A man 1.5m tall, observes a bird at the top of a tree 4.5m high. If the bird is 8m away from the man. Calculate the distance between the feet of the man and the base of the tree.

The effective height is the height of the tree minus the height of the man.

$$H_{eff} = 4.5\, \text{m} - 1.5\, \text{m} = 3\, \text{m}$$

Let $x$ be the horizontal distance between the man and the tree. The line of sight to the bird forms the hypotenuse of a right-angled triangle.

$$x^2 + H_{eff}^2 = (\text{distance to bird})^2$$ $$x^2 + (3\, \text{m})^2 = (8\, \text{m})^2$$ $$x^2 + 9\, \text{m}^2 = 64\, \text{m}^2$$

$$x^2 = 64\, \text{m}^2 - 9\, \text{m}^2$$ $$x^2 = 55\, \text{m}^2$$ $$x = \sqrt{55}\, \text{m}$$ $$x \approx 7.416\, \text{m}$$

The distance between the feet of the man and the base of the tree is approximately $\boxed{\text{7.416 m}}$.

2. The angle of elevation of a bird in the air from a hunter standing on the ground is $35^\circ$. If the bird is 81.7m away along the hunter's line of sight, how high is the bird from the ground? Correct to 1 decimal place.

We have the angle of elevation ($\theta = 35^\circ$) and the hypotenuse (line of sight distance, $d = 81.7$ m). We need to find the height ($h$), which is the opposite side.

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\sin(35^\circ) = \frac{h}{81.7\, \text{m}}$$

$$h = 81.7\, \text{m} \times \sin(35^\circ)$$ $$h = 81.7\, \text{m} \times 0.573576$$ $$h \approx 46.868\, \text{m}$$

$$h \approx 46.9\, \text{m}$$

The height of the bird from the ground is approximately $\boxed{\text{46.9 m}}$.

3. The angle of elevation of the top of a fence from a boy 1.2m tall is $57^\circ$. If the boy is 2.4m away from the fence, correct to 2 decimal places, the height of the fence.

Let $H_{eff}$ be the height of the fence above the boy's eye level. We have the angle of elevation ($\theta = 57^\circ$) and the adjacent side (distance to fence, $d = 2.4$ m).

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan(57^\circ) = \frac{H_{eff}}{2.4\, \text{m}}$$ $$H_{eff} = 2.4\, \text{m} \times \tan(57^\circ)$$ $$H_{eff} = 2.4\, \text{m} \times 1.53986$$ $$H_{eff} \approx 3.69566\, \text{m}$$

The total height of the fence ($H_{fence}$) is the height above the boy's eye level plus the boy's height.

$$H_{fence} = H_{eff} + 1.2\, \text{m}$$ $$H_{fence} = 3.69566\, \text{m} + 1.2\, \text{m}$$ $$H_{fence} = 4.89566\, \text{m}$$

$$H_{fence} \approx 4.90\, \text{m}$$

The height of the fence is approximately $\boxed{\text{4.90 m}}$.

4. The angle of elevation of an aircraft from a point on the horizontal ground is $30^\circ$. If the aircraft is 100m above the ground, how far is it from the point?

We have the angle of elevation ($\theta = 30^\circ$) and the opposite side (height of aircraft, $h = 100$ m). We need to find the direct distance from the point to the aircraft (line of sight, $d_{sight}$), which is the hypotenuse.

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\sin(30^\circ) = \frac{100\, \text{m}}{d_{sight}}$$

$$d_{sight} = \frac{100\, \text{m}}{\sin(30^\circ)}$$ $$d_{sight} = \frac{100\, \text{m}}{0.5}$$ $$d_{sight} = 200\, \text{m}$$

The aircraft is $\boxed{\text{200 m}}$ from the point.

5. The angle of elevation of the top of a communication mast from a point is $60^\circ$. If the mast is 120m high, how far is the point A from the foot of the mast? Correct to 1 decimal place.

We have the angle of elevation ($\theta = 60^\circ$) and the opposite side (height of mast, $h = 120$ m). We need to find the horizontal distance ($x$) from point A to the foot of the mast, which is the adjacent side.

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan(60^\circ) = \frac{120\, \text{m}}{x}$$

$$x = \frac{120\, \text{m}}{\tan(60^\circ)}$$ $$x = \frac{120\, \text{m}}{\sqrt{3}}$$ $$x = \frac{120\, \text{m}}{1.73205}$$ $$x \approx 69.282\, \text{m}$$

$$x \approx 69.3\, \text{m}$$

The point A is approximately $\boxed{\text{69.3 m}}$ from the foot of the mast.