To find the height of the building across the street, we can use trigonometric relationships based on the given angles of elevation and depression from a point on a window 31.0 ft above the street. Here are the steps:

1. Understand the Problem Setup:

   • The observer is at a height of 31.0 ft above the street.
   • Angle of elevation to the top of the building: $49.0^\circ$
   • Angle of depression to the base of the building: $16.0^\circ$

2. Break Down the Problem:

   • Let $d$ be the horizontal distance between the window and the building.
   • Let $h$ be the height of the building.

3. Calculate the Distance $d$ Using the Angle of Depression:

   • The angle of depression to the base of the building from the window is $16.0^\circ$.
   • Using the tangent function:
     $\tan(16.0^\circ) = \frac{31.0}{d}$ Solving for $d$: $d = \frac{31.0}{\tan(16.0^\circ)}$

4. Calculate the Additional Height of the Building Above the Window Using the Angle of Elevation:

   • The angle of elevation to the top of the building is $49.0^\circ$.
   • The additional height $h_{\text{top}}$ from the window to the top of the building is: $h_{\text{top}} = d \cdot \tan(49.0^\circ)$

5. Find the Total Height of the Building:

   • The total height $h$ of the building is the height of the window plus the additional height: $h = 31.0 + h_{\text{top}}$

Let's calculate these values step by step.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the calculation step-by-step so you can solve it manually:

1. Calculate the Distance $d$:

   • Use the formula $d = \frac{31.0}{\tan(16.0^\circ)}$.
   • Find $\tan(16.0^\circ)$ using a calculator and divide 31.0 by this value to find $d$.

2. Calculate the Additional Height $h_{\text{top}}$:

   • Use the formula $h_{\text{top}} = d \cdot \tan(49.0^\circ)$.
   • Multiply the value of $d$ you found earlier by $\tan(49.0^\circ)$ to find $h_{\text{top}}$.

3. Find the Total Height of the Building:

   • Add the window height (31.0 ft) to the additional height $h_{\text{top}}$ to get the total height of the building.

4. Round to the Nearest Whole Number:

   • Round the final result to the nearest whole number for your answer.

If you follow these steps using a calculator or computational tool, you should be able to find the height of the building.

Would you like to ask anything specific or need clarification on any step? Here are some related questions that could help:

1. How do you find the tangent of an angle using a calculator?
2. What is the relationship between angles of elevation and depression?
3. How do trigonometric functions relate to real-world problems like this?
4. What are some practical applications of using trigonometry in architecture?
5. How can you estimate angles of elevation and depression in everyday scenarios?

Tip: Always check the mode of your calculator to ensure it is set to degrees when working with degree angles.