Problem 4 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

A man in a boat rowing uniformly away from a cliff \(150 \mathrm{~m}\) high takes 2 minutes to change the angle of elevation of the top of the hill from \(\frac{\pi}{3}\) to \(\frac{\pi}{4}\). Find the speed of the boat.

Step-by-Step Solution

Answer

The speed of the boat should be calculated by following these steps: convert time to seconds, define distances based on changing angles, subtract these distances and divide by time to find the speed. The actual speed, however, would need a calculator to compute.

Step 1: Identify the knowns

The height of the cliff is given as 150m. Time taken for changing angle of elevation from \(\frac{\pi}{3}\) to \(\frac{\pi}{4}\) is 2 minutes. The task is to find the speed of the boat.

Step 2: Convert minutes to seconds

To get the speed in m/s, one needs to convert the time from minutes to seconds. So, this gives \(2 \times 60 = 120\) seconds.

Step 3: Define the distances based on angle of elevations

Let's define the distance from the base of the cliff to the boat at any instant as 'd'. When the angle of elevation is \(\frac{\pi}{3}\), the distance to the boat, which we will call d1, is calculated by: \(d1=150 / \tan(\frac{\pi}{3})\). Similarly, when the angle of elevation is \(\frac{\pi}{4}\), the distance to the boat, d2 is: \(d2=150 / \tan(\frac{\pi}{4})\).

Step 4: Calculate the distance travelled by the boat

The distance travelled by the boat means the distance between point d2 and d1. So, the travelled distance 'x' can be calculated by: \(x=d2-d1\).

Step 5: Compute the speed of the boat

The speed 'v' of the boat can be calculated by dividing the distance travelled by the time taken. So, \(v=x / 120\).

Key Concepts

Angle of Elevation

When solving trigonometry problems involving bodies in motion, the term 'angle of elevation' comes into play. An angle of elevation is the angle formed by the line of sight of an observer looking at an object above their level, with the horizontal line from the observer's eye.

For instance, if you imagine a man in a boat who looks up at the top of a cliff, the angle between his direct line of sight to the top of the cliff and the horizontal line is what we refer to as the angle of elevation. Typically, it is measured in degrees or radians, which are units of angular measurement. In the context of our exercise, the angle of elevation changes as the boat moves away from the cliff, starting at \(\frac{\pi}{3}\) radians and changing to \(\frac{\pi}{4}\) radians.

Understanding this concept is essential as it allows us to use trigonometric functions like tangent, cosine, and sine to calculate distances. These calculations are especially useful in navigation, architecture, and various branches of science.

Distance Calculation

Crucial calculations in trigonometry often involve finding distances that are not immediately measurable, such as the distance from a boat to the base of a cliff when only the height of the cliff is known. To compute such distances, trigonometric ratios are used, most commonly the tangent ratio in the context of angles of elevation.

In our original exercise, the tangent function relates the height of the cliff (the opposite side of a right triangle) to the distance from the boat to the base of the cliff (the adjacent side). The formula for calculating this distance 'd' at different angles of elevation is given by \(d = \frac{height}{\tan(angle)}\). Hence, by knowing the angle of elevation and the height of the cliff, one can calculate the horizontal distance from the boat to the cliff.

The understanding of how to manipulate these formulas for distance calculation is significant for anyone involved in fields that require precise measurements, and it is integral for solving various practical problems presented in the realm of trigonometry.

Time Conversion

Often in physics and mathematics problems, time conversion is a necessary step to ensure that all units match when performing calculations. It's typical to convert units of time between seconds, minutes, and hours to achieve a consistent unit throughout the equation.

In trigonometry problems like the one about the speed of a boat, we usually measure speed in meters per second (m/s). Therefore, if the time is given in minutes, as in our exercise, we must convert it to seconds to use it in the speed formula. The straightforward conversion is: 1 minute = 60 seconds. Thus, for a duration of 2 minutes, the conversion to seconds is \(2 \times 60 = 120\) seconds.

Understanding how to properly convert time units is crucial for accurately calculating speed, acceleration, or even for financial calculations involving interest over time. The ability to convert and manage different time units is a fundamental skill across various scientific and real-life applications.