Altitude Term In Math

Introduction about altitude term in math:

            Altitude or height term is defined based on the context in which it is used. As a general definition, the term altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. In this article we shall discus about altitude term based problems.




Triangle:


The total space inside the triangle is called as area of that triangle.

Formula to find Area:

            Area of right angle triangle (A) =1/2 (length x height) square unit

                                                               = 1/2 l x h square unit.

                        Here, the term height refers the altitude of the triangle.

Example problem:



A right angle triangle has length 5cm and altitude 13 cm. Find the area of that triangle.
Solution:

Given:

            Length (l) =5cm

            Altitude (h) =13cm

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (5 x 13)

                           = 1/2 (65)

                           =65/2

                           =32.5

Area of triangle = 32.5 cm2



A right angle triangle has length 7.5m and height 10 m. Find the area of that triangle.
Solution:

Given:

            Length (l) =7.5m

            Altitude (h) =10m

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (7.5 x 10)

                           = 1/2 (75)

                           =75/2

                           =37.5

Area of triangle = 37.5 cm2


Rhombus:


The altitude of rhombus is the distance between base and opposite side of the base.

Formulas:

Area of the rhombus (A) = b x a

                                    b – Base of rhombus.

                                    a – altitude of rhombus

If two diagonal lengths are given:

Area of the rhombus (A) = (d1 x d2)/2

Example problems:

1.      The altitude and base of rhombus are 11 cm and 6cm respectively. Find are of rhombus.

Solution:

      Given:

                  Altitude of rhombus (a) = 11 cm

                         Base of rhombus (b) = 6 cm

            Area of the rhombus (A) = b x a square units.

                                                      = 11 x 6

                                                      = 66

      Area of the rhombus (A) = 66 cm2

2.      The altitude and base of rhombus are 14 cm and 10cm respectively. Find are of rhombus.

Solution:

      Given:

                  Altitude of rhombus (a) = 14 cm

                         Base of rhombus (b) = 10 cm

            Area of the rhombus (A) = b x a square units.

                                                      = 14 x 10

                                                      = 140

      Area of the rhombus (A) = 140 cm2

How To Cross Multiply

Introduction of How to Cross Multiply:

Cross multiplication is the easiest method that can be done in the equation form of two fraction form. For Example: `(4)/(16)` = `(3)/(12)`. By using the cross multiply method, we can check this given example as 4 * 12 = 48 and 16 * 3 = 48. Now it be proved that  `(4)/(16)` = `(3)/(12)`. Let see how to do the cross multiply and example problem of cross multiply.


Rules – How to Cross Multiply:


Let take

`(P)/(Q)` ….. Fraction 1

`(R)/(S)` ………. Fraction 2

Now equate the two fractions as in the form of `(P)/(Q)` … = `(R)/(S)` ……fraction equation

Rule 1: How to cross multiply means, multiply denominator of the fraction 2 with numerator of the fraction 1 as well as multiply denominator of the fraction 1 with numerator of the fraction 2

P * S = Q * R

Rule 2: Multiply the term on both sides of the fraction equation, and then the value of the term remains unchanged.

PS = QR

Multiply term A on both sides then its value is not changed. PSA = QRA.

Let we see example problem of how to cross multiply.


Example Problem – How to Cross Multiply:


Example 1:

How to find the value of x in given equation `(3)/(x)` = `(4)/(8)`?

Solution:

Step 1: Multiply 8 with the numerator of the fraction `(3)/(x)`.

3 * 8 = 24

Step 2: Multiply the denominator of the fraction 3/x with numerator of the fraction 4/8.

x * 4 = 4 x

Step 3: Equate above two steps, 4 x = 24

Step 5: Divide 4 by both sides, we get 4 * (`(x)/(4)`) = `(24)/(4)`

Then the value of x = 6

Answer: x= 6

Example 2:

How to solve this equation `(X - 3)/(4)` = `(5)/(2)`?

Solution:

Step 1: Multiply 2 with the numerator of the fraction `(X - 3)/(4)`.

2 * (X - 3) = 2X - 6

Step 2: Multiply the denominator of the fraction `(X - 3)/(4)` with numerator of the fraction `(5)/(2)`.

4 * 5 = 20

Step 3: Equate above two steps, 2X – 6 = 20

2X = 14

Step 5: Divide 2 by both sides, we get 2 * (`(X )/(2)`) = `(14)/(2)`

Then the value of X = 7

Answer: x= 7

Math h Functions

Introduction:

The hyperbolic functions are defined as the analogs of common trigonometric functions. The fundamental hyperbolic functions include sin h called as the hyperbolic sine, cos h called the hyperbolic cosine and the tan h called the hyperbolic tangent function.

The hyperbolic functions are just the rational functions of the exponentials. The mathematical hyperbolic functions are also called as the math h functions. In this article we are going to see various math h functions.


Expressions of math h functions:


There are various mathematical algebraic expressions given to the math h functions.

The hyperbolic sine of x is given by sinh x = ½ (ex-e-x).

The hyperbolic cosine of x is given by cosh x = ½ (ex+e-x).

