Let's address both problems from the image one by one:

Problem 3: Find the height of the tree to the nearest foot.

You are given:

• A right triangle with one angle of 45°, another angle of 75°, and a base of 20 feet.

The tree's height corresponds to the vertical side of the right triangle, opposite the 45° angle.

To find the height ($h$), you can use the tangent function in trigonometry: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Here, the angle is $45^\circ$ and the adjacent side is 20 feet. So: $\tan(45^\circ) = \frac{h}{20}$ Since $\tan(45^\circ) = 1$: $1 = \frac{h}{20} \implies h = 20 \text{ feet}$ Therefore, the height of the tree is 20 feet.

Problem 4: Find the distance $d$ to the ship using triangulation.

You are given:

• A baseline distance of 20 miles between two lookouts.
• Two angles: $49^\circ$ and $38^\circ$ at Lookout A and Lookout B respectively.

This forms a triangle, and you can apply the Law of Sines: $\frac{d}{\sin(\theta_{\text{opposite}})} = \frac{\text{baseline}}{\sin(\text{included angle})}$ First, calculate the third angle of the triangle: $\theta_{\text{included}} = 180^\circ - 49^\circ - 38^\circ = 93^\circ$ Now, apply the Law of Sines: $\frac{d}{\sin(93^\circ)} = \frac{20}{\sin(49^\circ)}$ Since $\sin(93^\circ) \approx 1$, this simplifies to: $d = \frac{20}{\sin(49^\circ)}$

Let me calculate the exact distance.The distance $d$ from the shore to the ship is approximately 26.5 miles.

Would you like any more details or have any questions about the solution?

Here are five related questions to consider:

1. How can you calculate the height of an object using trigonometry with different angles?
2. What is the Law of Sines, and how does it apply in different scenarios?
3. How do different angles affect the accuracy of distance measurements in triangulation?
4. Can the Law of Cosines also be applied to find the distance $d$ in this case?
5. How does the angle of elevation influence the calculation of heights in real-world problems?

Tip: Always ensure that angles in trigonometric formulas are converted to radians when using calculators that require radian input.