,15, use the logic from part (c) to sketch *E = HN* on the same grid.

Exercise 5: Zoom in on the graph above to help you answer the following questions:

a) At what height is the energy expended minimized, on average?

b) How confident are you with your answer?

An interesting point to note here is that the plot we just created is not quite what we wanted. We are looking for the height at which energy expended is minimized, and so it would make sense to create a plot of energy versus height, not energy versus number of drops. However, this exercise has shown that there are often many different ways to find a solution to a problem.