91Ó°ÊÓ

In Exercises 31-50, use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$ \cot \theta=1,0 \leq \theta \leq 2 \pi $$

In Exercises 43-52, find the distance a point travels along a circle \(s\), over a time \(t\), given the angular speed \(\omega\), and radius of the circle \(r\). Round to three significant digits. $$ r=3.2 \mathrm{ft}, \omega=\frac{\pi \mathrm{rad}}{4 \mathrm{sec}}, t=3 \mathrm{~min} $$

In Exercises 31-50, use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$ \tan \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi $$

Planets. The Earth rotates every 24 hours (actually 23 hours, 56 minutes, and 4 seconds) and has a diameter of 7926 miles. If you're standing on the equator, how fast are you traveling in miles per hour (how fast is the Earth spinning)? Compute this using 24 hours and then with 23 hours, 56 minutes, 4 seconds as time of rotation.

Bicycle High Gear. If a bicycle has 26 -inch diameter wheels, the front chain drive has a radius of 4 inches, and the back drive has a radius of 1 inch, how far does the bicycle travel for every one rotation of the cranks (pedals)?

Music. Some people still have their phonograph collections and play the records on turntables. A phonograph record is a vinyl disc that rotates on the turntable. If a 12 -inch-diameter record rotates at \(33 \frac{1}{3}\) revolutions per minute, what is the linear speed of a point on the outer edge in inches per minute?

Bicycle. How fast is a bicyclist traveling in miles per hour if his tires are 27 inches in diameter and his angular speed is \(5 \pi\) radians per second?

For Exercises 67 and 68, refer to the following: To achieve similar weightlessness as that on NASA's centrifuge, ride the Gravitron at a carnival or fair. The Gravitron has a diameter of 14 meters, and in the first 20 seconds it achieves zero gravity and the floor drops. Gravitron. If the Gravitron rotates 24 times per minute, find the linear speed of the people riding it in meters per second.

Clock. What is the angular speed of a point on the end of a 10-centimeter second hand given in radians per second?

In Exercises 71 and 72, explain the mistake that is made. If the radius of a set of tires on a car is 15 inches and the tires rotate \(180^{\circ}\) per second, how fast is the car traveling (linear speed) in miles per hour? Solution: Write the formula for linear speed. \(\quad v=r \omega\) Let \(r=15\) inches and \(\omega=180^{\circ}\) per second. \(\quad v=(15\) in. \()\left(180^{\circ} / \mathrm{sec}\right)\) Simplify. \(\quad v=2700 \mathrm{in} . / \mathrm{sec}\) Let 1 mile \(=5280\) feet \(=63,360\) inches and \(\quad v=\left(\frac{2700 \cdot 3600}{63,360}\right) \mathrm{mph}\) 1 hour \(=3600\) seconds. Simplify. \(v \approx 153.4 \mathrm{mph}\) This is incorrect. The correct answer is approximately \(2.7\) miles per hour. What mistake was made?