Question

Radhey can walk along the boundary of a rectangular field and also along the diagonals of the field. His speed is \(53 \mathrm{~km} / \mathrm{h}\). The length of the field is \(45 \mathrm{~km}\). Radhey started from one corner and reached to the diagonally opposite comer in 1 hour. What is the area of the field? (a) \(860 \mathrm{~km}^{2}\) (b) \(1260 \mathrm{~km}^{2}\) (c) \(1060 \mathrm{~km}^{2}\) (d) can't be determined

Short answer

Answer: The area of the rectangular field is 1260 km².

Step-by-step solution

Identify the variables

Let: - l be the length of the field. - w be the width of the field. - d be the diagonal of the field. - v be the speed of Radhey. - t be the time taken to walk across the diagonal. Given in the question: - l = 45 km - v = 53 km/h - t = 1 hour We need to find w.

Use the Pythagorean theorem to find the width of the field

Since the field is rectangular, we can use the Pythagorean theorem to find the width of the field. \(d^2 = l^2 + w^2\) We know Radhey's speed and the time taken to walk across the diagonal, so we can find d as follows: \(v = \frac{d}{t} \Rightarrow d = vt\) \(d = 53 \cdot 1 = 53 \mathrm{~km}\) Now we can substitute the values of d and l in the Pythagorean theorem equation: \(53^2 = 45^2 + w^2\)

Solve for width of the field

Now, let's solve for w: \(53^2 - 45^2 = w^2 \Rightarrow w = \sqrt{2809-2025}\) \(w = \sqrt{784} = 28 \mathrm{~km}\) So, the width of the field is 28 km.

Calculate the area of the field

Now that we have both the length and the width of the field, we can find the area A: \(A = l \times w\) \(A = 45 \times 28\) \(A = 1260 \mathrm{~km}^{2}\) The area of the rectangular field is 1260 km². The correct answer is (b) \(1260 \mathrm{~km}^{2}\).

Key concepts

Pythagorean Theorem

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

To apply this theorem to a real-world problem, like calculating the area of a rectangular field using the diagonal, we can think of the length and width as the sides of a right triangle, and the diagonal as the hypotenuse. Once we know the length of two sides, we use the Pythagorean theorem to calculate the diagonal or vice versa. Knowing this theorem is essential for solving many geometric problems.

Speed Distance Time

The concept of speed, distance, and time is a fundamental aspect of physics that also plays a critical role in a variety of quantitative problems. The relationship between them can be neatly summarised by the formula \(s = \frac{d}{t}\), where \(s\) is speed, \(d\) is distance, and \(t\) is time. This formula can help us solve for any one of the three variables if we have the other two.

For instance, in the case of Radhey walking across a field, knowing his speed and the time taken allows us to calculate the distance he has traveled across the diagonal. This distance then serves as an essential component in using the Pythagorean theorem to determine the dimensions of the field. It’s a great example of how interconnected these fundamental mathematical concepts are in problem-solving.

Geometric Problem Solving

Geometric problem solving involves a systematic approach to finding the dimensions, areas, volumes, or other geometric properties of shapes and figures using known values and properties. It often requires the use of theorems, such as the Pythagorean Theorem, and formulas, like area and perimeter calculations, to uncover unknown measurements.

In the scenario provided, we approached the problem by first finding the missing dimension of the rectangle using the diagonal as a key to unlock the width. Once we have all necessary lengths, geometric problem solving leads us to calculate the area, which is a straightforward multiplication of length by width for rectangles. This process highlights the step-by-step logical thinking that is at the heart of geometric problem solving.

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numerical and mathematical calculations effectively. It is a skill highly valued in many fields, including education, science, engineering, and finance. This skill is also essential for standardized testing and competitive examinations where efficiency and accuracy in solving quantitative problems are assessed.

The problem of calculating the rectangular field’s area showcases quantitative aptitude by involving mathematical concepts such as the Pythagorean theorem and speed-distance-time relationship. It tests the students’ ability to translate a real-world situation into a mathematical model and then accurately perform calculations to reach the solution. The mastery of quantitative aptitude lies in understanding these underlying mathematical principles and applying them to various problems with precision.