A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the seating area is inclined at an angle of a = 20° above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats, so 0 < x< 60. Suppose you decide that the best place to sit is in the row where the angle 0 subtended by the screen at your eyes is a maximum. Let's also suppose that your eyes are 4 ft above the floor, as shown in the figure. 1. Show that a? + b2 - 625 arccos 2ab where a? = (9 + x cos a)² + (31 – x sin a)² and b? = (9 + x cos a)² + (x sin a – 6) 2. Use a graph of 0 as a function of x to estimate the value of x that maximizes 0. In which row should you sit? What is the viewing angle 0 in this row?
A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the seating area is inclined at an angle of a = 20° above the horizontal and the distance up the incline that you sit is x. The theater has 21 rows of seats, so 0 < x< 60. Suppose you decide that the best place to sit is in the row where the angle 0 subtended by the screen at your eyes is a maximum. Let's also suppose that your eyes are 4 ft above the floor, as shown in the figure. 1. Show that a? + b2 - 625 arccos 2ab where a? = (9 + x cos a)² + (31 – x sin a)² and b? = (9 + x cos a)² + (x sin a – 6) 2. Use a graph of 0 as a function of x to estimate the value of x that maximizes 0. In which row should you sit? What is the viewing angle 0 in this row?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Transcribed Image Text:A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first
row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the
seating area is inclined at an angle of a = 20° above the horizontal and the distance up the
incline that you sit is x. The theater has 21 rows of seats, so 0 < x< 60. Suppose you decide
that the best place to sit is in the row where the angle 0 subtended by the screen at your eyes
is a maximum. Let's also suppose that your eyes are 4 ft above the floor, as shown in the
figure.
1. Show that
a² + b2 – 625
arccos
2ab
where
a? = (9 + x cos a)² + (31 – x sin a)²
and
b? = (9 + x cos a)² + (x sin a – 6)²
2. Use a graph of 0 as a function of x to estimate the value of x that maximizes 0. In which
row should you sit? What is the viewing angle 0 in this row?
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