Out for a walk in Chicago, Morgan measures the angle of elevation to the distant

Willis Tower, and gets 3.6 degrees. After walking one km directly toward the building, Morgan

finds that the angle of elevation has increased to 4.2 degrees. Use this information to calculate

the height of the Willis Tower, and how far Morgan is from it now.

To solve this problem, we can use trigonometry, specifically the tangent function, to find the height of the Willis Tower and the distance from Morgan to the base of the building.

### Step-by-Step Solution:

1. **Define Variables:**

   Let's denote:
   - \( h \) = height of the Willis Tower (in meters).
   - \( d_1 \) = initial distance from Morgan to the base of the Willis Tower (in meters).
   - \( d_2 \) = distance from Morgan to the base of the Willis Tower after walking 1 km (in meters).

   Given:
   - Angle of elevation from the initial point: \( \theta_1 = 3.6^\circ \).
   - Angle of elevation after walking 1 km towards the building: \( \theta_2 = 4.2^\circ \).
   - Distance walked toward the building: \( 1 \) km = \( 1000 \) meters.

2. **Use Tangent for Each Position:**

   The tangent of the angle of elevation from each position is the ratio of the height of the Willis Tower to the horizontal distance from the tower:

   \[
   \tan(\theta_1) = \frac{h}{d_1}
   \]

   \[
   \tan(\theta_2) = \frac{h}{d_2}
   \]

   Since Morgan walked 1000 meters towards the building, we have:

   \[
   d_2 = d_1 - 1000
   \]

3. **Solve for Height (h):**

   Rearranging both equations for \( h \):

   \[
   h = d_1 \cdot \tan(\theta_1)
   \]

   \[
   h = d_2 \cdot \tan(\theta_2) = (d_1 - 1000) \cdot \tan(\theta_2)
   \]

   Set these two expressions for \( h \) equal to each other:

   \[
   d_1 \cdot \tan(\theta_1) = (d_1 - 1000) \cdot \tan(\theta_2)
   \]

   Solve for \( d_1 \):

   \[
   d_1 \cdot \tan(3.6^\circ) = (d_1 - 1000) \cdot \tan(4.2^\circ)
   \]

   Let's solve this equation for \( d_1 \), and then use it to find the height \( h \).It seems there was a mistake in solving for the distances, as the values obtained are negative, which doesn't make sense in the context of this problem. Let's carefully re-evaluate the equation:

Given:
\[
d_1 \cdot \tan(3.6^\circ) = (d_1 - 1000) \cdot \tan(4.2^\circ)
\]

We'll rearrange the equation again to ensure we correctly solve for \(d_1\).

### Corrected Step-by-Step:

1. Distribute and reorganize:
\[
d_1 \cdot \tan(3.6^\circ) = d_1 \cdot \tan(4.2^\circ) - 1000 \cdot \tan(4.2^\circ)
\]

2. Solve for \(d_1\):
\[
d_1 \cdot (\tan(3.6^\circ) - \tan(4.2^\circ)) = -1000 \cdot \tan(4.2^\circ)
\]

3. Now, solve for \(d_1\):
\[
d_1 = \frac{-1000 \cdot \tan(4.2^\circ)}{(\tan(3.6^\circ) - \tan(4.2^\circ))}
\]

I'll recalculate this to obtain the correct values.After recalculating, we find the correct values:

1. **Initial Distance from Willis Tower ( \( d_1 \) ):** \( 6980.06 \) meters.
2. **Height of Willis Tower ( \( h \) ):** \( 439.15 \) meters.
3. **Current Distance from Willis Tower ( \( d_2 \) ):** \( 5980.06 \) meters.

This means Morgan was initially 6980.06 meters away from the Willis Tower, and after walking 1 km (1000 meters) toward the building, she is now 5980.06 meters away. The calculated height of the Willis Tower is approximately 439.15 meters.

### Do you want more details or have any questions?

Here are 5 related questions you might consider:
1. How would the calculations change if Morgan walked a different distance toward the tower?
2. What if the angles of elevation were measured from different heights above ground level?
3. How does the curvature of the Earth affect such long-distance measurements?
4. Can you calculate the distance using other trigonometric functions like sine or cosine?
5. What tools could Morgan use to measure angles more accurately?

**Tip:** When solving problems involving angles of elevation, always ensure your angle units (degrees or radians) are correctly used in calculations.

Out for a walk in Chicago, Morgan measures the angle of elevation to the distant Willis Tower, and gets 3.6 degrees. After walking one km directly toward the building, Morgan finds that the angle of elevation has increased to 4.2 degrees. Use this information to calculate the height of the Willis Tower, and how far Morgan is from it now.

Learn how to calculate the height of the Willis Tower using angles of elevation and trigonometric functions. This example walks through calculating the height based on Morgan's measurements at 3.6 and 4.2 degrees, including distance traveled. Suitable for students in Grades 10-12.

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