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Some Applications Of Trigonometry
EXERCISE 9.1
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
Step-by-Step Solution
Key Concept: Use the definition of tangent in a right‑angled triangle: $\tan\theta = \frac{\text{opposite side}}{\text{adjacent side}}$.
1. Let the height of the pedestal be $h$ metres and the horizontal distance from the observation point to the foot of the pedestal be $x$ metres. 2. From the given angles of elevation: - For the top of the pedestal, $\tan45^{\circ}=\frac{h}{x}=1 \;\Rightarrow\; h = x$. - For the top of the statue, whose total height is $h+1.6$, $\tan60^{\circ}=\frac{h+1.6}{x}=\sqrt{3}$. 3. Substitute $x = h$ in the second equation: $$\frac{h+1.6}{h}=\sqrt{3} \;\Rightarrow\; 1+\frac{1.6}{h}=\sqrt{3}$$ 4. Hence $\frac{1.6}{h}=\sqrt{3}-1 \;\Rightarrow\; h=\frac{1.6}{\sqrt{3}-1}$. 5. Rationalising the denominator: $$h = \frac{1.6(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{1.6(\sqrt{3}+1)}{3-1}=0.8(\sqrt{3}+1)\text{ m}$$ 6. Numerically, $h \approx 0.8(1.732+1)=0.8\times2.732 \approx 2.19\text{ m}$. Thus the height of the pedestal is $0.8(\sqrt{3}+1)\,\text{m}$ (approximately $2.19\,\text{m}$).