We can solve this problem using trigonometry. Let $h$ be the height of the building and $x$ be the distance from the first point of observation to the base of the building.

Step-by-step solution:

1. From the first point, we have the angle of elevation of $35^\circ$. Using the tangent function: $\tan(35^\circ) = \frac{h}{x}$ This gives: $h = x \cdot \tan(35^\circ)$

2. From the second point, which is 150 feet closer, the angle of elevation is $51^\circ$. The distance from this point to the base of the building is $x - 150$. Using the tangent function: $\tan(51^\circ) = \frac{h}{x - 150}$ This gives: $h = (x - 150) \cdot \tan(51^\circ)$

3. Since both expressions equal $h$, we can set them equal to each other: $x \cdot \tan(35^\circ) = (x - 150) \cdot \tan(51^\circ)$

4. Solve for $x$: $x \cdot \tan(35^\circ) = x \cdot \tan(51^\circ) - 150 \cdot \tan(51^\circ)$ $x (\tan(35^\circ) - \tan(51^\circ)) = -150 \cdot \tan(51^\circ)$ $x = \frac{-150 \cdot \tan(51^\circ)}{\tan(35^\circ) - \tan(51^\circ)}$

5. Once $x$ is found, substitute it into the equation $h = x \cdot \tan(35^\circ)$ to find $h$.

Let me compute this. math

Angles in degrees

angle1 = 35 # First angle of elevation angle2 = 51 # Second angle of elevation distance_closer = 150 # Distance closer in feet

Convert angles to radians for computation

tan1 = math.tan(math.radians(angle1)) tan2 = math.tan(math.radians(angle2))

Solve for x

x = (-distance_closer * tan2) / (tan1 - tan2)

Solve for h

h = x * tan1

Round to the nearest hundredth

h_rounded = round(h, 2) h_roundedThe height of the building is 242.58.

Would you like me to go over any details or steps?

Here are 5 related questions you might find helpful:

1. How can tangent functions be used to solve real-world elevation problems?
2. What is the relationship between angles of elevation and trigonometric ratios?
3. How does changing the distance to the building affect the angles of elevation?
4. What are some other practical applications of trigonometry in construction or surveying?
5. How can we verify this calculation using another trigonometric method?

Tip: Always ensure angles are converted to radians when using programming tools for trigonometric calculations!