From the top, K of a building 320 m high, the angles of depression of the top, L and bottom, M of another building on the same horizontal ground are 29° and 41° respectively. (i) Illustrate the information in a diagram. (ii) Calculate, correct to the nearest metre, the height of the other building.

a) Step 1: Describe the diagram. For part (i), a diagram illustrating the situation would show:

• A vertical line segment representing the first building (Building A) with height $320$ m. Let its top be $K$ and its base be $A'$.
• A second vertical line segment representing the other building (Building B) with unknown height $H_B$. Let its top be $L$ and its base be $M$. Building B is parallel to Building A and on the same horizontal ground.
• A horizontal line segment $A'M$ connecting the bases of the buildings. Let the horizontal distance between the buildings be $x$.
• A horizontal line drawn from $K$, parallel to $A'M$.
• The angle of depression from $K$ to $L$ (the top of Building B) is the angle between the horizontal line from $K$ and the line segment $KL$, which is $29^\circ$.
• The angle of depression from $K$ to $M$ (the base of Building B) is the angle between the horizontal line from $K$ and the line segment $KM$, which is $41^\circ$.

Step 2: Set up trigonometric equations. Let $H_A$ be the height of the first building ($320$ m) and $H_B$ be the height of the other building. Let $x$ be the horizontal distance between the buildings. From the angle of depression to the base $M$ of the other building: $\tan(41^\circ) = \frac{H_A}{x} = \frac{320}{x}$ From the angle of depression to the top $L$ of the other building: The vertical distance from $K$ to the level of $L$ is $H_A - H_B$. $\tan(29^\circ) = \frac{H_A - H_B}{x} = \frac{320 - H_B}{x}$

Step 3: Calculate the horizontal distance $x$. Using the first equation: $x = \frac{320}{\tan(41^\circ)}$ $x \approx \frac{320}{0.86928674}$ $x \approx 368.110 \text{ m}$

Step 4: Calculate the height of the other building, $H_B$. Using the second equation: $320 - H_B = x \cdot \tan(29^\circ)$ Substitute the value of $x$: $320 - H_B \approx 368.110 \cdot \tan(29^\circ)$ $320 - H_B \approx 368.110 \cdot 0.55430905$ $320 - H_B \approx 204.050$ $H_B \approx 320 - 204.050$ $H_B \approx 115.950 \text{ m}$ Rounding to the nearest metre: $H_B \approx 116 \text{ m}$

The height of the other building is $\boxed{\text{*116 m*}}$.