In a mall, a shopper rides up an escalator between floors. At the top of the escalator, the shopper turns right and walks 13.33 m to a store. The magnitude of the shopper's displacement from the bottom of the escalator to the store is 16.0 m. The vertical distance between the floors is 5.87 m. At what angle is the escalator inclined above the horizontal?

To determine the angle at which the escalator is inclined above the horizontal, we can use the information provided to set up a right triangle. Here's a step-by-step breakdown:

### Known Information:
- The shopper's total displacement from the bottom of the escalator to the store is \( 16.0 \, \text{m} \).
- The horizontal distance the shopper walks after reaching the top is \( 13.33 \, \text{m} \).
- The vertical distance between floors is \( 5.87 \, \text{m} \).

### Step 1: Define the Right Triangle
From the information provided, we can imagine a right triangle where:
- The vertical distance (height) is \( 5.87 \, \text{m} \).
- The hypotenuse is the escalator itself, which is the displacement minus the horizontal walking distance. So, the length of the escalator forms the hypotenuse of the right triangle.

We need to find the length of the escalator using Pythagoras' theorem.

### Step 2: Length of the Escalator
The horizontal displacement due to the escalator is given by the total displacement minus the distance walked horizontally:
\[
\text{Escalator Length} = \sqrt{\text{Total Displacement}^2 - \text{Horizontal Distance}^2}
\]
\[
\text{Escalator Length} = \sqrt{16.0^2 - 13.33^2}
\]

Let's compute this.The length of the escalator is approximately \( 8.85 \, \text{m} \).

### Step 3: Calculate the Angle of Incline
Now that we have the length of the escalator and the vertical distance, we can calculate the angle \( \theta \) using the sine function:
\[
\sin(\theta) = \frac{\text{Vertical Distance}}{\text{Escalator Length}}
\]
\[
\theta = \arcsin\left(\frac{5.87}{8.85}\right)
\]

Let's compute the angle.The angle at which the escalator is inclined above the horizontal is approximately \( 41.55^\circ \).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the angle change if the vertical distance increased while keeping the same total displacement?
2. What would be the new angle if the escalator length changed but the vertical height remained constant?
3. How can the horizontal distance be calculated if only the angle and total displacement are known?
4. How does the length of the escalator affect the speed of the shopper if they take a constant time to ascend?
5. What would the angle be if the total displacement was less but the vertical distance remained the same?

**Tip:** When solving triangle problems, knowing at least one angle and two sides can give you full control over determining all remaining properties using trigonometric identities.

Learn how to calculate the angle of an escalator in a mall using trigonometric concepts. This problem involves applying Pythagoras' theorem and the sine function to determine the incline angle. Suitable for students in Grades 9-12, the solution includes a step-by-step breakdown.

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