Mensuration ,mathematics on right angled cone and on spheres

Students are  asking for the mathematics which they are not understanding in the email;    kaacconsultant@gmail.com . Solution is appearing in this page for helping others also. 

The problem is  The slant height of a right circular cone  is 10 m and its height is 8 m. find the area of the curved surface.


right circular cone means ,its base is a circle the meaning of the right is , its peak is straight  at 90 degrees. above the center of the base .

slant height means ,distance of the peak from the any point of base perimeter


height is   8 meter

slant height is 10 meter

from the above data we have to deduce another date , which is the radius of the base, otherwise we will not be able to find curved surface area.

radius of the base is 6 meter.  ( I have not deduced the same , students have to apply their skill) if not otherwise contact.

Now area of the curved surface is   pi * radius* slant height value of    pi has taken as = 3.142

which  will  become  187.52 sq meter     why  square meter ? because to units of meters has been multiplied.


Suppose there are two spheres  having radius R and r , is there any relation can be established   with this two radius. let us try

Surface area of the sphere with  radius  R  = 4 * pi* R2 ( R square)
similarly surface area of the sphere with radius r = 4*pi* r2 ( r square)

ratio is coming as R2 / r2 or  (R/r)2

Problem
A cylindrical shaped article has a height of 20 Cm.The radius of its base is 21 cm. and its volume is
2310  cm 3
It is open at the top
Find the cost of painting it from otside when rate is  Rs. 5 per Cm 2
Solution
Given data :
Height  H	20 Cm
Radius of the base R		21 Cm
Volume V		2310  cm 3
Now as the cylinder is open at the top , surface area we have to find out at the cylinder wall and the bottom
surface area of the cylinder			2 π R L
			2640	cm 2
surface area of the bottom			π R 2
			1386	cm 2
Total area to be painted			surface  + bottom
			4026	cm 2
And cost of painting is Rs. 5 per square Cm
That is one square Cm will cost  Rs. 5
But we have  4026		cm 2		Rs.	5*4026	cm 2
				Rs.	20,130	will be cost.

Mathematics, Polynomial MEMORIZE BY HEART THESE RULES

    If  p,q,r are the roots of the polynomial   a x 3 + b x 2 + c x - d  =  0

Then   (x - p) =0
           (x - q) =0
           (x - r) =  0

p + q + r = -b /a
p q +q r + r p  =    c/a

p q r =  d/a


Let us suppose  the equation

 x3 - 9 x2 + 26 x  -  24 = 0    and three roots are  2,3,4


sum of the three roots    2 +3 +4 = 9

from the equation  - (-9/1) = 9


2 *3 + 3*4+4*2 =  26
from the equation  c/a = 26

Product of the three roots = 2*3*4 = 24

from the equation  -(-24/1)= 24

ABOVE RULES ARE USED IN SO MANY SUMS . MEMORIZE AND PRACTICE SINCERELY

DO BOOND JINDAGI KI POLIO IMMUNIZATION

INDIA  is being declared as immune from POLIO in the year 2014. THANKS  to the Government and  the Organisations Volunteers. This is a great achievement. Two decades and more the country has to struggle to drive away POLIO.


TOP VACCINE
To eliminate POLIO , vaccine used was TOP  vaccine.  In our intestine  there  was antrovirus which caused polio. There was three types of Type 1, 2 and 3. TOP is effective with three types of  vaccine. Type 1 and 3 are severe infectious. and  these viruses cause paralytic effect. From 1999 , type 2 virus attack was not found.


INDIA , started activity against  POLIO  since 1995 . and  completed at 2014.   

Linear Equation of Two Variables, sum of straight lines

Students reading in class IX and X can send mathematics , which they are  not able to understand or  grasp, send an email with the mathematics at

kaacconsultant@gmail.com

that math will appear in this blog which will help others also







Two liner equation, both represents straight line

          a1 x + b1 y + c1 =    0
and     a2 x + b2 y +c2 =    0

Both the equation represents  two straight lines

General equation of an straight line    Y= M X +  C  where M is the inclination to the vertical axis called  gradient or slope and  C is the point where the line intersects with vertical  axis.


