Numerical Value of Pie

                             Numerical Value of Pie

 
 

Approximate value of pie is 22/7. This value is correct to 2 decimal places. The fraction 355/113 is a more correct value and this is correct to 6 decimal places.
[To remember the fraction 355/113, write down each of 1, 3, 5 in pairs and divide the last three digits by the first three.]
Approximate value of pie, correct to 10 decimal places, is 3.1415926536.
Remember that 1/pie = 0.31831 (approximately.)

Definition of radian

 

A radian is the measure of an angle at the centre of a circle subtended by an arc equal in length to the radius of the circle.
Let O be the centre of a circle of radius r and the circular arc AB be equal in length to the radius r of the circle.
Then by the definition of a radian, angle AOB = 1 radian.

      An important Theorem : Radian is a constant angle

 

Let PQR be a circle with centre O and radius r. Consider an arc AB of the circle so that the length of the arc and that of the radius r are equal. Join O and A and also O and B. Produce AO to meet the circumference of the circle at C.

It is evident, from the construction that angle AOB = 1 radian.
We know that the angle at the centre of a circle is proportional to the length of the corresponding arc.

Thus, angle AOB / angle AOC = arc AB / arc ABC
or angle AOB / 2 right angles = r /{(1/2) x circumference } =r/pie r
or angle AOB /2 right angles = 1/pie
Therefore angle AOB = 2 rt angles /pie = 180 degree/pie.
Thus, one radian = 180 degree/pie
or pie radians = 180 degree.

Therefore 180 degree/pie is not dependent on the radius of a circle and hence it is a constant quantity.
In other words, in any circle, the value of 180 degree/pie is always the same.
Hence, one radian = 180 degree/pie = a constant quantity.

Miscellaneous Examples based on Simple Interest

                         Miscellaneous Examples based on Simple Interest

i) A sum of money lent at simple interest amounts to Rs 3224 in 2 years and 4160 in 5 years. Find the sum and the rate of interest.

Solution

Amount in 5 years = (principal SI for 5 years) = Rs 4160.
Amount in 2 years = (principal SI for 2 years) = 3224.
On subtracting we get :
SI for 3 years = Rs (4160 - 3224) = Rs 936.

Or SI for 2 years = Rs (936 x 2 / 3 ) = Rs 624.

Or sum = (amount for 2 years) - (SI for 2 years) = Rs (3224 - 624) = Rs 2600.

Now, principal = 2600, SI = Rs 624, time = 2 years.

Therefore, rate = (100 x SI)/(P x T) = (100 x 624) / (2600 x 2)% = 12% p.a.


ii)The simple interest on a certain sum for 3 years at 8% per annum is Rs 96 more than the simple interest on the same sum for 2 years at 9% per annum. Find the sum.

Solution

Let the required sum be Rs X. Then,
SI on Rs X at 8% per annum for 3 years = Rs (X x 8 x 3) / 100 = Rs 6X / 25.
SI on Rs X at 9% per annum for 2 years = Rs (X x 9 x 2) / 100 = Rs 9X / 50.
Now according to the question
(6X / 25) - (9X / 50 ) = 96
or (12X - 9X) / 50 = 96
or 3X =50 x 96
or X = (50 x 96) / 3 = 1600.
Hence, the required sum is Rs 1600.

Word Problems on Percentage

i) A basket contains 350 eggs. If 12% of the eggs are rotten, find the number of eggs good enough to be sold.

Solution

Total number of eggs in the basket = 350.
Number of rotten eggs = 12% of 350 = (12 / 100) x 350 = 42.
Number of good eggs = 350 - 42 = 308.


ii) Mr. Pandey saves 12% of his salary. If he receives Rs 14700 per month as salary, find his monthly expenditure.

Solution

Total monthly salary = Rs 14700.
Monthly savings = 12% of Rs 14700 = Rs (12/100) x 14700 = Rs 1764.
Monthly expenditure = Rs (14700-1764) = 12936.

iii) In an examination, 96% of the students passed and 50 failed. How many students appeared?

Solution

Let the number of students appeared be Y.
Percentage of those who passed = 96%.
Percentage of those who failed = 4%.
Number of failures = 4% of Y.
Therefore, 4% of Y = 50.
Or, (4/100) x Y =50.
Or, Y = 50 x (100/4) =1250.
Hence, the number of students appeared is 1250.

iv) If 15% of the workers in a factory are females and the number of male workers is 272, find the total number of workers in the factory.

Solution

Let the total number of workers be X.
Percentage of females = 15%.
Percentage of males = 85%.
Number of males = 85% of X.
Therefore, 85% of X = 272.
Or (85/100) x X = 272.
Or X = 272 x (100/85) = 320.
Hence, the total number of workers in the factory is 320.

v) Out of her total monthly income, Mrs Pandey spends 30% on house rent and 60% of the rest on household expenses. If she saves Rs 6300 per month, what is her total monthly income?

Solution

Suppose her total monthly income is Rs 100.
Then, money spent on house rent = Rs 30.
Balance money = Rs 70.
Money spent on household expenses = 60% of Rs 70 = Rs 42.
Saving = Rs 28.
If she saves Rs 28, then her monthly income = Rs 100.
If she saves Rs 6300, then her monthly income = Rs (100/28) x 6300 = Rs 22500.
Hence, Mrs Pandey`s total monthly income is Rs 22500.

Definition of pie

Definition of pie

For any circle, the ratio of the circumference to its diameter is the same, i.e., this ratio is independent of the size of the circle.

That is, circumference of any circle/diameter of that circle = a constant quantity.

The constant ratio = circumference of a circle/diameter of that circle

                               = is an incommensurable (or irrational) number, denoted by the Greek letter pie (pronounced pi). The exact numerical value of pie can not be determined nor can it be expressed as a surd. But the value of pie correct to any number of decimal place can be found by methods given in higher trigonometry. With the aid of electronic computers, the value of pie correct to10000 decimal place has been calculated. This is occasionally used as a source of random numbers.

Another Supporting Blog

another supporting blog

Cartesian Product of two sets

 Q .   If G = { 7,8 }, H = { 5, 4, 2 }, find G x H and H x G.


Sol : 
        G x H ={ 7, 8 } x { 5, 4, 2 } ={ (7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8,2) }
and
        H x G = { 5, 4, 2 } x {7, 8 } = { (5,7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8) }