676 CHAPTER 6 The Circular Functions and Their Graphs (a) Plot the average monthly temperature over a two-year period. Let x = 1 correspond to January of the first year. (b) To model the data, determine a function of the form ƒ1x2 = a sin3b1x - d24 + c, where a, b, c, and d are constants. (c) Graph ƒ together with the data on the same coordinate axes. How well does ƒ model the data? (d) Use the sine regression capability of a graphing calculator to find the equation of a sine curve of the form y = a sin1bx + c2 + d that fits these data (over a two-year interval). 105. (Modeling) Maximum Temperatures The maximum afternoon temperature (in °F) in a given city can be modeled by the function t = 60 - 30 cos xp 6 , where t represents the maximum afternoon temperature in month x, with x = 0 representing January, x = 1 representing February, and so on. Find the maximum afternoon temperature, to the nearest degree, for each month. (a) January (b) April (c) May (d) June (e) August (f ) October 106. (Modeling) Average Monthly Temperature The average monthly temperature (in °F) in Chicago, Illinois, is shown in the table. Month °F Month °F Jan 24 July 74 Feb 28 Aug 72 Mar 38 Sept 65 Apr 49 Oct 53 May 59 Nov 40 June 69 Dec 28 Data from National Climatic Data Center. Solve each problem. 103. Viewing Angle to an Object Suppose that a person whose eyes are h1 feet from the ground is standing d feet from an object h2 feet tall, where h2 7h1. Let u be the angle of elevation to the top of the object. See the figure. h2 h1 d u (a) Show that d = 1h2 - h12 cot u. (b) Let h2 = 55 and h1 = 5. Graph d for the interval 0 6u … p 2 . 104. (Modeling) Tides The figure shows a function ƒ that models the tides in feet at Clearwater Beach, Florida, x hours after midnight. (Data from Pentcheff, D., WWW Tide and Current Predictor.) (a) Find the time between high tides. (b) What is the difference in water levels between high tide and low tide? (c) The tides can be modeled by the function ƒ1x2 = 0.6 cos 30.5111x - 2.424 + 2. Estimate the tides, to the nearest hundredth, when x = 10. 0 4 8 1216202428 1 2 3 4 x y Time (in hours) Tides (in feet) (2.4, 2.6) (14.7, 2.6) (27, 2.6) (21, 1.4) (8.7, 1.4)
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