Surface area of an ellipse cross section

Area of ellipse:
Like a circle or any other closed figure, an ellipse is also a closed figure. Therefore it is possible to find its area. Area of an ellipse can be found using the values of a and b. But what are these a and b? That can be understood using the general equation of an ellipse.

General equation of an ellipse:
[(x-h)/a]^2 + [(y-k)/b]^2, where a is the semi major axis and b is the semi minor axis. (h,k) is the centre of the ellipse.

Area of ellipse formula:
From the above general equation we know that the semi major axis is a and the semi minor axis is b. The formula for the area of an ellipse can now be written as follows:
Area = A =  pi * ab. The semi major axis can also be termed as the distance between the centre and the vertex, which is ‘a’ here. The semi minor axis can also be terms as the distance between the centre and the co-vertex, which is ‘b’ here.  See the picture below:

Implicit equation of an ellipse:

Another method for finding the area of an ellipse is using its implicit equation. The general form of the implicit equation of an ellipse is:

Px^2 + Qxy + Ry^2 = 1

To calculate area of an ellipse if the equation of the ellipse is given in this form, we use the following formula:

Area = A = 2 pi/√(4PR – Q^2)

Let us now try to understand how to calculate the area of an ellipse using an example.

Example 1: 
Find the area of an ellipse with centre at (2,3), length of major axis being 6 cm and length of minor axis being 4 cm.

Solution:
 In this problem, major axis = 6, therefore semi major axis = a = 6/2 = 3 cm
Minor axis  = 4, therefore semi minor axis = b = 4/2 = 2 cm
Using the formula for area of ellipse,
A =pi* ab =  pi * 3 * 2 = 6 pi cm^2

Statistics Mode

Mode definition (math):
In statistics, a branch of mathematics, mode of a set of observations (also called data) is the particular observation that occurs most number of times. In other words, the observation with the maximum frequency is the mode of the data.

Finding mode:
The following steps are to be followed when calculating the mode of a set of observations of ungrouped data.
Step 1: Arrange the data in ascending order
Step 2: Look for the entry that occurs the most number of times.
Step 3: The number found in the step 2 above is your mode.

Mode Math examples:
To properly understand what is a mode in math or statistics, let  us try the following examples:

Example 1: The height of 10 students in centimeters is as follows, find the mode of the data.
145, 142, 143, 146, 144, 146, 143, 141, 140, 143
Solution:
Step 1: Arrange the data in ascending order. So now the numbers would look like this:
140, 141, 142, 143, 143, 143, 144, 145, 146, 146
Step 2: Look for the entry that occurs the most number of times. In the above data, the number 143 occurs thrice, whereas all the other numbers occur either once or twice.
Step 3: The number found in step 2 is the mode. Therefore for this problem, mode = 143 since it occurs most number of times.

Example 2: The lengths of screws manufactured by a firm in inches are as follows:
1.55, 1.50, 1.52, 1.53, 1.50, 1.52, 1.51, 1.54, 1.56
Find the mode of the data.
Solution:
Step 1: Arranging the data in ascending order we have:
1.50, 1.50, 1.51, 1.52, 1.52, 1.53, 1.54, 1.55, 1.56
Step 2: From the above arrangement we see that two numbers 1.50 and 1.52, both occur twice.
Step 3: Therefore this data has two modes: 1.50 and 1.52. Such a data is called bimodal.

Example 3: 
The ages of students in college algebra class is as follows:
25, 23, 24, 22, 23, 25, 22, 21, 27, 28, 20, 29
Find the modes.
Solution:
Step 1: Arrange the data in ascending order
20, 21, 22, 22, 23, 23, 24, 25, 25, 27, 28, 29
Step 2: There are three numbers, each occurring twice. They are 22, 23 and 25.
Step 3: These three numbers are therefore the modes. Such a data is called trimodal.
If there are more than three modes in a data set, it is called multimodal data.

Identifying conic sections

See the picture above. Consider a double cone as shown in the picture. Also consider a plane. Let this plane intersect the double cone. The way in which the plane intersects the double cone, would give rise to conic sections. As we can see in the picture the axis of the cone is vertical.

1. Conic sections circles:  If the intersecting plane is perfectly horizontal, that means that if the intersecting plane is perpendicular to the vertical axis, then the conic section thus produced would be a circle.

2. Conic sections ellipse: If the intersecting plane is inclined to the vertical at an angle that is between α to 90 degrees, where α is the angle between the vertical axis and the slant length of the cone, then the conic section thus produced would be an ellipse.

3. Conic sections parabola: If the intersecting plane is inclined to the vertical axis at an angle that is between 0 to α (α is same as described above), then the conic section produced would be a parabola.

4. Conic sections hyperbola: If the intersecting plane is parallel to the vertical axis, then the conic section thus produced would be a hyperbola.

