Download A movie screen on a wall is 20 feet high and 10 feet

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