Sec.6.5 - Where to Sit at the Movies

Applied Project in Sec.6.5, Calculus by Stewart
Chinese version 電影院要坐哪?

A movie theater has a screen that is positioned $3$m off the floor and is $10$m high. The first row of seats is placed $3$m from the screen and the rows are set $1$m apart. The floor of the seating area is inclined at an angle of $\alpha=20^{\circ}$ above the horizontal and the distance up the incline that you sit is $x$. The theater has $21$ rows of seats, so $0 \leq x \leq 20$. Suppose you decide that the best place to sit is in the row where the angle $\theta$ subtended by the screen at your eyes is a maximum. Let’s also suppose that your eye are $1.2$m above the floor.

Question 1:

Show that $\theta = \cos^{-1}\left(\frac{a^2+b^2-100}{2ab}\right)$, where $$ a^2 = (3+x\cos{\alpha})^2+(11.8-x\sin{\alpha})^2 $$ and $$ b^2 = (3+x\cos{\alpha})^2+(x\sin{\alpha} - 1.8)^2 $$

Question 2:

Use a graph of $\theta$ as a function of $x$ to estimate the value of $x$ that maximizes $\theta$. In which row should you sit? What is the viewing angle $\theta$ in this row? What is the angle $\theta$ subtended by the screen at your eyes?

Answer:

The maximum of $\theta$ is 0.943356, you should sit at third row.

Question 3:

Use your computer algebra system to differentiate $\theta$ and find a numerical value for the root of the equation $\frac{d\theta}{dx} = 0$. Does this value confirm your result in Problem 2?

Answer:

According to the graph of $y = \frac{d\theta}{dx}$

we found that there is a root at $\approx 2.84556$

Question 4:

Use the graph of $\theta$ to estimate the average value of $\theta$ on the interval $0 \leq x \leq 20$. Then use your CAS to compute the average value. Compare with the maximum and minimum values of $\theta$.

Answer:

average value:

$$ avg = \frac{1}{20}\int_{0}^{20} \theta(x) dx \approx 0.71941 $$

There exists three critical points at $x=0$, 2.84556 and 20, the maximum value is approximately 0.94336, the minimum value is approximately 0.45594