Identifying conic sections:
Conic sections can be identified by two methods. They are, graphical and algebraic.
Graphical method of identifying conic sections:
The graphs of various conic sections are as shown below:

1. Graph of a circle:

The graph of a circle would look like above, it would have a centre and the radius on all sides would be equal.

2.Graph of an ellipse:

The graph of an ellipse can look like any of the two above figures. One is a horizontal ellipse and other is a vertical ellipse. An ellipse would have a major and a minor axis, two vertices and two co-vertices.

3.Graph of a parabola:

A parabolic curve would look like above. It would essentially have a vertex, a focus, a directrix and an axis of symmetry.

A hyperbola can be horizontal or vertical. Algebraic method for identifying conic sections:
In a conic sections practice problem, the equation of the curve should be similar to any of the following general equations:

Equation of the curve	Type of conic section
(x-h)^2 + (y-k)^2 = r^2	Circle
((x-h)/a)^2 + ((y-k)/b)^2 = 1	Ellipse
y-k = 4a(x-h)^2	Parabola
((x-h)/a)^2 - ((y-k)/b)^2 = 1	Hyperbola

Introduction to discrete math applications

Discrete mathematics is defined as the study of mathematical structures that are logically discrete rather than continuous. Discrete mathematics had a concept in topics in "continuous mathematics" such as calculus and analysis

The set of conepts studied in discrete mathematics can be finite or infinite. Some of the mathematical relations often considered i part of discrete mathematics are Boolean algebra, the mathematics of social choice, linear programming, and number theory

Discrete mathematics includes sets, functions and relations, matrix algebra, combinatory and finite probability, graph theory, finite differences and recurrence relations, logic, mathematical induction, and algorithmic thinking.

Applications of Discrete Mathemetics:

It can be applicable in various fields such as combinational analysis, functional system theory, codings, crptology,etc

It can also be used in graph theory, probabilistic problems of discrete mathemetics, and it has the applications of algorithms and their complexity and computational problems of number theory and of algebra

It can be also used in vital, exciting and useful information about mathemetics that will be very useful to taught the lower and higher grade

Problems by Using Discrete Math Applications:

1. By using discrete mathematics find the value of n.

(1). (n+ 1)! = 10 * 9(n!)

Solution:

(n+1)! = 10 * 9(n!)

here expanding the factorization that gives

(n + 1) n! = 10 * 9n!

now cancel n! on both the sides

we get n+1 = 90

n = 90 – 1 = 89

therefore n = 90 .

(2). 1/4! + 1/5! = n/6!

Solution:

1/4 + 1/5(4!) = n/6.5(4!)

now cancel 4! On both sides

1 + 1/5 = n/6.5

6/5 = n/30

n = (30 * 6)/5 = 180 /5 = 36

Therfore we get n = 36.

2. Find the following factorial question by using discrete mathemetic

Where n = 4 and r = 2 find the values of

(1). n! / r!

Solution:

n! / r! = 4! / 2! = 4.3.2.1 / 2.1 = 12.

Therfore n! / r! =12

(2). n! /r!(n- r )!

Solution:

n! / r!(n-r)! = 4! / 2!(4-2 )!

= 4.3.2.1 / 2!

= 4.3.2.1 / 2.1

therefore n! / r!(n-r)! =12.

Divergence theorem

Divergence theorem commonly known as divergence theorem of gauss. This theorem also help to integrate the functions, divergence theorem is a combination of vector and integral. To understand the theorem we have to know about divergence of vector field.
Vector calculus is very important in engineering and physics to the gradient, divergence and curl. Lets take vector V(x,y,z), be a differential vector function where x y z are Cartesian coordinate and if v1,v2,v3 are component of vector V then the function div.V is called divergence of vector field. The physical meaning of divergence is value of a function that characterize a physical or geometrical property must be independent of the particular choice of coordinate that means those values must be invariant with respect to coordinate transformation.
Now divergence-theorem in integrals, this theorem is used for triple integration. Divergence-theorem, which transforms surface integral to triple integral, it also involves the divergence of vector function. The triple integration is a generalization the double integration. triple integral can be transform in to surface integral, over the boundary surface of a region in space and conversely. This is a practical interest because one of the two kinds of integral is often simpler than other. It also helps in establishing fundamental equation in fluid flow, heat conduction etc. the transformation is done by gauss divergence-theorem which include the divergence of a vector field function.
Divergence-theorem of gauss means also transformation between volume integral and surface integral. Let constant T be a closed bounded region in space whose boundary is a piece wise smooth Orientale surface and let F(x,y,z) be a vector function that is continuous and continuous first partial derivatives in some domain containing T. then we proof the divergence-theorem of gauss. In this theorem a unit normal vector of surface which is pointing to the out side of surface can also used.
Let’s clear this theorem with some Divergence Theorem Examples. Suppose we have to evaluate the surface integral by this divergence-theorem. In our problem surface S is a closed surface consisting cylindrical coordinates and also coordinates of circular disc are given.  To solve this problem we need to know triple integration. Then we can solve this type of problem, when we solve this types of problem we mostly use polar coordinates, which is defined by X=rcosθ and y = rsin θ
Let’s take one more Divergence Theorem Example that is transformation of volume integral in to surface integral. Suppose function which is in vector form is given and function also involve unit normal vector. Here in place of surface volume of sphere is given, now by using gauss divergence-theorem we integrate the function from volume integral to surface integral.

