Double Triangle Models

Real-world problems rarely involve a single, perfect triangle. Whether you are observing a moving shadow on the ground or looking at a flagstaff perched on a building, you will often need to take two observations from different angles.

Problem. From a point P on the ground the angle of elevation of the top of a $10\text{ m}$ tall building is $30^\circ$. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is $45^\circ$. Find the length of the flagstaff and the distance of the building from the point P. (You may take $\sqrt{3} = 1.732$)

Problem. The shadow of a tower standing on a level ground is found to be $40\text{ m}$ longer when the Sun’s altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower.

Problem. The angles of depression of the top and the bottom of an $8\text{ m}$ tall building from the top of a multi-storeyed building are $30^\circ$ and $45^\circ$, respectively. Find the height of the multi-storeyed building and the distance between the two buildings.