A man stands on a tree 150cm high and sees a boat at an angle of depression of 74°. Find the distance of the boat from the base of the tree.

1. Analyze the question

We are given the height of a tree ($h = 150\text{ cm}$) and the angle of depression from the top of the tree to a boat on the ground ($\theta = 74^\circ$). We need to find the horizontal distance ($d$) from the base of the tree to the boat.

2. Explain the concept

In trigonometry, the angle of depression from an observer to an object is equal to the angle of elevation from the object to the observer (due to alternate angles).

We can visualize this as a right-angled triangle where:

• The height of the tree is the opposite side to the angle of elevation.
• The distance from the base of the tree is the adjacent side.
• We use the tangent ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

3. Show the solution

• Given: • Height (Opposite) = $150\text{ cm}$ • Angle ($\theta$) = $74^\circ$ • Distance (Adjacent) = $d$

• Formula: $$\tan(74^\circ) = \frac{150}{d}$$

• Rearrange to solve for $d$: $$d = \frac{150}{\tan(74^\circ)}$$

• Calculation: Using a calculator, $\tan(74^\circ) \approx 3.4874$ $$d = \frac{150}{3.4874}$$ $$d \approx 43.01\text{ cm}$$

Rounding to the nearest whole number, we get 43 cm.

4. Identify the answer

Comparing our result to the given options: A. 52cm B. 43cm C. 40cm D. 15cm

The calculated value matches option B.

5. Exam Tip

When dealing with angles of depression, always remember that the angle is measured from the horizontal line of sight downwards. If you draw a diagram, the angle of depression is outside the triangle at the top, but it is equal to the angle inside the triangle at the base of the boat. This simple "Z-angle" rule (alternate angles) will save you from making mistakes in your setup!