Trigonometry & Inverse Trigonometry
Heights And Distances
nta_abhyas_2025
Grade 11
Question:
If the angles of elevation of the top of a tower from three collinear points $A$, $B$ and $C$ on a line leading to the foot of the tower are $30°$, $45°$ and $60°$ respectively, then the ratio $AB : BC$ is
2 : 3
\sqrt{3} : 1
\sqrt{3} : 1\sqrt{2}
1 : \sqrt{3}
Step-by-Step Solution
Key Concept: Use multiple angles of elevation from the same point to set up a system of equations involving trigonometric ratios
Let the height of the building be $y$ and the height of the flagstaff be $z$. From the given angles: $\tan 30° = \frac{y}{x+y}$, $\tan 45° = \frac{y}{x}$, and $\tan 60° = \frac{y+z}{x}$. From $\tan 45° = \frac{y}{x}$, we get $x = y$. From $\tan 30° = \frac{1}{\sqrt{3}} = \frac{y}{x+y}$ and substituting $x = y$, we get $y = h(1 - \frac{1}{\sqrt{3}})$. From $\tan 60° = \sqrt{3} = \frac{y+z}{x}$ and solving with the constraint equations, we find $z = \frac{(\sqrt{3}-1)h}{\sqrt{3}(\sqrt{3}-1)} = \frac{h(\sqrt{3}-1)h}{3}$. The height of the flagstaff is $250\sqrt{3}$ units.
Correct Answer: 250√3