Rectangle Geometry

Introduction to rectangle geometry:

Geometry is a theoretical subject, but easy to learn, and it has many real practical applications. Eventually, geometry has developed into a skillfully arranged and sensibly organized body of knowledge. Rectangle is defined as the four sided shape polygon which every angle is 90 degree or right angle. It opposite sides are parallel and also equal length. I like to share this Definition of Rectangle with you all through my article.

Formulas of Rectangle in Geometry:

Area of the rectangle formula as,

Area A= l*w

Where l is the length of the rectangle and w is the width of the rectangle.

Perimeter of a rectangle formula as,

Perimeter P= 2(l +w)

Circumradius of the rectangle formula as,

Circumradius R=`(sqrt(l^2+w^2))/(2)`

Inradius of the rectangle formula as,

Inradius r=`(lw)/(l+w)`

Examples of Rectangle in Geometry:

In geometry rectangle problem 1:

Find the area of the following figure:

Solution:

We can find the area of a rectangle by using the following formula

Area A= l*w

In the above figure, the value of length l=4 cm and width w= 6 cm

Substitute the values of l and w into the above formula. Then we get

Area A= 4* 6

=24

Answer: 24 cm2

In geometry rectangle problem 2:

Find the perimeter of the following figure:

Solution:

We can find the Perimeter of a rectangle by using the following formula

Perimeter P= 2(l+w)

In the above figure, the value of length l= 7cm and width w= 12cm

Substitute the values of l and w into the above formula. Then we get

Perimeter P= 2(7+12)

Here we can add the 7 and 12 and then multiplying with 2.

Then we get,

Perimeter P=2(19)

= 38

Answer: 38 cm

In geometry rectangle problem 3:

Find the circumradius of a rectangle with the values of length and width are 3 cm and 4 cm.

Solution:

We can find the circumradius of a rectangle by using the following formula

Circumradius R=`(sqrt(l^2+w^2))/(2)`

Substitute the values of l and w into the above formula. Then we get

Circumradius R =     `(sqrt(9+16))/(2)`


=    `(sqrt(25))/(2)`

=      ` 5/2`

= 2.5 cm

Answer: 2.5 cm


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Practice problems of rectangle in  geometry:

Find the area of rectangle with the length= 5 cm and width= 7 cm.
Find the perimeter of rectangle with the radius length= 6cm and width= 8 cm
Find the circumradius of a rectangle with the values of length and width are 6 cm and 8 cm.
Answer keys:

35 cm2
28 cm
10 cm

Integration Reduction Formula

Introduction to Integration Reduction Formula:

There are many functions whose integrals cannot be reduced to one or more the other of the well known standard forms of integration . However , in some cases these integrals can be connected algebraically with integrals of other expressions which can either be directly integrable or which may be easier to integrate than the original functions . Such connecting algebraic relations are called 'reduction formulae' . These formu;ae connect an integral with another which is of the same type , but is of lower degree or order or at any rate easier to integrate than the original one .

Example of Integration Reduction Formula

Ex:1  Find the reduction formula for  `int` xn eax dx , n being a positive integer and hence evaluate `int` x^3 eax dx .

Sol: Let  In  =  `int` xneax   dx   .

On using the formulae for integration by parts , we get

In   =   `(x^n e^(ax))/(a)`    -  `int` n xn-1  `(e^(ax))/(a)`   dx

=   `(x^n e^(ax))/(a)`    -   `(n)/(a)` `int` xn-1  eax  dx   .

=     `(x^n e^(ax))/(a)`    -  `(n)/(a)` In-1   .

This is called reduction formulae for  `int` xn eax dx   .   Now  In-1  in turn can be connected to  In-2   .  By successive reduction of  n  , the original integral In finally depends on I0  ,  where  I0 =  `int` eax dx    =   `(e^(ax))/(a)`     .

To evaluate `int` x^3 eax dx  , we  take a = 5  and use  the reduction formula  for  n = 3 , 2 , 1 in that order . Then we have I have recently faced lot of problem while learning how to solve linear equations word problems, But thank to online resources of math which helped me to learn myself easily on net.

I3  =   `int` x^3 e5x dx   =    `(x^3 5^(ax))/(5)`   -  `(3)/(5)` I2    .

I2   =   `(x^2 e^(5x))/(5)`   -  `(2)/(5)` I1

I1  =  `(xe^(5x))/(5)`   -  `(1)/(5)` I0

I0   =   `(e^(5x))/(5)`    +    c

Hence    I3   =  `(x^3 e^(5x))/(5)`   -  `(3)/(5^2)` x2e5x   +  `(6)/(5^3)` xe5x    -   `(6)/(5^4)` e5x  +  c

Integration Reduction Formula- Example

Q:1 Find the reduction formula for  `int` tannx  dx   for an integer  n`>=` 2  and hence  find `int` tan6x dx .

