A building is 24m away from a mast. The angle of depression to the foot of the mast is 45 degrees. The angle of elevation to the top of the mast is 60 degrees. Calculate the height of the mast.

Using the angle of depression to the foot of the mast:

$$\tan(45^\circ) = \frac{H}{x}$$ $$H = x \tan(45^\circ)$$ $$H = 24 \times 1$$ $$H = 24 \text{ m}$$

Using the angle of elevation to the top of the mast:

$$\tan(60^\circ) = \frac{h_1}{x}$$ $$h_1 = x \tan(60^\circ)$$ $$h_1 = 24 \times \sqrt{3}$$ $$h_1 \approx 24 \times 1.73205$$ $$h_1 \approx 41.5692 \text{ m}$$

The total height of the mast is the sum of the building's height and the mast's height above the building.

$$h = H + h_1$$ $$h = 24 + 24\sqrt{3}$$ $$h = 24(1 + \sqrt{3})$$ $$h \approx 24(1 + 1.73205)$$ $$h \approx 24(2.73205)$$ $$h \approx 65.5692 \text{ m}$$

Rounding to two decimal places:

$$\boxed{\text{65.57 m}}$$

b) Calculate the distance between the top of the building and the top of the mast.

This distance is the hypotenuse of the right-angled triangle formed by the horizontal distance ($x$) and the height $h_1$.

$$\cos(60^\circ) = \frac{x}{D}$$ $$D = \frac{x}{\cos(60^\circ)}$$ $$D = \frac{24}{0.5}$$ $$D = 48 \text{ m}$$ $$\boxed{\text{48 m}}$$

:
Let $x$ be the distance from point H to the foot of the tower, $x = 20$ m.
Let $h_T$ be the height of the tower.
Let $h_C$ be the total height from the ground to the top of the mast.
The angle of elevation to the top of the tower (T) is $30^\circ$.
The angle of elevation to the top of the mast (C) is $50^\circ$.
We need to correct all answers to the nearest whole number.

a) Calculate the height of the tower.

Using the angle of elevation to the top of the tower:

$$\tan(30^\circ) = \frac{h_T}{x}$$ $$h_T = x \tan(30^\circ)$$ $$h_T = 20 \times \frac{1}{\sqrt{3}}$$ $$h_T = \frac{20\sqrt{3}}{3}$$ $$h_T \approx \frac{20 \times 1.73205}{3}$$ $$h_T \approx 11.547 \text{ m}$$

Rounding to the nearest whole number:

$$\boxed{\text{12 m}}$$

b) Calculate the height of the mast.

Using the angle of elevation to the top of the mast:

$$\tan(50^\circ) = \frac{h_C}{x}$$ $$h_C = x \tan(50^\circ)$$ $$h_C = 20 \tan(50^\circ)$$ $$h_C \approx 20 \times 1.19175$$ $$h_C \approx 23.835 \text{ m}$$

Rounding to the nearest whole number:

$$h_C \approx 24 \text{ m}$$

The height of the mast is the difference between the total height to the top of the mast and the height of the tower.

$$h_M = h_C - h_T$$ $$h_M \approx 23.835 - 11.547$$ $$h_M \approx 12.288 \text{ m}$$

Rounding to the nearest whole number:

$$\boxed{\text{12 m}}$$