The hyperbolic tangent of x is given by tanh x = sinh x/cosh x

= (ex-e-x) / (ex+e-x)

= e2x-1 / e2x+1

The hyperbolic cosecant of x is given by csch x = (sinh x)-1

= 2 / ex-e-x

The hyperbolic secant of x is given by sech x = (cosh x)-1

= 2 / ex+e-x

The hyperbolic cotangent of x is given by coth x = cosh x / sinh x

= (ex+e-x) / (ex- e-x)

= e2x+1 / e2x-1 .


Math h functions with respect to circular functions:


The hyperbolic functions or math h function with respect to the circular functions is given by

x = a cos t and y = a sin t

Here the circle is given as a rectangular hyperbola.

The math h functions exist in many applications of mathematics which involve the integrals with v (1+x2) and the circular functions with v (1-x2)

The math h functions also include many identities similar to that of the trigonometric identites. The identites of the math h functions includes

Cosh2 x – sinh2 x = 1

Cosh x + sinh x = ex

Cosh x – sinh x = e-x

The identities for the complex arguments includes

Sinh (x+iy) = sinh x cos y + i cosh x sin y

Cosh (x+iy) = cosh x cosy + i sinh x sin y.

Symbols That Represent Me

Introduction to symbols that represent me:

There are numerous mathematical symbols that can be used in mathematics mode. This is a listing of common symbols found within all branches of Mathematics.

SYMBOL		REPRESENTS
Symbol	=	Represents	Is equal to
Symbol	+	Represents	Plus or Addition
Symbol	-	Represents	Minus or subtraction
Symbol	/	Represents	Division
Symbol	X or *	Represents	Multiplication
Symbol	≠	Represents the	Inequality
Symbol	< and >	Represents the	“Is less than” and “is greater than”
Symbol	≤ and ≥	Represents the	“Is less than or equal to” and “is greater than or equal to”
Symbol	є	Represents the	Is an element of
Symbol	∑	Represents the	summation
Symbol	α	Represents the	Alpha
Symbol	β	Represents the	Beta
Symbol	γ	Represents the	Gamma
Symbol	δ	Represents the	Delta
Symbol	ζ	Represents the	Zeta
Symbol	η	Represents the	Eta
Symbol	θ	Represents the	Theta
Symbol	λ	Represents the	Lamda or Lambda
Symbol	μ	Represents the	Mu
Symbol	π	Represents the	Pi
Symbol	σ	Represents the	Sigma
Symbol	φ	Represents the	Phi
Symbol	χ	Represents the	Chi
Symbol	ω	Represents the	Omega
Symbol	#	Represents the	Number  Sign
Symbol	±	Represents the	Plus Minus
Symbol	Ω	Represents the	Omega
Symbol	ι	Represents the	Iota
Symbol	℮	Represents the	Estimate sign
Symbol	√	Represents the	Square Root
Symbol	∞	Represents the	Infinity
Symbol	∫	Represents the	Integral
Symbol	≈	Represents the	Almost equal to
Symbol	∂	Represents the	Partial differential
Symbol	∆	Represents the	Increment
Symbol	w.r.t	Represents the	With respect to
Symbol	log	Represents the	Logarithm
Symbol	!	Represents the	Factorial
Symbol	%	Represents the	Percentage

History for symbols that represent me

A very elongated form of the modern equality symbol (=) was first introduced in print in The Whetstone of Witte (1557) by Robert Recorde (1510-1558) the man who first introduced algebra into England.  He justified the symbol by stating that no two things can be more equal than a pair of parallel lines...

The infinity symbol was first given its current mathematical meaning in "Arithmetica Infinitorum" (1655) by the British mathematician John Wallis (1616-1703).

Symbols that represent me : Further history

Gottfried Wilhelm Leibniz (1646-1716) viewed integration as a generalized summation, and he was partial to the name "calculus summatorius" for what we now call [integral] calculus.  He eventually settled on the familiar elongated ‘s’  for the sign of integration, after discussing the matter with Jacob Bernoulli (1654-1705) who favored the name "calculus integralis" and the symbol  I  for integrals...  Eventually, what prevailed was the symbol of Leibniz, with the name advocated by Bernoulli...

In Math What Does Range Mean

Introduction to range in math:

• In math, Data set is a collection of data, which is usually presented, in tabular form. Each column represents a variable. In math, Range is generally defined as the value we obtained as a result of difference between a greater value and a smaller value.
• In other words, range is defined as the difference between a maximum and minimum value.

Steps to learn the range in math:

     In order to learn the range of data set the following steps are necessary.
     Step 1: Arrange the numbers from ascending to descending order.
     Step 2: Identify the greater value in the given set
     Step 3: Identify the smaller value in the given set
     Step 4: Find the difference between the greater value and the smaller value to identify the range of the given data set.

