A movie screen on a wall is 20 feet high and 10 feet above the floor. What is the maximum viewing angle (in radians) that can be achieved by positioning yourself at the optimal distance from the wall?

Step-by-step solution

Let x be the distance of the observer from the wall. Let be the angle of elevation to the bottom of the screen and be the angle of elevation to the top of the screen. The height of the bottom of the screen is 10 feet and the height of the top of the screen is 10 + 20 = 30 feet. From the geometry of the setup, we have = 10 x and = 30 x . The viewing angle is = - . = ( - ) = - 1 + Substituting the values, we get: = 30 x - 10 x 1 + ( 30 x ) ( 10 x ) = 20 x x^2 + 300 x^2 = 20x x^2 + 300 To maximize , we must maximize s