Watching a Ferris wheel An observer stands 20 m from the hot tom of a Ferris wheel on a line that is perpendicular to the 1 the wheel, with her eves at the level of the bottom of the wheel. The wheel revolves at a rate of ? rad/min and the observer's line of Sight with a specific seat on the Ferris wheel makes a?n angle ?? with the horizontal (see figure). At what time during a full revo? lution is ?? changing most rapidly?
Solution 43E Step 1: Consider that an observer stands 20 m from the bottom of the Ferris wheel on a line that is perpendicular to the face of the wheel. Consider the line with observer sight and a seat makes angle and the line length is L m. Now, consider the diameter of the wheel is r m. Also, consider that the radius of the wheel is r m. Now, the revolving rate is So, the height of the seat from the ground is: Here t is the time in minute. Draw the wheel accordingly Step 2 Now, the height is also described as sin = h L h = Lsin So, compare with Lsin = r + r sin 2t Find the derivative as L cos d = r(2)cos (2t) dt Also, from the diagram you get cos = 20 L L = 20 cos d Put this intoL cos dt = r(2)cos (2t) 20 d cos os dt= r(2)cos (2t) r(2)cos (2t) dt = 20 d rcos (2t) dt = 10 So, the change of is expressed as d= rcos (2t) dt 10