Sol : Let  In  =  `int` tannx  dx

=     `int` tann-2x  tan2x    dx

=    `int` tann-2x   sec2x   dx     -     `int` tann-2x   dx

=     `(tan^(n-1)x)/(n-1)`   -   In-2   ,

which is the required reduction formula .

When n is even  , In will finally depend on

I0   =   `int` dx      =    x + c1

When  n is odd  ,   In  will finally depend on

I1   =   `int` tanx  dx    =   log (secx)   +   c2

Now   ,    I6    =    `int` tan6x  dx

=     `(tan^5x)/(5)`   -  `int` tan4x  dx

=   `(tan^5x)/(5)`   -   `(tan^3x)/(3)`   +  `int` tan2x   dx

=     `(tan^5x)/(x)`   -   `(tan^3x)/(3)`   +  tanx  -  x  +  c .

Solving the Proportion Math

Introduction of solving the proportion math:

A part considered in relation to its whole. Statement of equality between 2 or more ratios like a/b=c/d. When we multiply we get ad=bc. The two ratios are equivalent. And we can say, two set of numbers are proportional, if one set is a constant times the other. Here we are going to see solving proportions with variable.I like to share this Perpendicular Lines Calculator with you all through my article.

Example Problems of Solving Proportion in Math:

Example 1:

Given:  `(45)/(h) = (35)/(21)`

Solution:

Step 1: We need to find h value.

Step 2: Here we are using cross multiplication method to solve h.

Step 3: So, when we cross multiply we get 945 = 35h

Step 4: Now we divide using 35 on both the sides.

Step 5: The value of h=27.

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Example 2:

Given:  `(15)/(t) = (20)/(44)`

Solution:

Step 1:  From this question we need to find t value

Step 2: Using cross multiplication we get, 660= 20t.

Step 3: Now we divide using 20 on both the sides

Step 4:  `(660)/(20)=(20t)/20`

Step 5: t = 33.

Example 3:

Given:  `(21)/(9) = (42)/(g)`

Solution:

Step 1:  From this question we need to find g value.

Step 2: Using cross multiplication we get, 21g = 378.

Step 3: Now we divide using 21 on both the sides

Step 4: `(21g)/(21) = (378)/(21)`

Step 5: 18 = g.

These are the simple examples of solving proportion in math.

Solving Word Problem Using Proportions in Math:

Example 1:

Sam read 40 pages of a book in 50 minutes. How many pages should he be able to read in 80 minutes?

Solution:

Step 1: We know that Sam read 40 pages of book in 50 minutes.

Step 2: And we need to find how many pages Sam read within 80 minutes. Let us assume the unknown number is x.

Step 5: So,  `(40)/(x) = (50)/80`

Step 6: Using cross multiplication we get, 3200 = 50x.

Step 7: Divide using 50 on both the sides. `(3200)/(50)=(50x)/(50)` .

Step 8: 64 =x. Therefore Sam read 64 pages within 80 minutes.



Example 2:

A car travels 48 miles in 20 minutes. And how many miles travel the car within 30 minutes?

Step 1: We know that a car travels 48 miles in 20 minutes.

Step 2: And we need to find how many miles travel the car in 30 minutes. Let us consider the unknown number is x.

Step 5: So,  `(48)/(20) = (x)/(30)`

Step 6: Using cross multiplication we get, 1440 = 20x.

Step 7: Divide using 50 on both the sides.  `(1440)/(20) = (20x)/(20)` .

Step 8: 72 =x. Therefore a car travels 72 miles in 30 minutes.

These are the example word problems of solving proportions in math.

Pattern Block Fractions

Introduction :

Fraction is one of a part in math numerical values. A fraction number is an integer value. Fraction number is a one branch of the whole number value in decimals.  A Fraction number is consisting of a two parts. That is numerator value and denominator value. In online few websites are providing fraction pattern tutoring. In this article we shall discuss the pattern block fractions. I like to share this Types of Fractions with you all through my article.

Types of Pattern Block Fractions:

Proper Fractions:

In proper fractions having numerator value is less than the denominator value is called as proper fractions.

Example: `5/9` and `6/13`

Improper Fractions:

In improper fractions having numerator value is larger than or equal to their denominators value is called as improper fractions.

Example:  `6/5` and `9/7`

Mixed Fractions:

In mixed fraction Numbers having a two part, which is a whole number part and a fractional part is called as mixed numbers.

Example: `3(2/5)` and `4(1/3)`

Decimal Fractions:

In decimal fraction number having denominator value as 10, 100 or 1000 or any other higher power of 10 is called as decimal fractions.

