To solve this problem, we need to calculate the height of the building based on the information given. Here's a step-by-step approach:

Step 1: Define the problem

Let:

• $h$ be the height of the building (in feet).
• $x$ be the horizontal distance from the first observation point to the base of the building (in feet).

We are given:

• First angle of elevation: $32.4^\circ$
• Second angle of elevation: $25.8^\circ$
• The distance between the two observation points: $56.8 \, \text{feet}$.

Step 2: Use trigonometry

From the first observation point: $\tan(32.4^\circ) = \frac{h}{x}$ So: $h = x \cdot \tan(32.4^\circ)$

From the second observation point (56.8 feet farther back): $\tan(25.8^\circ) = \frac{h}{x + 56.8}$ So: $h = (x + 56.8) \cdot \tan(25.8^\circ)$

Step 3: Equate the two expressions for $h$

$x \cdot \tan(32.4^\circ) = (x + 56.8) \cdot \tan(25.8^\circ)$

Step 4: Solve for $x$

Expand and rearrange: $x \cdot \tan(32.4^\circ) = x \cdot \tan(25.8^\circ) + 56.8 \cdot \tan(25.8^\circ)$ $x \cdot (\tan(32.4^\circ) - \tan(25.8^\circ)) = 56.8 \cdot \tan(25.8^\circ)$ $x = \frac{56.8 \cdot \tan(25.8^\circ)}{\tan(32.4^\circ) - \tan(25.8^\circ)}$

Step 5: Calculate $x$

Using a calculator:

• $\tan(32.4^\circ) \approx 0.634181$
• $\tan(25.8^\circ) \approx 0.484321$

Substitute: $x = \frac{56.8 \cdot 0.484321}{0.634181 - 0.484321}$ $x \approx \frac{27.521 \, \text{feet}}{0.14986}$ $x \approx 183.6213 \, \text{feet}$

Step 6: Calculate $h$

From $h = x \cdot \tan(32.4^\circ)$: $h = 183.6213 \cdot 0.634181$ $h \approx 116.4192 \, \text{feet}$

Final Answer:

The height of the building is approximately: $\boxed{116.4192 \, \text{feet}}$

Would you like a breakdown of any step or further clarification?

Related Questions:

1. How is the tangent function defined in trigonometry?
2. Can you use the same method if the angles and distances change?
3. What would happen if the surveyor moved closer instead of farther away?
4. Can this problem be solved graphically or using software?
5. How does the angle of elevation affect the height calculation?

Tip:

Always double-check the angle measurements to ensure they are in degrees when using a calculator, as some calculators default to radians.