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PROBLEM 14 : A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible ? (See diagram.) SOLUTION Let variable be the viewing angle and variable x the distance as denoted in the diagram. We seek to write angle as a function of distance x . Introduce angle as in the diagram below. It follows from basic trigonometry that so that (Equation 1) . In a similar fashion so that , or (Equation 2) . Use from Equation 1 to substitute into Equation 2, getting . We wish to MAXIMIZE angle THETA given in this equation. Differentiate this equation, getting =0, so that , 30 x2 + 3000 = 10 x2 + 9000 , 20 x2 = 6000 , x2 = 300 , for . But since variable x measures distance and . If (These are well-known values from basic trigonometry.) radians or degrees . See the adjoining sign chart for . If ft. ft. then degrees is the largest possible viewing angle. radians , then