Example: `7/10` and `65/100`

Simple Fractions:

In simple fraction number having both the numerator and denominator as whole numbers are called simple fractions.
Example: `4/5` , `12/95` .

Sample Problem for Pattern Block Fractions:

Pattern block fraction problem 1:

Evaluate the value of given fraction numbers `(7)/(9)` + `(5)/(9)`

Solution:

In the proper fraction number a denominator values are same. So we are directly add or subtract the numerator values.

Step 1: In the given problem denominator values are same.

Step 2: Add the numerator values of given numbers, we get.

`(7)/(9)` + `(5)/(9)` = `(7 + 5)/(9)`

=`(12)/(9)`

Step 3: Now we are simplify the fraction values.

= `(4)/(3)`

Pattern block fraction problem 2:

Evaluate the value of given fraction numbers`(9)/(11)` + `(8)/(11)`

Solution:

In the proper fraction number a denominator values are same. So we are directly add or subtract the numerator value.

Step 1: In the given problem denominator values are same.

Step 2: Add the numerator values of the given numbers, we get.

`(9)/(11)` + `(8)/(11)` = `(9+8)/(11)`

=  `(17)/(11)`

Definition of Division in Math

Introduction:

The definition of division steps are one of the easiest techniques in math. The Division is the one of the major arithmetic operation in math. Division is mathematically expressed by using of given symbol (/). In math division definition is inverse of the multiplications. In math division is the equal portion or group.  Consider X and Y are two value the product of two value is equal to Z means the mathematical expression is

`X*Y=Z`

The Y value is not equal to zero means

`X=Z/Y`

Types of Division Definition:

The given step is basic form of the divisions. These Division consists the basic three parts

Consider these equation `X=Z/Y`

In these

1] The upper part of the above equation is called as Dividend that is Z

2] The lower part of the above equation is called as Divisor that is Y

3] The X is called as quotient

Dividend/divisor= quotient

Division symbol definition specifies the various form some of them are given in below

1] `X=Z % Y`

2] `X=Z/X`

3] `X=Z) X` or `X=Z) bar(X)` [It is the US Notation]

Divisible:

The one number can be divided with another number and the result is whole number is called as divisible number.Understanding Writing Piecewise Functions is always challenging for me but thanks to all math help websites to help me out.

Division is used in various areas some of the names are given in below. That are

1] Division of Integers

2] Division of Rational Number

3] Division of Zero

4] Division of Complex Numbers

5] Division of Polynomials

6] Division of Matrices

7] Division of abstract algebra

8] Division and Calculus

Example for Division Definition:

Consider this example. There are 4 students in the class and 12 boxes are available then each student gets how many boxes?

Steps 1:

Write the all the given values

Total Students `(S) =4` and Total Boxes `(B) =12`

Finding Shared box (F)

Steps 2:

Form the equation using Division symbol

F=B/S

Steps 3:

Write the value based on above equation

`F=12/4`

Steps 4:

`F=3`

Each student gets the `3` Boxes.

Number Theory Research

Introduction :

In this we will study about number theory research. Number theory research is one of the most olden areas of research in math. Number theory is a study of all types of numbers in math  such as integers, whole number, and so on. Let us start our research in number theory.

Example Problems for Number Theory Research:

Example problem 1: How do you write 88,000 in scientific notation?

Solution:

Shift the decimal point to the left until the number is between 1 and 10. Calculate the places we move the decimal point.

88,000  ?  8.8

We moved the decimal point 4 places to the left. The power of 10 is 104.

88,000 = 8.8 × 104

Answer: 88,000 = 8.8 × 104

Example problem 2: Which of the following numbers are primes?

5, 18,17, 8

Solution:

The factors of 5 are 1 and 5. Since the only factors of 5 are 1 and itself, 5 is prime number.

The factors of 18 are given as1, 2, 3, 6, 9, and 18. Since 18 has more factors than just 1 and itself, 18 is not prime number.

The factors of 17 are given as 1 and 17. Since the only factors of 17 are 1 and itself, 17 is prime number.

The factors of 8 are given as1, 2, 4, and 8. Since 8 has more factors than just 1 and itself, 8 is not prime number.

Therefore, the prime numbers are 5 and 17.

Answer: The prime numbers are 5 and 17.

Practice Problems for Number Theory Research:

Practice problem 1: Which number is prime number? 20, 7, 16, 10

Practice problem 2: What is the prime factorization of 16?

Practice problem 3: Write is the greatest common factor of 10 and 6.

Solutions for number theory research:

Solution 1: The prime number is 7.

Solution 2: The prime factorization of 16 is 2 * 2* 2* 2.

Solution 3: The greatest common factor of 10 and 6 is 2.