Let's solve this problem using trigonometry.

Given:

• A flagstaff of height 3 m is on top of a pole.
• The angles subtended by the pole and the flagstaff to a point on the ground are $45^\circ$ and $15^\circ$, respectively.

Let:

• $h$ be the height of the pole.
• $d$ be the horizontal distance from the point on the ground to the pole.

Step 1: Using the angle subtended by the pole

From the problem, the angle subtended by the pole alone is $45^\circ$.

Using the tangent of the angle: $\tan(45^\circ) = \frac{h}{d}$

Since $\tan(45^\circ) = 1$: $1 = \frac{h}{d} \implies h = d$

Step 2: Using the angle subtended by the flagstaff and pole

The total height from the ground to the top of the flagstaff is $h + 3$ meters.

The angle subtended by the total height is $15^\circ$. So, $\tan(15^\circ) = \frac{h + 3}{d}$

Substituting $d = h$ from Step 1: $\tan(15^\circ) = \frac{h + 3}{h}$

Step 3: Solving for $h$

Using the value of $\tan(15^\circ) \approx 0.2679$: $0.2679 = \frac{h + 3}{h}$

Simplify this equation: $0.2679h = h + 3$

$0.2679h - h = 3$

$-0.7321h = 3$

$h = \frac{3}{-0.7321} \approx 4.1 \, \text{m}$

So, the height of the pole is approximately 4.1 m.

Would you like to see more details, or do you have any other questions? Here are some related questions:

1. How would the height change if the angle subtended by the flagstaff were different?
2. Can you find the horizontal distance from the point on the ground to the pole?
3. What is the effect of increasing the flagstaff's height on the angles subtended?
4. How can we verify the height calculation using different trigonometric functions?
5. What are the assumptions made in this problem regarding measurements and angles?

Tip: When solving trigonometric problems involving angles and heights, always sketch a diagram to visualize the relationships between the elements given in the problem.