Question

question 7 (multiple choice worth 1 points) (05.03 hc) elmon is standing at the top of a store escalator that leads to the ground floor below. the angle of depression from the top of the escalator to the floor is 39.81°, and the escalator is 15.5 feet long. how far is the top of the escalator from the ground floor? round your answer to the nearest foot. 15 feet 38 feet 10 feet 12 feet

Explanation:

Step1: Identify the problem type

This is a trigonometry problem involving the angle of depression, the length of the escalator (hypotenuse), and we need to find the horizontal distance (adjacent side) or vertical distance? Wait, the question is "How far is the top of the escalator from the ground floor?" Wait, no, wait. Wait, the angle of depression is from the top of the escalator to the floor. Wait, the escalator length is 15.5 feet, angle of depression is 39.81 degrees. Wait, maybe it's a right triangle where the escalator is the hypotenuse, the vertical distance (height from top to ground) is opposite side, or horizontal? Wait, no, the angle of depression: the angle between the horizontal line from the top and the line of sight to the ground. So the triangle formed has the escalator as the hypotenuse (15.5 ft), angle of depression 39.81 degrees, so the angle inside the triangle (at the top) is also 39.81 degrees (alternate interior angles). We need to find the vertical distance (opposite side) or horizontal? Wait, the question is "How far is the top of the escalator from the ground floor?" So that's the vertical distance? Wait, no, maybe the horizontal? Wait, no, let's re-read.

Wait, the problem says: "Elton is standing at the top of a store escalator that leads to the ground floor below. The angle of depression from the top of the escalator to the floor is 39.81°, and the escalator is 15.5 feet long. How far is the top of the escalator from the ground floor?" Wait, "from the ground floor" – so the vertical distance? Wait, no, maybe the horizontal? Wait, no, the escalator is the hypotenuse. Let's define:

Let’s denote:

- Hypotenuse (escalator length) = \( c = 15.5 \) ft

- Angle of depression \( \theta = 39.81^\circ \)

The angle of depression is equal to the angle of elevation from the ground to the top, so in the right triangle, the angle at the ground (angle of elevation) is \( 39.81^\circ \). Wait, no, the angle of depression is from the top's horizontal line down to the ground. So the triangle has:

- The vertical side (height from top to ground) = \( h \) (opposite side to the angle of depression)

- The horizontal side (distance from top's horizontal projection to the ground point) = \( x \) (adjacent side)

- Hypotenuse (escalator) = \( 15.5 \) ft

We need to find \( h \) (vertical distance, how far the top is from the ground floor). So using sine: \( \sin(\theta) = \frac{h}{c} \), so \( h = c \times \sin(\theta) \)

Wait, but let's check the options. The options are 15, 38, 10, 12. Wait, 15.5 is the escalator length. Wait, maybe I misread. Wait, maybe the angle is 39.81 degrees, and we need to find the vertical distance. Let's calculate:

\( \theta = 39.81^\circ \)

\( c = 15.5 \) ft

\( h = 15.5 \times \sin(39.81^\circ) \)

First, calculate \( \sin(39.81^\circ) \). Let's convert to radians or use calculator. \( \sin(39.81^\circ) \approx \sin(40^\circ) \approx 0.6428 \), but more accurately, 39.81 degrees:

Using calculator: \( \sin(39.81^\circ) \approx \sin(39.81) \approx 0.6406 \)

Then \( h = 15.5 \times 0.6406 \approx 15.5 \times 0.6406 \approx 9.93 \), which rounds to 10 feet. Wait, but let's check cosine. Wait, maybe it's the horizontal distance? No, the question is "how far is the top of the escalator from the ground floor" – that should be vertical. Wait, but 15.5 * sin(39.81) ≈ 10, which is one of the options (10 feet). Let's verify:

\( \sin(39.81^\circ) \approx 0.6406 \)

\( 15.5 \times 0.6406 = 15.5 \times 0.6406 \approx 9.93 \approx 10 \) feet. So the answer should be 10 feet.

Wait, but let's check the angle again. Maybe I made a mistake. Let's use more precise calculation.

39.81 degrees:

\( \sin(39.81^\circ) = \sin(39 + 0.81^\circ) = \sin(39^\circ) \cos(0.81^\circ) + \cos(39^\circ) \sin(0.81^\circ) \)

\( \sin(39^\circ) \approx 0.6293 \), \( \cos(39^\circ) \approx 0.7771 \)

\( 0.81^\circ = 0.81 \times \pi / 180 \approx 0.01414 \) radians

\( \sin(0.01414) \approx 0.01414 \), \( \cos(0.01414) \approx 1 \)

So \( \sin(39.81^\circ) \approx 0.6293 \times 1 + 0.7771 \times 0.01414 \approx 0.6293 + 0.0110 \approx 0.6403 \), which matches the earlier approximation.

Then \( 15.5 \times 0.6403 \approx 15.5 \times 0.6403 \approx 9.92465 \approx 10 \) feet. So the answer is 10 feet.

Step2: Calculate the vertical distance

Using the sine function in a right triangle, where the hypotenuse is the escalator length (\( c = 15.5 \) ft) and the angle of depression (\( \theta = 39.81^\circ \)) corresponds to the angle in the triangle. The vertical distance (\( h \)) from the top of the escalator to the ground floor is the opposite side to the angle \( \theta \).

The formula for sine is:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Substituting the known values:
\[ \sin(39.81^\circ) = \frac{h}{15.5} \]
Solving for \( h \):
\[ h = 15.5 \times \sin(39.81^\circ) \]
Using a calculator to find \( \sin(39.81^\circ) \approx 0.6406 \):
\[ h \approx 15.5 \times 0.6406 \approx 9.93 \]
Rounding to the nearest foot, \( h \approx 10 \) feet.

Answer:

10 feet (corresponding to the option "10 feet")