Math 125-103 Fall 2012 Quiz 23, Oct. 26 Carter Name A hot air balloon rising straight up from a level field is tracked by a range finder 150 meters from the liftoff point. At the moment that the range finder’s elevation angle is π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment? Solution. • Let θ denote the angle of inclination. • Let y denote the vertical displacement. • Given the lift-off point of the balloon is 150 meters from the range finder. • Given • Find dθ dt |θ=π/4 = 014 rad/min. dy dt |θ=π/4 . Evidently from the figure y = 150 tan (θ). Thus 150 meters dy dθ = 150 sec2 (θ) . dt dt At θ = π/4, we obtain: dy = 150 sec2 (π/4)0.14 = 150 · 2 · 0.14 = 300 · 0.14 = 3 · 14 = 42. dt The units are Meters per minute.