Question

The town park does an outdoor movie night every Saturday during the summer on a large screen. Kate is sitting 36 feet from the base of the screen, watching a movie with her family. If the angle of elevation from Kate to the top of the screen is 24°, how tall is the movie screen? Round to the nearest whole number if necessary.

Answer 1

To determine the height of the movie screen, we can use the tangent function from trigonometry. The tangent of an angle in a right triangle is equal to the opposite side (height of the screen, \( h \)) over the adjacent side (distance from Kate to the base of the screen, \( d \)).

We have the following information:

- The angle of elevation, \( \theta = 24^\circ \)
- The distance from Kate to the base of the screen, \( d = 36 \) feet

The relationship can be expressed as:
\[
\tan(\theta) = \frac{h}{d}
\]
Substituting the known values:
\[
\tan(24^\circ) = \frac{h}{36}
\]

Next, we can solve for \( h \):
\[
h = 36 \cdot \tan(24^\circ)
\]

Now, we will calculate \( \tan(24^\circ) \) and then find \( h \).

Using a calculator:
\[
\tan(24^\circ) \approx 0.4452
\]
Now substitute this value back into the equation for \( h \):
\[
h = 36 \cdot 0.4452 \approx 16.0312
\]

Rounding to the nearest whole number, the height of the movie screen is approximately:
\[
\boxed{16} \text{ feet}
\]