We can solve this problem using trigonometry, specifically the tangent function.

Step 1: Define Variables

Let:

• $h$ be the height of the tree.
• $d$ be the horizontal distance from the first observation point to the base of the tree.
• The two observation points be 200 feet apart.

From the problem:

• The first observer sees the tree at an angle of $45^\circ$.
• The second observer (200 feet further away) sees the tree at an angle of $30^\circ$.

Step 2: Use the Tangent Function

From the first observation point: $\tan(45^\circ) = \frac{h}{d}$ Since $\tan 45^\circ = 1$, we get: $h = d$

From the second observation point: $\tan(30^\circ) = \frac{h}{d+200}$ Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we get: $\frac{h}{d+200} = \frac{1}{\sqrt{3}}$

Step 3: Solve for $h$

We substitute $h = d$ into the second equation: $\frac{d}{d+200} = \frac{1}{\sqrt{3}}$ Multiply both sides by $d+200$: $d = \frac{d+200}{\sqrt{3}}$ Multiply by $\sqrt{3}$ to eliminate the fraction: $d \sqrt{3} = d + 200$ Rearrange: $d \sqrt{3} - d = 200$ Factor out $d$: $d (\sqrt{3} - 1) = 200$ Solve for $d$: $d = \frac{200}{\sqrt{3} - 1}$

Step 4: Rationalize the Denominator

Multiply numerator and denominator by $\sqrt{3} + 1$: $d = \frac{200 (\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$ Since $(\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2$, we simplify: $d = \frac{200 (\sqrt{3} + 1)}{2}$ $d = 100 (\sqrt{3} + 1)$ Approximating $\sqrt{3} \approx 1.732$: $d \approx 100 (1.732 + 1) = 100 (2.732) = 273.2$

Since $h = d$, we get: $h \approx 273 \text{ feet}$

Final Answer:

The height of the tree is 273 feet.

Would you like further explanation or a diagram? Here are some related questions:

1. How would the answer change if the angles of inclination were different?
2. Can we solve this problem using a different trigonometric approach?
3. How does changing the distance between the two observation points affect the calculation?
4. What happens if the ground is not level between the observers and the tree?
5. How does this problem relate to real-world surveying techniques?

Tip: Always check the reasonableness of your answer by considering the real-world scale of the object being measured!