MAT Exam 2021 Paper

Concept:In a right triangle, $\tan \theta = \frac{\text{height}}{\text{horizontal distance from the base of the tower}}$.Explanation:Let the height of the tower be $h$ m.Let the distance of the second point from the tower base be $x$ m.Then the first point is $192$ m farther away, so its distance from the base is $x + 192$ m.At the first point:$$\tan \theta_1 = \frac{h}{x+192} = \frac{5}{12}$$So,$$\frac{x+192}{h} = \frac{12}{5} \quad \text{...(1)}$$At the second point:$$\tan \theta_2 = \frac{h}{x} = \frac{3}{4}$$So,$$\frac{x}{h} = \frac{4}{3} \quad \text{...(2)}$$Subtracting equation (2) from equation (1):$$\frac{x+192}{h} - \frac{x}{h} = \frac{12}{5} - \frac{4}{3}$$$$\frac{192}{h} = \frac{36 - 20}{15} = \frac{16}{15}$$Therefore,$$h = \frac{192 \times 15}{16} = 180 \text{ m}$$Answer:The height of the tower is $180$ m.Correct option: C.