Question

A statue 1.6 m tall, stands on top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point, the angle of elevation of the top of the pedestal is $45^\circ$. Find the length of the pedestal.

Hint:

In height and distance problems, always form right triangles using tangent ratios and substitute $\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

Answer 1

Step 1: Let the height of the pedestal be $h$ m and the distance from the point on the ground to the pedestal be $x$ m.

Step 2: Using the given information.
Angle of elevation to the top of pedestal = $45^\circ$ Angle of elevation to the top of statue = $60^\circ$ Height of statue = $1.6$ m

Step 3: Apply trigonometric ratios.
For the pedestal (at $45^\circ$): \[ \tan 45^\circ = \dfrac{h}{x} \Rightarrow 1 = \dfrac{h}{x} \Rightarrow x = h \] For the top of the statue (at $60^\circ$): \[ \tan 60^\circ = \dfrac{h + 1.6}{x} \Rightarrow \sqrt{3} = \dfrac{h + 1.6}{h} \Rightarrow \sqrt{3}h = h + 1.6 \]
Step 4: Simplify for $h$.
\[ (\sqrt{3} - 1)h = 1.6 \Rightarrow h = \dfrac{1.6}{\sqrt{3} - 1} \] Step 5: Rationalize the denominator.
\[ h = \dfrac{1.6(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \dfrac{1.6(\sqrt{3} + 1)}{2} \Rightarrow h = 0.8(\sqrt{3} + 1) \] \[ h = 0.8(1.732 + 1) = 0.8 \times 2.732 = 2.1856 \approx 2.14 \, \text{m} \] Step 6: Conclusion.
Hence, the height of the pedestal is 2.14 m.