Homework 8 Exercises on angles and trig functions

Exercise 8.1 . Convert to radians: (a) [Maple Math] , (b) 1 [Maple Math] , (c) a quarter revolution, (d) 10 revolutions.

\vfill

Exercise 8.2: A bicycle wheel has radius [Maple Math] cm. Find the rate in rad/sec at which the wheel is turning if the bicycle is traveling at (a) 10 m/sec, (b) 10 km/hr

\vfill

Exercise 8.3 : A drawbridge whose two spans are each 30 meters long is opened as shown below. Express in terms of [Maple Math] : (a) the height y of the end of a span, (b) the slope of the left span, (c) the distance x between the ends of the spans.

code for diagram

[Maple Plot]

\newpage

Exercise 8.4. Using the diagram show that :


[Maple Math] ,.

code foe diagram

>

[Maple Plot]

\newpage

Exercise 8.5: Sketch the graph of each of the following functions:

code to generate graph paper

(a) [Maple Math] ,

[Maple Plot]

(b) [Maple Math] ,
 [Maple Plot]

\newpage

(c) [Maple Math] ,

[Maple Plot]

(d) [Maple Math] .

[Maple Plot]

\newpage

Exercise 8.6: The mean daily temperature in Fairbanks, Alaska was tabulated over a 30-year period, the average was taken over the 30 values for each day, and the result was found to be very close to the sinusoidal function [Maple Math] , where t is the day of the year. (Here we are NOT assuming 30 days per month.) Sketch the graph of this temperature function, find the maximum and minimum mean daily temperatures, and find the warmest and coolest days of the year.

[Maple Plot]

Acknowledgement: This example is from an article by B. M. Lando and C. A. Lando in Mathematics Teacher ), Sept. 1977.

Exercise 8.7: Studies are made of the variation in population of foxes and rabbits in a certain large forest. The number changes from year to year. Roughly speaking, when there are a lot of rabbits, the foxes will start to increase in number, because they can eat rabbits plentifully; but then this decreases the number of rabbits, causing the foxes to be short of food, and this causes a decrease in the fox population; and, finally, when there are fewer foxes the rabbits will be able to increase in number without being eaten, so we are back to where we started. Thus, we might expect a sinusoidal oscillation in the number of foxes and in the number of rabbits (This is the so-called `` predator-prey '' problem in mathematical biology .)

Suppose that t stands for the number of years since the study began, F stands for the number of foxes, and R stands for the number of rabbits. Suppose that F and R are each a sinusoidal function of t. Use the following table of data (which gives the maximum and minimum population of each) to find a formula for F and for R in terms of t:

code for table

[Maple Math]

\vfill\newpage

Exercise 8.8: Write a formula for each of the following functions of t, assuming that the function is sinusoidal (also suppose that a year consists of 12 months of 30 days each):

(a) the mean low daily temperature, if the coldest is [Maple Math] F on January 19 and the highest low temperature reading is [Maple Math] F on July 19;
 \vfill


(b) the time of sunset each day, if the sun sets at 9 p.m. on June 21 and at 5 p.m. on December 21;
 \vfill


(c) the temperature in a well-insulated building, which varies between [Maple Math] C at 5 p.m. (which is 17:00 on the 24-hour clock) and [Maple Math] C at 5 a.m;
 \vfill


(d) the velocity of a weight attached to the end of a spring , which is extended and released at time [Maple Math] , if the weight reaches its maximum velocity of 10 cm/sec after 0.1 sec;
 \vfill

(e) the height above the ground of a pebble that is picked up in the tread of a bicycle tire at time [Maple Math] , if the tire has radius 40 cm and the bicycle is traveling at 10 m/sec.

 \vfill


table of contents