SECTION 6.4 Graphs of the Sine and Cosine Functions 441 97. Ferris Wheel The function π( ) ( ) = − + h t t 100cos 50 105 represents the height h, in feet, of a seat on a Ferris wheel as a function of time t, where t is measured in seconds. (a) How high does a seat on the Ferris wheel go? (b) How close to the ground does a seat get? (c) If a ride lasts for 5 minutes, how many times will a passenger go around? (d) What is the linear speed of the Ferris wheel in miles per hour? Round to one decimal place. 98. Holding Pattern The function π( ) ( ) = + d t t 50cos 39 60 represents the distance d, in miles, from the airport after t minutes of an airplane asked to fly in a circular holding pattern. (a) What is the plane’s average distance from the airport over one cycle? (b) How long does it take the plane to complete one cycle in the holding pattern? (c) What is the plane’s speed, in miles per hour, while in the holding pattern? (d) If the plane travels 1.8 miles per gallon of fuel, how much fuel is used in one cycle of the holding pattern? 99. Biorhythms In the theory of biorhythms, a sine function of the form ω ( ) ( ) = + P t t 50sin 50 is used to measure the percent P of a person’s potential at time t, where t is measured in days and = t 0 is the day the person is born. Three characteristics are commonly measured: Physical potential: period of 23 days Emotional potential: period of 28 days Intellectual potential: period of 33 days (a) Find ω for each characteristic. (b) Using a graphing utility, graph all three functions on the same screen. (c) Is there a time t when all three characteristics have 100% potential? When is it? (d) Suppose that you are 20 years old today ( ) = t 7305 days . Describe your physical, emotional, and intellectual potential for the next 30 days. Credit: Srdjan Draskovic/Alamy Stock Photo 92. Tunnel Clearance A one-lane highway runs through a tunnel in the shape of one-half a sine curve cycle. The opening is 28 feet wide at road level and is 15 feet tall at its highest point. (a) Find an equation for the sine curve that fits the opening. Place the origin at the left end of the opening. (b) If the road is 14 feet wide with 7-foot shoulders on each side, what is the height of the tunnel at the edge of the road? Source: en.wikipedia.org/wiki/Interstate_Highway_standards and Ohio Revised Code. 93. Modeling Blood Pressure Several research papers use a sinusoidal graph to model blood pressure. Suppose an individual’s blood pressure is modeled by the function π ( ) ( ) = + P t t 20 sin 7 3 100 where the maximum value of P is the systolic pressure, which is the pressure when the heart contracts (beats), the minimum value is the diastolic pressure, and t is time, in seconds. The heart rate is the number of beats per minute. (a) What is the individual’s systolic pressure? (b) What is the individual’s diastolic pressure? (c) What is the individual’s heart rate? 94. Modeling Tides The function below models the water height H, in feet, at a monitoring station in Charleston, South Carolina. π ( ) ( ) = + + H t t 2.91sin 24 149 1.360 2.97 where t is the number of hours after midnight. (a) What is the height of the water at high tide? (b) What is the height of the water at low tide? (c) What is the time between high and low tide? 95. Modeling Average Monthly Temperature The function below models the average monthly temperature T, in °F, for Indianapolis, Indiana. π π ( ) ( ) = − + T x x 23.65sin 6 2 3 51.75 where x is the month ( = = January 1, February 2, etc.). (a) What is the highest average monthly temperature? (b) What is the lowest average monthly temperature? (c) What is the time between the highest and lowest average temperatures? 96. Modeling Hours of Daylight The function below models the number of hours of daylight in Miami, Florida. π ( ) ( ) = − + D x x 1.615sin 2 365 1.39 12.135 where x is the day of the year. (a) How many hours of daylight are there on the longest day? (b) How many hours of daylight are there on the shortest day? (c) What is the time between the longest and shortest days? 15 ft 14 ft 28 ft 1 2 Credit: Jag_cz/Shutterstock
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