R. Zach reported data on the foraging behavior of crows. The crows break whelk shells by carrying them to a rocky area and dropping them onto rocks. Zach measured the probability of a shell breaking in a fall from a height of x m. (a) The energy E that a crow expends on an upward flight is proportional to the total height x of the flight. Which of the following statements are true? Circle all that apply. (b) It seems reasonable that crows would want to minimize the total energy cost of breaking a shell. Since the energy that a crow expends on an upward flight is proportional to the total height of the flight, the total energy expended to break a shell can be modeled by a function: f(x) = xn(x) where n(x) is the average number of drops needed to break a shell falling from a height of x, where x is measured in meters. Find an expression for f′(x) in terms of n(x) and n′(x); write your answer in the box below. (c) Joe Mahaffy proposed the model n = 1 + (a/(x-b)) for the average number of drops needed to break a shell falling from a height of x, for positive parameters a and b where a > b, x > b, and a corresponds to the hardness of the surface below. In this model, does a larger value of a indicate a harder surface or a softer one? Circle one answer, and justify your answer using a derivative computation. A. A larger value of a indicates a harder surface. B. A larger value of a indicates a softer surface.
R. Zach reported data on the foraging behavior of crows. The crows break whelk
shells by carrying them to a rocky area and dropping them onto rocks. Zach measured the
probability of a shell breaking in a fall from a height of x m.
(a) The energy E that a crow expends on an upward flight is proportional to the
total height x of the flight. Which of the following statements are true? Circle all that
apply.
(b) It seems reasonable that crows would want to minimize the total energy cost of
breaking a shell. Since the energy that a crow expends on an upward flight is proportional
to the total height of the flight, the total energy expended to break a shell can be modeled
by a function:
f(x) = xn(x)
where n(x) is the average number of drops needed to break a shell falling from a height
of x, where x is measured in meters. Find an expression for f′(x) in terms of n(x) and
n′(x); write your answer in the box below.
(c) Joe Mahaffy proposed the model
n = 1 + (a/(x-b))
for the average number of drops needed to break a shell falling from a height of x, for
positive parameters a and b where a > b, x > b, and a corresponds to the hardness of the
surface below. In this model, does a larger value of a indicate a harder surface or a softer
one? Circle one answer, and justify your answer using a derivative computation.
A. A larger value of a indicates a harder surface.
B. A larger value of a indicates a softer surface.
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