Chapter 6

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta=4 \beta-2 \alpha\) where $$\sin \alpha=k \sin \beta$$ and \(\alpha=59.4^{\circ}\) and \(k=1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow. (For a mathematical explanation of rainbows see Calculus, 5 th Edition, by James Stewart, pages \(288-289 .\) )

Using a Calculator To solve a certain problem, you need to find the sine of 4 rad. Your study partner uses his calculator and tells you that $$\sin 4=0.0697564737$$ On your calculator you get $$\sin 4=-0.7568024953$$ What is wrong? What mistake did your partner make?

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has latitude \(35^{\circ} \mathrm{N}\) and New Orleans, \(30^{\circ} \mathrm{N}\) . Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. [The path of the earth around the sun is actually miles. [The path of the earth around the sun is actually an ellipse, hith the sun at one focus (see Section 11.2\()\) . This ellipse, however, has very small eccentricity, so it is nearly circular.

Nautical Miles Find the distance along an arc on the sur- face of the earth that subtends a central angle of 1 minute (1 minute \(=\frac{1}{60}\) degree). This distance is called a nautical mile. (The radius of the earth is 3960 \(\mathrm{mi.}\) )

Irrigation An irrigation system uses a straight sprinkler pipe 300 \(\mathrm{ft}\) long that pivots around a central point as shown. Due to an obstacle the pipe is allowed to pivot through \(280^{\circ}\) only. Find the area irrigated by this system.

Windshield Wipers The top and bottom ends of a wind- shield wiper blade are 34 in. and 14 in. from the pivot point, respectively. While in operation the wiper sweeps through \(135^{\circ} .\) Find the area swept by the blade.

Winch A winch of radius 2 \(\mathrm{ft}\) is used to lift heavy loads. If the winch makes 8 revolutions every 15 \(\mathrm{s}\) , find the speed at which the load is rising.

Fan A ceiling fan with 16-in. blades rotates at 45 rpm. (a) Find the angular speed of the fan in rad/min. (b) Find the linear speed of the tips of the blades in in./min.

Radial Saw A radial saw has a blade with a 6-in. radius. Suppose that the blade spins at 1000 rpm. (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in ft/s.

Speed at Equator The earth rotates about its axis once every 23 h 56 min 4 s, and the radius of the earth is 3960 mi. Find the linear speed of a point on the equator in mi/h.

Speed of a Car The wheels of a car have radius 11 in. and are rotating at 600 rpm. Find the speed of the car in mi/h.

Truck Wheels A truck with 48-in.-diameter wheels is traveling at 50 mi/h. (a) Find the angular speed of the wheels in rad/min. (b) How many revolutions per minute do the wheels make?

Speed of a Current To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 0.20 m and rotates at 100 rpm, find the speed of the current in m/s.

Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 40 rpm. (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

Conical Cup A conical cup is made from a circular piece of paper with radius 6 \(\mathrm{cm}\) by cutting out a sector and joining the edges as shown. Suppose \(\theta=5 \pi / 3 .\) (a) Find the circumference \(C\) of the opening of the cup. (b) Find the radius \(r\) of the opening of the cup. [Hint: Use \(C=2 \pi r . ]\) (c) Find the height \(h\) of the cup. IHint: Use the Pythagorean Theorem.] (d) Find the volume of the cup.

Different Ways of Measuring Angles The custom of measuring angles using degrees, with \(360^{\circ}\) in a circle, dates back to the ancient Babylonians, who used a number system based on groups of \(60 .\) Another system of measuring angles divides the circle into 400 units, called grads. In this system a right angle is 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disad- vantages of these two systems and the radian system of measuring angles. Which system do you prefer?

Clocks and Angles In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through \(\frac{1}{12}\) of a circle. Through how many radians do the minute and the hour hand move between \(1 : 00 \mathrm{P.M.}\) and \(6 : 45\) P.M. (on the same day)?