प्र.18 एक चिमनी और मीनार एक ही तल पर खड़े हैं। मीनार की चोटी से चिमनी के शीर्ष एवं आधार के अवनमन कोण क्रमशः 30° एवं 60° हैं। यदि चिमनी की ऊँचाई 100 मी हो, तो मीनार की ऊँचाई ज्ञात कीजिए।

Step-by-step solution

1. 1

   Given Information

   Chimney height $h = 100 \text{ m}$. Angles of depression from tower top to chimney top and base are $30^\circ$ and $60^\circ$.
2. 2

   Trigonometric Ratios

   In a right-angled triangle, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.

   $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$
3. 3

   Formulate Equations

   Let tower height be $H$, distance between them be $x$. Form two equations using $\tan 60^\circ$ and $\tan 30^\circ$.
4. 4

   Equation 1 (Base)

   From the tower top to the chimney base: $\tan 60^\circ = \frac{H}{x}$.

   $$\tan 60^\circ = \frac{H}{x}$$
5. 5

   Equation 2 (Top)

   From the tower top to the chimney top: $\tan 30^\circ = \frac{H-h}{x}$.

   $$\tan 30^\circ = \frac{H-h}{x}$$
6. 6

   Substitute Values

   We know $\tan 60^\circ = \sqrt{3}$ and $\tan 30^\circ = \frac{1}{\sqrt{3}}$.

   $$H = x\sqrt{3} \quad \text{and} \quad H-100 = \frac{x}{\sqrt{3}}$$
7. 7

   Solve for x

   From first equation, $x = \frac{H}{\sqrt{3}}$. Substitute into second equation.

   $$H-100 = \frac{H/\sqrt{3}}{\sqrt{3}}$$
8. 8

   Simplify and Solve for H

   $H-100 = \frac{H}{3} \implies 3H - 300 = H \implies 2H = 300 \implies H = 150 \text{ m}$.

   $$H-100 = \frac{H}{3}$$
9. 9

   Final Answer

   The height of the tower is $150 \text{ m}$.

   $$H = 150 \text{ m}$$