Worked Examples for range in math:

Example 1:
Find the range of the data set given below:
                    8 , 15 , 13 , 7 , 24 , 37 , 6.
    Step 1: Arranging the numbers given in data set from least to greatest.
                    6 , 7 , 8 , 13 , 15 , 24 , 37.
    Step 2: Identify the greater value in the given set
                  From the given set, we can identify 37 as the greater value.
    Step 3: Identify the smaller value in the given set
                  From the given set, we can identify  6 as the smaller value.
    Step 4: Find the Range.
                          Range     =   Greater value – Smaller value
                                           =   37 - 6
                                           =  31
                   Hence, the range of the given data set is 31.
Example 2:
Find the range of the data set shown below:
                    33 , 12 , 79 , 24 , 34 , 53 , 27 , 61
     Step 1: Arranging the numbers given in data set from least to greatest.
                    12 , 24 , 27 , 33 , 34 , 53 , 61 , 79.
     Step 2: Identify the greater value in the given set
                    From the given set, we can identify 79 as the greater value.
     Step 3: Identify the smaller value in the given set
                    From the given set, we can identify  12 as the smaller value.
    Step 4: Find the Range.
                              Range    =   Greater value – Smaller value
                                               =   79 - 12
                                               =   67
                        Hence, the range of the given data set is 67.

Practice problems for range in math:

1) Find the range of the data set shown below:
                    7 , 9 , 19 , 31 , 37 , 43 , 6 , 37
     Answer: 37
2) Find the range of the data set shown below:
                    14 , 11 , 32 , 10 , 32 , 77 , 27 , 43
     Answer: 67

Geometrical Meaning of the Zeroes of a Polynomial

We know that  a real number k is a zero of the polynomial p(x) if p(k) = 0 in a geometrical way.

The x-coordinate of the point, where the graph of a polynomial intersect the x-axis is called the zero of the polynomial.

An nth - degree polynomial intersects the x-axis of n points and therefore, has a maximum of n zeros in geometrical graph

In a quadratic polynomial ax2 + bx +c,

If a>0, then the graph is a parabola that open upwards.

If a<0, then the graph is a parabola that open downwards.

Special case to the zeroes of a polynomial in geometrical meaning


In a geometrical way the evaluation of zeroes with polynomial evaluation of the equation in geometrical meaning is follows

Case (i) :

Here, the graph cuts x-axis at two distinct points A and A′.
The x-coordinates of A and A′ are the two zeroes of the quadratic polynomial ax2 + bx + c in this case is shown in the following figure



Case (ii) :

The graph given here cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (i) coincide here to become one point A is shown in the following figure.



The x-coordinates  of the  A is the only zero for the quadratic polynomial ax2 + bx + c in this case.

No zero case

The graph given here is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point

So, the quadratic polynomial ax2 + bx + c has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also refers that the polynomial of degree 2 has at-most two zeroes is shown in the following figure.

Concept of the Derivative Online Help

In online, concept of derivative is clearly explained with the help of solved example problems. In online so many websites explain the concepts with the help of math sites. The derivative concept is clearly explained in calculus whereas derivative concept helps to find the rate of change for the given function with respect to change in the input function. The following are the example problems with detailed solution helps to explain the concept of derivative in online.


Concept of derivative online help example problems:


The following example problems explain the concept of derivative in online.

Example 1:

Determine the derivative by differentiating the polynomial function.

f(t) = 3t 3 +4 t 4  + 5t

Solution:

The given function is

f(t) = 3t 3 +4 t 4  + 5t

The above function is differentiated with respect to t to find the derivative

f '(t) = 3(3t 2 )+4(4t 3  ) + 5

By solving above terms

f '(t) = 9t 2 +  8t 3 + 5

Example 2:

Determine the derivative by differentiating the polynomial function.

f(t) = 6t6 + 5 t5 + 4 t4 + t

Solution:

The given equation is

f(t) = 6t6 + 5 t5 + 4 t4 + t

The above function is differentiated with respect to t to find the derivative

f '(t) =  6(6t 5)  +5 (5 t4 ) +4(4 t3) + 1

By solving above terms

f '(t) =  36t 5  + 25 t4  + 16 t3 + 1

Example 3:

Determine the derivative by differentiating the polynomial function.

f(t) = 2t 2 +4 t 4  + 15

Solution:

The given function is

f(t) = 2t 2 +4t 4  + 15

The above function is differentiated with respect to t to find the derivative

f '(t) = 2(2t  )+4(4 t 3 ) + 0

By solving above terms

f '(t) = 4t +16t3

Example 4:

Determine the derivative by differentiating the polynomial function.

f(t) = 5t5 +4t 4 +3t 3  + 2

Solution:

The given function is

f(t) = 5t5 +4t 4 +3t 3  + 2

The above function is differentiated with respect to t to find the derivative

f '(t) = 5(5t 4 )+4(4t 3 ) +3( 3t 2) +0

By solving above terms

f '(t) = 25t 4 +16t 3  +9 t 2


Concept of derivative online help practice problems:


1) Determine the derivative by differentiating the polynomial function.

f(t) = 2t 3 +3 t 4  + 4 t 5

Answer: f '(t) = 6t 2 +12 t3 + 20 t 4

2) Determine the derivative by differentiating the polynomial function.

f(t) = t 3+t5 + 4 t 6

Answer: f '(t) = 3t2 + 5t4 + 24 t 5