Question

Amit walked \(12 \mathrm{~m}\) toward east then be turned to his right and walked \(18 \mathrm{~m}\). He then-turined to his right and walked 12 \(\mathrm{m}\). He again turned to his right and walked \(28 \mathrm{~m}\) then he again turned to his right and walked 24 m: At what distance is he from the starting point and in which direction? (a) \(23 \mathrm{~m}\) north-cast (b) \(26 \mathrm{~m}\) inorth-cast (c) \(26 \mathrm{~m}\) west (d) \(34 \mathrm{~m}\) north-east

Short answer

Answer: Amit is 26 m away from the starting point in the North-East direction (approximately 22.62° north of east).

Step-by-step solution

Represent Amit's movements in coordinate form

Firstly, we'll need to represent Amit's movements in a coordinate system. Let's assume north as the positive y-direction, east as the positive x-direction. Amit's movements can be written as: 1. 12 m East: (12, 0) 2. 18 m South (right turn): (0, -18) 3. 12 m West (right turn): (-12, 0) 4. 28 m North (right turn): (0, 28) 5. 24 m East (right turn): (24, 0)

Calculate the total displacement in x and y directions

Now, we'll find the net displacement by summing up the x and y components of Amit's movements: X-axis displacement: 12 - 12 + 24 = 24 Y-axis displacement: 0 - 18 + 28 + 0 = 10 So, Amit's net displacement is (24, 10)

Find the distance from the starting point

Now, we use the Pythagorean theorem to find the distance: Distance = \(\sqrt{(X-axis\ displacement)^2 + (Y-axis\ displacement)^2}\) Distance = \(\sqrt{(24)^2 + (10)^2} = \sqrt{576 + 100} = \sqrt{676} = 26\ \mathrm{m}\)

Determine the direction in which Amit is

To find the direction, we will use the tangent formula: \(\tan\theta = \frac{Y-axis\ displacement}{X-axis\ displacement}\) \(\tan\theta = \frac{10}{24}\) Now, use the inverse tangent function to find the angle: \(\theta = \tan^{-1}(\frac{10}{24}) \approx\) 22.62° This angle is measured counter-clockwise from the east direction. So, Amit is in the North-East direction, approximately 22.62° north of east.

Conclusion

Therefore, Amit is at a distance of 26 m from the starting point, in the North-East direction (approximately 22.62° north of east). The correct answer is option (b) 26 m North-East.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as Cartesian geometry, is a branch of mathematics that allows us to precisely locate any point in a two-dimensional plane using numerical coordinates. It is an essential tool in various fields like engineering, astronomy, physics, and of course, in solving quantitative aptitude problems.

In the case of Amit's path, we use a two-dimensional coordinate system to analyze his movements. The key idea is to assign coordinates to each stage of his journey, with each turn resulting in a change along the x-axis (East or West direction) or the y-axis (North or South direction). By tracing Amit's steps and noting down the coordinates, we can visualize his path and use mathematics to find out his final position relative to his starting point.

Coordinate geometry not only simplifies the problem-solution process but also gives a clear visual representation of scenarios, which is crucial for better understanding and solving geometrical problems.

Displacement Calculation

Displacement is a vector quantity that refers to the shortest distance from the initial to the final position of a point and is a key concept in physics. It takes into account the magnitude and the direction. In the context of quantitative aptitude problems involving paths and movements, displacement allows us to calculate the straight-line distance between starting and ending points, irrespective of the path taken to get there.

In our problem, we calculate Amit's displacement by considering the sum of changes in the x-direction (East-West axis) and y-direction (North-South axis). After summing the movements along each axis, we end up with a net displacement that points directly from the start to the finish. This resultant vector tells us not only how far Amit is from his starting point but in which direction he's facing. By using displacement calculations, we simplify complex paths into a single line that can be easily quantified.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is mathematically expressed as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

In our scenario with Amit, once we have calculated the horizontal and vertical displacements, we effectively have two sides of a right-angled triangle. These sides are perpendicular to each other, which makes the situation perfect for applying the Pythagorean theorem. We can calculate the direct straight-line distance (hypotenuse) from Amit's starting point to his final position using this elegant mathematical formula. It's a powerful tool that works behind the scenes in everyday situations, from construction to navigation.

Trigonometry in Navigation

Trigonometry is often used in navigation to find direction and distance, particularly in fields like aviation, nautical navigation, and even in hiking. It involves the study of the relationships between the angles and lengths of triangles. Core to trigonometry is the concept of sine, cosine, and tangent, which are ratios that relate the angles of a triangle to the lengths of its sides.

In navigation, the tangent function can provide the angle of a path in relation to a reference direction. In our exercise, we use trigonometry to determine Amit's final direction. By taking the inverse tangent (arctan or tan^-1) of the ratio of the y-axis displacement to the x-axis displacement, we calculate the angle of Amit's displacement vector relative to the east, which gives us the precise direction he's facing. This application of trigonometry is instrumental in ensuring accurate navigation whether you're on land, at sea, or in the air.