RATIO AND PROPORTION

Ratio and Proportion are mathematical entities that help in comparison of numbers. In other words a ratio can be defined as a relationship between two numbers that helps in defining the quantity of the first number with the second number. You can learn more about it by downloading the Ratio and Proportion PDF below.

In solving any kind of problems it is important to learn ratio and proportion formulas, as this helps in solving any kind of mathematical problems one may encounter. If we consider two numbers as a and b, then the ratio of a to b is written as a : b = a / b. Here a is antecedent and b is the consequent. This is what as known as a ratio.

Proportions, on the other hand can simply be defined when two ratios are equal to each other. These ratios can be of definable quantities such as shapes and sizes or similar numbers such as fractions. Proportions can be used to calculate metrics such as length, weight, sizes and percentages as well.

#Surface Area Of Sphere 

Have you ever wondered how the Surface Area of a Sphere was derived?

Well here is a great visualization to alter your perception.

Step 1: Cut the sphere in the following way.

Step 2: Spread the cut out part across the paper.

Step 3: Collate the pieces together in the following way

Step 4: Spread the areas out separately to form a sine curve

Step 5: The area of the sine curve is the surface area of the sphere.

                          CUBOID

In everyday life, objects like a wooden box, a matchbox, a tea packet, a chalk box, a dice, a book etc are encountered. All these objects have a similar shape. In fact, all these objects are made of six rectangular planes. The shape of these objects is a cuboid.

Cuboid:

A cuboid is a closed 3-dimensional geometrical figure bounded by six rectangular plane regions.

Figure 1

Face – A Cuboid is made up of six rectangles, each of the rectangle is called the face. In the figure above, ABFE, DAEH, DCGH, CBFG, ABCD and EFGH are the 6-faces of cuboid. The top face ABCD and bottom face EFGH form a pair of opposite faces. Similarly, ABFE, DCGH, and DAEH, CBFG are pairs of opposite faces.

Any two faces other than the opposite faces are called adjacent faces.

Consider a face ABCD, the adjacent face to this are ABFE, BCGF, CDHG, and ADHE.

Base and lateral faces: Any face of a cuboid may be called as the base of the cuboid. The four faces which are adjacent to the base are called the lateral faces of the cuboid.

Usually the surface on which a cube rest on is known to be the base of the cube.

In Figure (1) above, EFGH represents the base of a cuboid.

Edges – The edge of the cuboid is a line segment between any two adjacent vertices.

There are 12 edges, they are AB,AD,AE,HD,HE,HG,GF,GC,FE,FB,EF and CD and the opposite sides of a rectangle are equal.

Hence, AB=CD=GH=EF, AE=DH=BF=CG and EH=FG=AD=BC.

Vertex – The point of intersection of the 3 edges of a cuboid is called vertex of a cuboid.

A cuboid has 8 vertices A,B,C,D,E,F, G and H represents vertices of cuboid in fig 1.

By observation, the twelve edges of a cuboid can be grouped into three groups such that all edges in one group are equal in length, so there are three distinct groups and the groups are named as length, breadth and height.

                      DIFFERENTIAL EQUATION

A differential equation can simply be termed as an equation with a function and one or more of its derivatives. You can read more about it from the differential equations PDF below. The functions usually represent physical quantities. The simplest ways to calculate quantities is by using differential equations formulas.

Differential Equations are used to solve practical problems like Elmer Pump Heat Equation

Differential Equations first came into existence by Newton and Leibniz who also invented Calculus. The three kinds of equations Newton initially conceptualized were: