Pendulum Length

The length of the pendulum in the clock can be found by knowing the angle between the maximum points of the pendulum and the distance covered by the pendulum. The length of the pendulum can be calculated using the arc length formula where the known values are the central angle and the distance covered by the pendulum between the maximum points. In the following article we will see in detail about the topic pendulum length.

pendulum
More about Pendulum Length:

The pendulum oscuillstes from a single point and the pendulum covers a maximum position on the each side and the angle between the maximum positions can be measured and the distance covered by the pendulum between the maximum positions is also measured. Using all the values and substituting these values in the arc length formula we can calculate the length of the pendulum from the formula.

The formula for the arc length L = (theta/360)*2*pi*r

Here the θ is the angle between the maximum positions and r is the length of the pendulum and L is the distance covered by the pendulum between the maximum positions.
Example Problems on Pendulum Length:

1. The angle covered between the maximum positions of pendulum and the distance covered between the maximum positions are 110 degrees and 20 cm. Find the length of the pendulum.

Solution:

The length of the pendulum = (360*L)/(2*pi*theta)

= (360*20)/(2*pi*110)

= 360/11*pi

= 10.4 cm

2. The angle covered between the maximum positions of pendulum and the distance covered between the maximum positions are 120 degrees and 25 cm. Find the length of the pendulum.

Solution:

The length of the pendulum = (360*L)/(2*pi*theta)

= (360*25)/(2*pi*120)

= (3*25)/(2*pi)

= "12 cm"
Practice problems on pendulum length:

1. The angle covered between the maximum positions of pendulum and the distance covered between the maximum positions are 100 degrees and 18 cm. Find the length of the pendulum.

Answer: 10.3 cm.

Quadrilaterals Kite Tutor

The students are learn mathematics by using tutoring in online.  The tutor and students are communicated and the tutor share the information about related topics in online. The kites are quadrilaterals because it is drawn in geometry with four sides. The quadrilaterals have convex property in all types. So the kites also has convex property. Now we are going to learn about quadrilaterals kite by tutor.

Explanation for Quadrilaterals Kite Tutor

Tutor description for kite quadrilateral:

The quadrilateral kite has two pair of equal sides. So it is also known as deltoids. The adjacent sides are present next side. The parallelogram is a regular polygon. The kite also regular polygon.

The kite quadrilaterals property:

The right angle measurement is based on diagonals intersection.

The adjacent sides angles are equal.

The area of kite is find out by half of the diagonals product.

The perimeter of kite is find out by sum of length.

Quadrilaterals kite formula:

Area based on diagonals method – `(d_(1)d_(2))/2` .
Area based on trigonometry method – ab sin C.
Perimeter based on side’s sum – 2a + 2b.

More about Quadrilaterals Kite Tutor

Example problems for quadrilaterals kite tutor:

Problem 1: Calculating the kite area with diagonals 5.1 cm and 6.3 cm.

Tutor solution:

The diagonals of quadrilateral kite are d1 = 5.1 cm and d2 = 6.3 cm.

The area of quadrilaterals kite is `(d_(1)d_(2))/2` = `(5.1 * 6.3)/2` = 16 cm2.

Problem 2: Calculating the quadrilateral kite perimeter with  length of sides 21 cm and 10.8 cm.

Tutor solution:

The length of sides are a = 21 cm and b = 10.8 cm.

The quadrilateral kite perimeter is 2a + 2b = 2 x 21 + 2 x 10.8 = 63.6 cm.

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Exercise problems for quadrilaterals kite tutor:

1. The quadrilaterals kite with diagonals 14 cm and 12.5 cm. Calculating the kite area.

Tutor solution: The quadrilateral kite area is 87.5 cm2.

2. Calculating the kite perimeter with sides 18.3 cm and 13.8 cm.

Tutor solution: The kite perimeter is 64.2 cm.

Fixed Rate Calculator

A loan in which the interest rate do not alter through the whole expression of the loan reverse of adjustable rate A loan in which the interest rate does not modify through the whole term of the loan for an entity taking out a loan when rates are low, the fixed rate loan would permit him or her to "lock in" the short rates and not be worried with fluctuations. On the another hand, if interest rates were in the past elevated at the time of the loan, he or she would profit from a floating rate loan, since as the prime rate cut down to historically normal levels, the rate on the loan would reduce. Reverse of adjustable rate. A calculator is a small electronic device; it is used to achieve the fundamental operations of arithmetic. Modern calculators are more convenient than computers, while most PDAs are similar in size to handheld calculators.

Fixed rate calculator
Examples for the Fixed Rate Calculator:
Fixed rate calculator – Example 1:

Determine the payments and interest for a fixed rate loan, using monthly interest compounding and monthly payments. The purchase price $150, no of monthly payments is 2 months, and interest rate 5.000%, and the payment calculator computes the payment amount for you.
Solution:

Fixed rate calculator

The down payment amount is $11.00

The Loan amount is $ 139.00

Payment amount is $ 69.93

Interest rate is      5.000 %

Interest compounding: Monthly

Total amount financed: $139.00

Total payments: $139.87

Total finance charge: $0.87

Payment schedule:

Date           Payment InterestPrincipal Balance

Loan     08-26-2010                            139.00

1           09-26-2010   69.93       0.58    69.35    69.65

2           10-26-2010   69.94       0.29    69.65

2010 Total                139.87      0.87

Grand Total              139.87      0.87
More Examples for the Fixed Rate Calculator:
Fixed rate calculator – Example 1:

Determine the payments and interest for a fixed rate loan, using monthly interest compounding and monthly payments. The purchase price $100, no of monthly payments is 5 months, and interest rate 5.000%, and the payment calculator computes the payment amount for you.
Solution:

Fixed rate calculator

The down payment amount is $18.02

The Loan amount is $ 89.00

Payment amount is $ 18.02

Interest rate is      5.000 %

Interest compounding: Monthly

Total amount financed: $89.00

Total payments: $90.11

Total finance charge: $1.11

Payment schedule:

Event      Date             Payment Int    PrincipalBalance

Loan       08-26-2010                          89.00

1             09-26-2010  18.02    0.37    17.65  71.35

2             10-26-2010  18.02    0.30    17.72  53.63

3             11-26-2010  18.02    0.22    17.80  35.83

4             12-26-2010  18.02    0.15    17.87  17.96

2010 Total                  72.08    1.04    71.04

5             01-26-2011  18.03    0.07    17.96

2011 Total             18.03    0.07    17.96

Grand Total            90.11    1.11

Score as in Mathematical Terms

“Score” means 20! .The term "score" was came from the old English word “scoru”. It is the Middle age English word. Also, in turn, it came from the Norse skor. In mathematical terms score carried a value 20. Abraham Lincoln's celebrated Gettysburg consists of the phrase, "Four score and 7 years ago". In bible the term mentioned as “three score years and ten”.

Orgin of score:

In mathematical terms "Scores" way "groups of 20", just as "dozens" way "groups of 12". "Score" is eventually from the Proto-Indo-European root (s)ker-, sense "to cut" . Another English offspring is "shear". English still uses the word "score" to submit to an indentation or line made by a sharp instrument. Scores cut in wooden tally firewood were used in including, a fact we still make the unwitting suggestion to today when we talk regarding "keeping score". For a while in the history of the Germanic languages, the word for the cut used to record the number 20 became a word for the number itself.
Examples

4 score

Score mean 20 in mathematical terms

`4xx20=80`

Therefore four score is 80

3 score and 10

Score mean 20 in mathematical terms

`3 x 20` and 10

60+10=70.

Therefore 3 score and 10 is 70.

In Abraham Lincoln's celebrated Gettysburg consists of the phrase, "Four score and 7 years ago".

The phrase denotes, in mathematical terms

`4xx20` and 7

80+7

87

The phrase means 87 years ago.

Three score difference 10

Three score

`3xx20`

60 difference 10

60-10

50

Some mistakes arises that in some cases score is taken as 10. It wii lead to wrong answer

We cant assume any value to score. It means to 20 alone.

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Related examples like score:

Like score there are a lot of examples are in mathematical terms.

Foursquare

It means symmetrical in all dimensions or equal in all dimensions like cubical.

