Viewing angles An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle?

Solution 41E Step 1: Suppose the eye is x feet away from the screen then the angle to the top of the screen is tan = 10 (since angle of elevation is given by tan = opposite si)e 1 x adjacent side = tan ( 1 10) 1 x angle to bottom of the screen is tan = 3 (since angle of elevation is given by tan = opposite si)e 2 x adjacent side 1 3 =2tan ( ) x So the viewing angle from x feet away is given by = -1 2 1 10 1 3 = tan ( x )-tan ( ) x Step 2: In order to get maximum viewing angle,we need to find critical point in order to find critical number we need to make first derivative equal to 0 d dx= 0 d 1 10 1 3 dx( tan ( x )-tan ( ))x0 1 d 10 1 d 3 1+(2)2dx x) 1+(2)2dx x)=0 10 3 x x x2 102 1+(2)2=0 1+(x2) x x2 x3 x +100 x +9=0 ( x2 ) (x2 ) 10 3 (x +100)+ x +9 =0