91Ó°ÊÓ

MATHEMATICAL CONNECTIONS The Tropic of Cancer is the circle of latitude farthest north of the equator where the Sun can appear directly overhead. It lies \(23.5^{\circ}\) north of the equator, as shown. \begin{tabular}{l|l} Tropic of & North Pole \\ Cancer \\ equator & South Pole \end{tabular} a. Find the circumference of the Tropic of Cancer using 3960 miles as the approximate radius of Earth. b. What is the distance between two points on the Tropic of Cancer that lie directly across from each other?

The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point \(P\) in the diagram equals the degree measure of \(\operatorname{arc} P E\). At what latitude \(\theta\) is the circumference of the circle of latitude at \(P\) half the distance around the equator?

PROBLEM SOLVING You measure the angle of elevation from the ground to the top of a building as 32°. When you move 50 meters closer to the building, the angle of elevation is 53°. What is the height of the building?

MAKING AN ARGUMENT Your friend claims it is possible to draw a right triangle so the values of the cosine function of the acute angles are equal. Is your friend correct? Explain your reasoning.

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\).. \(f(x)=\frac{1}{3} \cos \pi x\); translation 1 unit down, followed by a reflection in the line \(y=-1\)

The average wind speed \(s\) (in miles per hour) in the Boston Harbor can be approximated by $$ s=3.38 \sin \frac{\pi}{180}(t+3)+11.6 $$ where \(t\) is the time in days and \(t=0\) represents January 1. Use a graphing calculator to graph the function. On which days of the year is the average wind speed 10 miles per hour? Explain your reasoning.

Find the average rate of change of each function over the interval \(0

You are riding a Ferris wheel that turns for 180 seconds. Your height \(h\) (in feet) above the ground at any time \(t\) (in seconds) can be modeled by the equation $$ h=85 \sin \frac{\pi}{20}(t-10)+90 $$ a. Graph the function. b. How many cycles does the Ferris wheel make in 180 seconds? c. What are your maximum and minimum heights?

Write an expression in terms of the integer \(n\) that represents all the \(x\)-intercepts of the graph of the function \(y=\cos 2 x\). Justify your answer.

Your friend states that for functions of the form \(y=a \sin b x\) and \(y=a \cos b x\), the values of \(a\) and \(b\) affect the \(x\)-intercepts of the graph of the function. Is your friend correct? Explain.