Like the words numbers also holds some meaning, Let take an example

223

It bibbiblically mean things involving to an order of survival beyond the visible, demonstrate-able space; that which is above and away from the laws of nature; and/or matters concerning the stroke or influence of the ghostly or psychic powers and the top-secret knowledge of them.

General Probability

General probability is the part of mathematics that learns the feasible outcomes of given events together with the outcomes' relative possibilities and distributions. In general usage, the word "probability" is used to mean the opportunity that a particular event will occur expressed on a linear scale from 0 to 1, also expressed as a percentage between 0 and 100.Now we will see the examples of probability.

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Examples- General Probability

Example 1

In a class there are 8 students got top eight marks in English. The marks are 78,83,86,88,92,93,95,98.  What is the probability for the following outcomes?

i) Select the marks are below 85.

ii) Select the marks between 80 and 90.

Solution:

i) Take P(A) is the probability for the marks are below 85.

Given marks are 78,83,86,88,92,93,95,98.

Total numbers n(S)=8

Here the following marks are below 85 n(A)={78,83}=2

So P(A)=(n(A))/(n(S))

=2/8

=1/4 .

ii) Take P(B) is the probability for the marks between 80 and 90.

The marks 83,86,88  are available between the 80 and 90.So n(B)=3

Total outcomes n(S)=8

So P(B)=(n(B))/(n(S))

= 3/8 .

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Example 2

What is the probability for select the letter ‘G’ from the word ‘GENERAL KNOWLEDGE’?

Solution:

Given word is GENERAL KNOWLEDGE.

Total letters n(S)=16

Number of ‘G’ letter n(A)=2

So the probability=2/16

=1/8 .

Example 3

Paul has 20 caps and Alex has 15 caps. What is the probability for select the Alex’s caps?

Solution:

Paul’s caps n(A)=20

Alex’s caps n(B)=15

Total number of caps n(S)=20+15

=35

Probability for select the Paul’s caps= 15/35

= 3/7 .
Practice Problems- General Probability

1) What is the probability for select the letter ‘E’ from the word ‘ELEMENT’?

Answer:

Probability=3/7 .

2) Tom has 5 white balls, Joseph has 7 yellow balls. What is the probability for getting the yellow balls?

Answer:

Probability=7/12 .

These examples and practice problems are used to study the general probability.

Solving Summation Notation

A summation notation (Σ) is used for sum the numbers or quantities indicated. We can sum all types of numbers such as real, integer, complex, rational numbers using sigma notation. Summation notation is also called as sigma notation. Using summation notation we can solve the problems easily. Summation a concise and convenient way of writing long sums.


I like to share this Fractional Notation with you all through my article.

Solving Summation Notation – Solving Example Problems

Example 1: Find the sum for the given expression and explain the terms of summation notation;

7

∑ (n + 1)3 =?

n=1

Solution:

7

∑ (n + 1)3 =?

n=1

Lower Bound: ‘n’ is the lower bound of the summation notation.

Upper Bound:  ‘7’ is the upper bound of the summation notation.

Formula: (n + 1)3 is the formula to describe the each term of the summation notation.

Start: n = 1, here 1 is indicating the first number of the terms of summation notation. Usually value of start is zero (0) or one (1).

End: 7 is indicating the last umber of the terms of the summation notation.

Index Variable: Index variable ‘n’ is used to labeled each term of summation notation.

7

∑ (n + 1)3 = 8 + 27 + 64 + 125 + 216 + 343 + 512 = 1295

n=1

7

Therefore, ∑ i3 = 1295

i=1

Example 2: Solve and find the sum for the given expression;

6

∑ 3(n + n2) =?

n=1

Solution:

6

∑ 3(n + n2) =?

n=1

Formula: 3(n + n2), starting term: 1, end term: 6, and index variable: n

6

∑ 3(n + n2) / n = 3(1 + 12) + 3(2 + 22) + 3(3 + 32) + 3(4 + 42) + 3(5 + 52) + 3(6 + 62)

i=1


= 3(1 + 1) + 3(2 + 4) + 3(3 + 9) + 3(4 + 16) + 3(5 + 25) + 3(6 + 36)

= 3(2) + 3(6) + 3(12) + 3(20) + 3(30) + 3(42)