Introduction to algebra test questions

Algebra is one of a part of arithmetic that uses letters in place of some unknown numbers. It is concerning about learn the rules of operations and dealings, and the constructions. It also was including the conditions, polynomial expressions, equations and algebraic arrangement. In other words, we use numerals like 1, 2, 3, etc to represent numbers. In algebra we use numbers as well as letters of the alphabet such as a, b, c, etc for any numerical values we choose. In this article we shall discuss about algebra test questions.First we will solve some sample questions for algebra test and then do some practice questions.

Sample Questions for Algebra Test :

Ex 1: Solve the linear questions

19x - 7 = 21x – 9

Solution: Subtract 21 x from both sides of the above equation, we get the below term
            -2x – 7 = -9
Add 7 to both sides of the above equation, we get the below term

-2x – 7 + 7 = - 9 + 7

-2x = - 2

Divide both sides by -2 of the equation, we get  

        X = 1

The answer is x = 1

Test the answer for the given equation. Replace with in the above given equation for x. If the left side of the equation sum value is equal to the right side of the equation sum value after the replacement of x value, you have got the exact answer.

Left side of the equation:

 19(1) - 7 = 12

Right side of the equation:

 21(1) - 9 = 12

Ex 2: Solve the given Questions

12− (3x + 7) = 2x

Solution: In the given equation is expanding and the terms are out the bracket.

12 − (3x + 7) = 2x

12 −3 x − 7 = 2x

(12 – 7) − 3x = 2x

5 – 3x = 2x

Now we are identify that it is easier to get all the x value on the right side of the given equation, by adding x to both sides of the above equation.

5 = 5x

Now we divide both sides by 5 and swap the sides:

We get the x value is 1

Test the answer for the given equation. Replace with in the above given equation for x.

We get

12 – 3(1) + 7 = 2(1)

12 – 3 - 7 = 2

Practice Questions for Algebra Test :

Solve 17 x - 6 = 23 x – 18
                    Ans: x = 2

Solve 13− (5x + 9) = -3x
                    Ans: x = 2

Intermediate algebra test

Intermediate algebra is mainly used for solving the equations. Intermediate algebra mainly uses the simplification technique for solving the equations. Intermediate algebra uses the letters and variables. Intermediate algebra also deals with the expressions and exponents of polynomials. Algebra test problems are very simple and easy. Intermediate algebra test covers the topics,

• linear Equations
• Equation of a line
• Quadratic Equations
• Polynomial Functions
• Slope formula

Intermediate Algebra Test Problems

Algebra test Problem: 1

Solve the equation x + y = 6, x - 3y = 2.

A) x = 2, y = 4 B) x = 5, y = 2 C) x = 4, y = 2 D) x = 5, y = 1.

Answer: D

Algebra test Problem: 2

Find the slope of the equation 4x + 3y = 21

A) M = - 4 / 5 B) M = - 4 / 3 C) M = - 3 / 4 D) M = - 3 / 5.

Answer: B

Algebra test Problem: 3

Factorize the equation 2x2 - 5x + 3.

A) X = 2, X = 4 B) X = - 2, X = - 3 C) X = 1, X = 3 / 2 D) X = 1, X = 2 / 3.

Answer: C

Algebra test Problem: 4

Find the vertex of the equation Y = x2 + 4x + 7

A) (2, 4) B) ( -2, 3) C) (-3, 2) D) (1,-1).

Answer: B

Algebra test Problem: 5

Find the distance between the two points (1, 5) and (3, 9).

A) d = 6 B) d = 3 C) d = 7 D) d = 5.

Answer: D

Problems for Intermediate Algebra Test

Algebra test Problem: 6

 Find the value of x for the expression: 4x + (4 - 7x) = (2x + 3) - 4.

A) 3 B) 2 C) 8 D) 1

Answer: D

Algebra test Problem: 7

Add the two expressions (3x + 2y - 8) and (5x - 3y -4)

A) 5x + 4y - 12 B) 8x + y + 12 C) 8x - y - 12 D) 2x - y - 4.

Answer: C

Algebra test Problem: 8

The sum of the number and 32 is 67. Find the unknown number.

A) 32 B) 33 C) 31 D) 35

Answer: D

Algebra test Problem: 9

Multiply (3a2 + 4b) (2a + 6b)

A) 6a3 + 8ab + 18a2b + 24b2

B) 3a3 + 8ab + 18a2b + 24b2

C) 4a3 + 6ab + 16a2b + 24b2

D) 12a3 + 9ab +18a2b + 24b2

Answer: A