Tower Height Calculation Using Angles
Let the height of the vertical tower be $h$ meters. Let the base of the tower be point $O$ and the top be point $P$. So, $OP = h$.
A person stands at point $A$ on the horizontal ground, due north of the tower. The angle of elevation of $P$ from $A$ is $45^\circ$. In the right-angled triangle $\triangle PAO$, we have:
- $\tan(\angle PAO) = \frac{OP}{OA}$
- $\tan(45^\circ) = \frac{h}{OA}$
- $1 = \frac{h}{OA}$
- $OA = h$
Another person stands at point $B$, which is $50$ m west of $A$. The angle of elevation of $P$ from $B$ is $30^\circ$. In the right-angled triangle $\triangle PBO$, we have:
- $\tan(\angle PBO) = \frac{OP}{OB}$
- $\tan(30^\circ) = \frac{h}{OB}$
- $\frac{1}{\sqrt{3}} = \frac{h}{OB}$
- $OB = h\sqrt{3}$
Ground Geometry and Pythagorean Theorem
Point $A$ is due North of the tower base $O$. Point $B$ is $50$ m West of $A$. This means the angle $\angle OAB$ is a right angle ($90^\circ$) because North and West directions are perpendicular. We can apply the Pythagorean theorem to the right-angled triangle $\triangle OAB$ on the ground:
- $OB^2 = OA^2 + AB^2$
- We know $OA = h$ and $AB = 50$ m.
- $OB^2 = h^2 + 50^2$
- $OB^2 = h^2 + 2500$
Solving for Height
Now, substitute the expression for $OB$ from the trigonometry step ($OB = h\sqrt{3}$) into the Pythagorean equation:
- $(h\sqrt{3})^2 = h^2 + 2500$
- $3h^2 = h^2 + 2500$
- $3h^2 - h^2 = 2500$
- $2h^2 = 2500$
- $h^2 = \frac{2500}{2}$
- $h^2 = 1250$
- $h = \sqrt{1250}$
- $h = \sqrt{625 \times 2}$
- $h = 25\sqrt{2}$
The height of the tower is $25\sqrt{2}$ meters.
Similar Questions
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
Let $\vec{a_k} = (\tan \theta_k) \hat{i} + \hat{j}$ and $\vec{b_k} = \hat{i} - (\cot \theta_k) \hat{j}$, where $\theta_k = \frac{2^{k - 1}\pi}{2^n + 1}$, for some $n \in \mathbb{N}, n > 5$. Then the value of $\frac{\sum_{k=1}^n |\vec{a_k}|^2}{\sum_{k=1}^n |\vec{b_k}|^2}$ is _____.
- If $k = \tan\left(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right) + \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1) = \sin^{-1}x - \cos^{-1}x$ is ________
- Let the maximum value of $(\sin^{-1} x)^2 + (\cos^{-1} x)^2$ for $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n} \pi^2$, where $\gcd(m, n) = 1$. Then $m + n$ is equal to __________.
- Let $S = \{\theta \in [0,2\pi] - (\frac{\pi}{2}, \frac{3\pi}{2}) \mid \sin^{-1}(\sin\theta) + \cos^{-1}(\cos\theta) + \tan^{-1}(\tan\theta) = \frac{4\pi}{5}\}$ Then $\frac{30}{\pi} \sum_{\theta \in S} \theta$ is equal to ________
Consider the following two statements :-
Statement p :
The value of $sin120^\circ$ can be derived by taking $\theta = 240^\circ$ in the equation $2sin \frac{\theta}{2} = \sqrt{1+sin \theta} - \sqrt{1-sin \theta}$
Statement q :
The angles A, B, C and D of any quadrilateral ABCD satisfy the equation $cos \left( \frac{1}{2}(A+C) \right) + cos \left( \frac{1}{2}(B+D) \right) = 0$
Then the truth values of p and q are respectively :-
- A tower $T_1$ of height 60 m is located exactly opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevaion of the top of $T_2$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is :-
The number of solutions of $sin3x = cos2x$, in the interval $\left[ \frac{\pi}{2}, \pi \right]$ is :-
Important Questions from Trigonometry
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
Let $\vec{a_k} = (\tan \theta_k) \hat{i} + \hat{j}$ and $\vec{b_k} = \hat{i} - (\cot \theta_k) \hat{j}$, where $\theta_k = \frac{2^{k - 1}\pi}{2^n + 1}$, for some $n \in \mathbb{N}, n > 5$. Then the value of $\frac{\sum_{k=1}^n |\vec{a_k}|^2}{\sum_{k=1}^n |\vec{b_k}|^2}$ is _____.
- If $k = \tan\left(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right) + \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1) = \sin^{-1}x - \cos^{-1}x$ is ________
