Question

In 1979, a biologist Reto Zach published a study on how crows drop whelks,
a type of mollusk, from a height that minimized the amount of energy spent to break open the
shells. Drop from too low a height, and the bird has to pick the shell up many times before it
breaks. Drop from too high a height, and the bird spends more energy than necessary flying
that high.
Toy Model: Instead of looking at the original crow data, imagine a hypothetical “peanut
hummingbird” that picks up peanuts and drops them until they break open. If the peanut is
dropped from a height of 20 cm, it takes an average of 9 drops before breaking. If the peanut
is dropped from a height of 40 cm, it takes an average of 4 drops before breaking.
Let N be the number of average number drops required to break the peanut. Let h be the
height from which the peanut is dropped. It always takes at least one drop to break the
peanut, so that N − 1 is the number of extra drops.
Let R be the reciprocal of the number of extra drops, R =1/N − 1
. Assume that the graph
(h, R) is a linear relation.
(a) Use the data and the assumption to find the equation relating h and R. Use this to find
the equation relating h and N.
(b) The energy spent to open the peanut is proportional to the distance traveled. Let D be
the average distance traveled to break open a peanut. Write an equation for D in terms
of h and N.
(c) Find the height h the minimizes D.