= 6 + 18 + 36 + 60 + 90 + 126

= 336

6

Therefore, ∑ 3(n + n2) = 336

i=1

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Example 3: Solve and find the sum for the given expression;

6

∑ n(2n + 1) / 2 =?

n=1

Solution:

6

∑ n(2n + 1) / 2 =?

n=1

Formula: n(2n + 1) / 2, starting term: 1, end term: 6, and index variable: n

6

∑ n(2n + 1) / 2 = (1(2(1) + 1) / 2) + (2(2(2) + 1) / 2) + (3(2(3) + 1) / 2) + (4(2(4) + 1) / 2)

n=1  + (5(2(5) + 1) / 2) + (6(2(6) + 1) / 2)


= (1(3) / 2) + (2(5) / 2) + (3(7) / 2) + (4(9) / 2) + (5(11) / 2) + (6(13) / 2)

= (3 / 2) + (10 / 2) + (21 / 2) + (36 / 2) + (55 / 2) + (78 / 2)

= 1.5 + 5 + 10.5 + 18 + 27.5 + 39

= 101.5

6

Therefore, ∑ n(2n + 1) / 2 = 101.5

n=1
Solving Summation Notation – Solving Practice Problems

Problem 1: Solve and calculate the sum for the given expression;

5

∑ n2(n) =?

n=1

Answer: 225

Problem 2: Solve and calculate the sum for the given expression;

5

∑ (2n + 2) / n =?

n=1

Answer: 14.57

Define Expanded Form

Expanded Form is a method to break up a number to illustrate how much each digit in the number represents and in other words, expanded form is the method of pulling a number separately and expressing it as a sum of the values of every digit. In this article we study about expand form define of and develop the knowldge of the math .

Example on Define Expanded Form

Choose the expanded form of the number 15,793.

Solution:

The expanded form of the number 15,793 can be found using a place value chart.

Now, let’s aim to write the number in words and define how it helps us identify the expanded form.

The number 15,793 is read as: 15 thousands 7 hundreds 93 which is the same as 1 ten thousand  5 thousands 7 hundreds 9 tens and 3 ones.

So, the expanded form of 15,793 is 10,000 + 5,000 + 700 + 90 + 3.

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Example to Define Expand Form Math:

Example 1:

57  can be expand as 57 = 50 + 7.

226 can be expand as 296 = 200 + 20 + 6.

5467 can be expand as 5467 = 5000 + 400 + 60+7.

89192 can be expand as 89192 = 80000 + 9000 + 100+90+2

Example 2:

What will we get when we expand form 23,368?

Choices:
A. 2 ten thousands, 3 thousands, 3 hundreds, 6 tens, and 8 ones
B. 2 ten thousands, 3 thousands, 6 hundreds, 6 tens, and 7 ones
C. 5 thousands, 5 ten thousands, 6 hundreds, 6 tens, and 8 ones
D. 5 thousands, 8 hundreds, 3 tens, and 68 ones

Correct Answer: A

Solution:

Step 1: 23,368 = 20,000 + 3,000 + 300 + 60 + 8
Step 2: = (2 × 10,000) + (3 × 1,000) + (3 × 100) + (6 × 10) + (8 × 1)
Step 3: So, when we expand 23,368, we will get ‘2 ten thousands, 3 thousands, 3 hundreds, 6 tens, and 8 ones’.

Example 3:

What will we get when we expand form 13,369?

Choices:
A. 1 ten thousands, 3 thousands, 3 hundreds, 6 tens, and 9 ones
B. 2 ten thousands, 3 thousands, 3 hundreds, 6 tens, and 5 ones
C. 3 thousands, 7 ten thousands, 9 hundreds, 6 tens, and 9 ones
D. 4 thousands, 7 hundreds, 9 tens, and 68 ones

Correct Answer: A

Solution:

Step 1: 13,369 = 10,000 + 3,000 + 300 + 60 + 9
Step 2: = (1 × 10,000) + (3× 1,000) + (3 × 100) + (6 × 10) + (9 × 1)
Step 3: So, when we expand 13,368, we will get ‘1 ten thousands, 3 thousands, 3 hundreds, 6 tens, and 9 ones’.