UNIVERSITY OF BRITISH COLUMBIA

Science 1 Math/Phys

Second Midterm

18 November 1999

Time: 50 minutes

Instructors: Mark MacLean, Jess H. Brewer & Domingo Louis-Martinez

1.
Hummingbird Helper   [30 marks] In considering the strategy a hummingbird should follow to maximize the total energy R gained per total time spent foraging, we derived the following expression:

\begin{displaymath}R(t) \; = \; {n \, f(t) \over 1 + n \, t} \; = \; {f(t) \over t + 1/n} \; , \end{displaymath}

where n was the number of (identical) patches, t is the time spent in each patch and f(t) is the total energy gained from one patch as a function of the time spent in that patch. In class we postulated a specific form for the function F(t) and carried out the optimization procedure on R(t) for that particular function f(t).

Show how to optimize R(t) for a function f(t) whose general shape is known but whose functional form is unknown. If you wish, you may choose to draw two or three differently shaped graphs for possible functions f(t); try to make them biologically reasonable.

Hint: Consider the geometry illustrated below:
 

\begin{figure} \epsfysize 1.667in \begin{center} \mbox{\epsfbox{Hummingbird.ps}} \end{center}\end{figure}

2.
Radiation Effects   [40 marks] The harmful effects of a 2 Gray (200 rad) whole-body dose of gamma rays can be crudely modelled by stating that the affected person's probability ${\cal P}$ of still being alive at a time t after the exposure is approximated by the function

\begin{displaymath}{\cal P}(t) \; = \; e^{-\lambda t} + \lambda t e^{-\lambda t} . \end{displaymath}

(This function is, of course, invalid for very large t.)
(a)
Sketch the function ${\displaystyle {\cal P}(t)}$.
(b)
Explain in words what is represented by the function ${\displaystyle {\cal Q}(t) \equiv - {d{\cal P} \over dt} . }$
(c)
Sketch the function ${\displaystyle {\cal Q}(t)}$.
(d)
If the most probable time of death (in this model) is 10 years after the exposure, what is the value of  $\lambda$?

3.
Swinging Rod   [30 marks]
 
Figure: - A thin, uniform rod of mass m = 1 kg and length L = 1 m is attached to a fixed wall bracket at A by a frictionless pin through its end. The rod is initially at rest in an upright vertical position ( $\theta_o = 0$). It is then given a slight ``nudge'' so that it begins to fall, as shown.
\begin{figure} \epsfysize 2.0in \begin{center} \mbox{\epsfbox{rod-swing.eps}} \end{center} \end{figure}


 
At the instant when the rod is oriented vertically downward ( $\theta = \pi$),
what is the force exerted on the rod by the pin?


Jess H. Brewer
2000-02-25