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If The Percentage Error In The Surface Area Of Sphere Is k Then The Percentage Error

Referencing Styles : APA | Pages : 1

Percentage Error

Surface Area of a Sphere

The surface area of the sphere is defined by the formula-

A=4πr2

Where radius r is described as radius of the sphere. Any change in the measurement of the radius of a sphere can significantly affect the surface area of the sphere. The formula of finding out the surface area was discovered by a Greek philosopher Archemedes. It is realized that the surface are of the sphere is exactly equal to the rea of the curved wall related to the circumscribed cylinder. Circumscribed cylinder can be described as the smallest cylinder than can contain the sphere. The above formula can be rearranged in order to find the radius of a cylinder.

r=√a/4π

In the above formula, the “a” denotes the surface area of the sphere.

The surface area of sphere can be easily described as the number of square units that can exactly cover the surface of a sphere.

For a particular volume, a sphere can be described as a shape that has the smallest surface area and therefore, it generally appears in the nature, for example, in form of water drops, bubbles and planets.

The surface area of a sphere is generally considered to be exactly four times the area of a circle which is having the same radius. The area of a circle is πr2 while the surface area of the sphere is 4 πr2

A sphere is a perfectly round geometrical 3-dimensional object that can be characterised as perfectly symmetrical having no edge or vertices. Therefore a sphere having the radius r has a volume of 4/3​πr3 and a surface area of 4πr2. A sphere has a number of interesting properties, which include shape with same surface area.

Archimedes' Hat-Box Theorem

Archimedes' hat-box theorem states that “for any sphere section, its lateral surface will equal that of the cylinder with the same height as the section and the same radius of the sphere”.

With an aim of finding out the surface area of the sphere, the area of a disk enclosed by a circle of radius R is πr2

The formula of finding out the circumferences of the circle having a radius r is 2 πr, Similarly for finding out the volume of the ball that is enclosed by a sphere of radius r is 4/3​πr3.

For a sphere, a small change in the radius can result in a significant change in the volume of the sphere. The volume of the spherical shell of radius R and thickness is has significant effect on the volume of the sphere.

It was discovered by the Greek Mathematician Archimedes that the surface area of a sphere is same as that of the lateral surface area of a cylinder that has the same radius as similar to the sphere and a height and length of the diameter of the sphere.

The lateral surface area of the cylinder is 2πrh where h=2r.

Lateral Surface Area of the Cylinder =2πr (2r)=4πr2

Therefore, the Surface Area of a Sphere with radius r equals 4πr2

In this context, an example can be cited. For example, if we need to find out the surface area of the sphere having a radius of 5 inches, the surface area would be-

S.A. = 4π (5)2 = 100 π inches2 ≈ 314.16 inches2

Relation between surface area and the volume-

The surface area to the volume ratio can be considered as an object that defines the relationship between particular two measurements. It can be described as the ratio of surface area to volume. Larger objects might have a smaller surface area in comparison to the volume so they have a smaller surface area to the volume ratio.

Small and thin objects can significantly have a large surface area in comparison to the volume of the object. This in turn gives a large ratio of surface to volume. Since the larger objects have small surface area in comparison to the volume, they generally have a small surface area to volume ratio.

Therefore, it can be said that although the volume of an object may increase rapidly, the surface area may not rapidly increase.

Similarly in order to get the surface area of a 6 side cube, it is required to multiply the length times with the width, which would approximately be 4 square meters, then it is to be multiplied 6 times for 6 sides, which would finally be 24 square meters. On the contrary, in order to get the volume of the cube, the length and width of the cube is multiplied with the height of the cube.

The relationship between the surface area and the volume of a sphere can be explained by taking an example of the cell. As the size of the cell increases, the area and the volume of that cell does not increase as proportionally to the length. It is mainly observed that greater the diameter of a single celled organism, the lower is the surface area to the volume ratio. However, this relationship may not be true in all the cases and it is restricted to the particular cell that is considered for an example.

Answer-

S. A = 4πr2

V = 4/3 πr3

 ⅆS/S =k = 4π * 2rⅆr/ 4 πr2

=ⅆr/r = k/2

ⅆV/V = 4/3 π 3r2 * ⅆr/ 4/3 π r2

ⅆV/V= 3 * k/2 = 3k/2 

From the above calculation, if any percentage error in the surface area of the sphere is k, then the percentage error in the volume of that sphere will be 3k/2.

This is particularly because even for a small change in the surface area of a sphere, there is a significantly larger change in the volume of the sphere. This is calculated on basis of the formula of the surface area of a sphere and the formula of the volume of a s

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