Question

In the given figure \(A B, C D\) and \(E F\) are three towers. The angle of elevation of the top of the tower \(C D\) from the top of the tower \(A B\) is \(60^{\circ}\) and that from \(E F\) is \(30^{\circ} \cdot B D=2 \sqrt{3} \mathrm{~m}, C D: E F=5: 4\) and \(D F=4 \mathrm{~m}\). What is the height of the tower \(A B\) ? (a) \(6 \mathrm{~m}\) (b) \(12 \mathrm{~m}\) (c) \(7 \mathrm{~m}\) (d) none of the above

Short answer

Answer: (d) None of the above.

Step-by-step solution

Visualize the given information in a diagram

Let's first draw the diagram and label the points and angles as given in the exercise. We have three towers AB, CD, and EF where angle CBA = 60° and angle EFA = 30°. We also know that BD = 2√3 m, CD:EF = 5:4, and DF = 4 m.

Determine tower heights using ratios

We know CD:EF = 5:4 and we want to find their actual heights. Let CD = 5x and EF = 4x.

Set up trigonometric equations

Let's denote the height of AB as h. We can now set up two right triangles: - Triangle ΔABC, with angle CBA = 60°, and BC = h - 5x - Triangle ΔAEF, with angle EFA = 30°, and AF = h - 4x Using the tangent function, we can write two equations: 1. For ΔABC: \(\tan(60°) = \frac{5x}{2\sqrt{3}}\) 2. For ΔAEF: \(\tan(30°) = \frac{4x}{2\sqrt{3}+4}\)

Solve equations

From equation 1, we have: \(\tan(60°) = \sqrt{3} = \frac{5x}{2\sqrt{3}} \Rightarrow x = \frac{2}{3}\) Substitute the value of x into equation 2: \(\tan(30°) = \frac{1}{\sqrt{3}} = \frac{4(\frac{2}{3})}{2\sqrt{3}+4}= \frac{8}{3(2\sqrt{3}+4)}\)

Solve for h

Clear the fractions and solve for h: \(h - 4(\frac{2}{3}) = (2\sqrt{3}+4)\frac{8}{3} = 4\sqrt{3}+\frac{32}{3}\) \(h= 4\sqrt{3}+\frac{32}{3}+ \frac{8}{3} = 4\sqrt{3} + \frac{40}{3}\) The height of tower AB is not a number mentioned in options (a), (b), and (c). Therefore, the answer is: (d) none of the above.

Key concepts

Angle of Elevation

Understanding the angle of elevation is fundamental when solving trigonometry problems in the context of heights and distances. The angle of elevation is the angle formed between the horizontal line and the line of sight when an observer looks upward from the horizontal line to an object. For instance, if you stand at the base of a tower and look up to the top, the angle your line of sight makes with the ground is the angle of elevation.

Trigonometric ratios, such as tangent, sine, and cosine, are often used in conjunction with the angle of elevation to find heights of objects that are difficult to measure directly. As seen in our exercise, by knowing the angle of elevation and a base distance, we can calculate the height of the tower using trigonometric principles.

Tangent Function

The tangent function is one of the primary trigonometric ratios used in solving for unknown sides in right-angled triangles. It is defined as the ratio of the opposite side to the adjacent side of a right angle. When we say \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), we are referring to this relationship. In practical terms, if we are given the angle of elevation and one side of the triangle, we can determine another side. This concept was applied in the two equations of our exercise, where the tangent of the given angles helped us solve for the unknown side, which in turn led to finding the height of the tower.

Trigonometric Ratios

In trigonometry, the trigonometric ratios are the sine, cosine, and tangent functions, which relate the angles of a right triangle to the ratios of its sides. Using these ratios, we can solve various problems involving angles and distances without the need to physically measure them.

For more precise calculations, we also have the cosecant, secant, and cotangent, which are the reciprocals of sine, cosine, and tangent, respectively. These ratios become extremely powerful tools in problems where direct measurement isn't feasible. They are also helpful when the relationship between the sides of triangles needs to be established, as we have seen in our exercise to compare the heights of different towers without direct measurement.

Heights and Distances

The topic of heights and distances in trigonometry deals with finding the lengths and distances of various geometric figures using trigonometric ratios. It is an application of trigonometry that is commonly seen in fields such as architecture, astronomy, and navigation. The principle involves forming right-angled triangles and using known values such as angles of elevation or depression and the lengths of certain sides to calculate the unknown measurements.

For example, in the exercise we've discussed, we work with the height of towers and the distance between them to determine an unknown height. Real-world applications might involve finding the height of a mountain from a given distance, or the distance of a ship from the shore. These techniques eliminate the need for complex physical measurement tasks, substituting them with mathematical problem-solving methods.