Institution: Bank Asia Ltd Exam: Bank Asia Ltd - Management Trainees - 17.06.2011 ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

The correct formula for determining the surface area of a sphere is D) \(4\pi r^2\). This formula arises from the study of geometrical properties of spheres. Here, \(r\) represents the radius of the sphere, which is the distance from the center to any point on the surface.

To understand why this formula is used, we can think about how a sphere can be conceived. A sphere is a perfectly symmetrical three-dimensional object. The surface area is the total area that covers the outside of the sphere. The constant \(\pi\) is a mathematical constant representing the ratio of the circumference of a circle to its diameter, and it plays a crucial role in formulas involving circular shapes, including spheres.

The term \(r^2\) indicates that the area is proportional to the square of the radius, meaning if the radius doubles, the surface area will increase by a factor of four. The factor of 4 arises because a sphere has a rotational symmetry and covers all directions in three-dimensional space.

In contrast, the other options presented relate to different geometric shapes or properties: \(πr^2l\) and \(2πr^2l\) could represent aspects of a cylindrical shape, while \( \frac{4}{3}πr^3\) represents the volume of a sphere, not its surface area. Therefore, option D) \(4πr^2\) is the correct and definitive formula for the surface area of a sphere.