If we try to rearrange the equations in Y= MX + C form


a1x + b1y +c1 = 0
or ,    b1y  = -a1x - c1
or          y = -a1/b1 x  - c1/b1

Now the equation took the shape of  Y = MX +C

so     M= -a1/b1
and   C = -c1/b1





Suppose two linear equation are as follows

 6x-2y +9 = 0          ------------    1
3x -  y +12  = 0        ------------   2

equation 1      6x -2y +9 = 0
                     or,  -2y = -6x -9
                     or      y =  -6/-2 x  -9/-2
                     or      y = 3 x +9/2


From equation 1  slope or gradient value is  3

For  equation 2,   3x - y + 12 =0
                          or   -y = -3x -12
                          or    y =   3x +12

In this equation slope or gradient is  also 3

So both the equations gradient is same  so the lines are parallel.

When two lines are parallel , both the lines can not intersect at a common point. So the intersection point is unreal.


In the other example
Two equations are      2x +3y -2 = 0 ------------- 1
                                    x  - 2y  - 8 = 0  ------------2


slopes or gradients are     2x +3y -2 = 0

                                       or,   3y =   -2x +2
                                       or,     y =  -2/3 x + 2/3

so     slop is   -2/3

For second equation         x -2y -8 = 0

                                     or,   -2y =   -x +8

                                      or      y =  1/2 x  - 8/2

so slop of the second line is  1/2

When both the lines slope that is angle of inclination or gradient is different  the lines will intersect at a common point

Now if we  try to solve the equation we will get the point of intersection that is at the point of intersection both the equation  is satisfied


From first equation    y =  -2/3 x  + 2/3

From second equation y= 1/2 x  -  8/2

if the value of  y  is put   in equation 1

                            1/2 x  -8/2 = -2/3 x  + 2/3

                         or, 1/2 x + 2/3 x =   2/3 + 4
                         or, x = 4

Putting x =4 ,
 y = 1/2*4 - 4
or y =   -2                      So the point of intersection  (4,-2 )


Now in other conditions , let two lines are coincidence that is one line is lying on other , then how come the equations are different, It happens , when one line length is bigger or shorter than the other line.

Two equations are  5x -15y = 8 ------------ 1
other equation is     3x - 9y  = 24/5 -----------2

slope or gradient will be same and simultaneously   intersection point with the vertical axis will also be the same
                               rearranging the lines in y= mx +c pattern
                     
                                                                    5x -15 y = 8
                                                             or,       -15 y =  -5x +8
                                                             or,             y=  1/3 x - 8/15

 For other equation                                       3x -9y = 24/5
                                                              or,        -9y = -3x + 24/5
                                                              or,           y =  1/3 x - 8/15


So for both the equations  slope and intersection point at the vertical axis  are same . So both the lines are coincidence.





Now another interesting aspect
If the equation        x  =  a   this is also an equation of a straight line  which is parallel to vertical y axis

Similarly if the equation is  y = b , this straight line is parallel to  horizontal x axis. .

this both straight lines will met at the intersection point at (a,b)

On data theft, if not aware your money will be lost

you have an ATM card through which you can draw your money at any time. If you approach bank counter in their office hours and want to withdraw money as like old days, they will see you as worlds biggest fool.
Everyday lakhs of ATM cards details are going to wrong hand(s) and one day you will  lose your own money. You will run to the Bank , but bank will not take the responsibility. Again they will see you as worlds greatest fool with some sympathy , "thats why we tell to change passwords" will be their statement
Details of your ATM card, which you can not see through your naked eye , but ATM machine can  recognize when it is inserted into the ATM machine. This ATM machine will relay the details to a bigger computer system(server) and get acknowledge to proceed further. The process is no doubt fast.
Details of your ATM card can be leaked from the machine itself if suitable reader is installed. It can be leaked through the bigger server system if security program is not checked or updated. That can be seen and downloaded if suitable computer is installed by anyone who knows how to do that. There is a big market where this data can be sold out. A suitable duplicate card is made , this is called card cloning. Now the job is half done. Your pass word can also be known by the same procedure. You do  not know which hidden camera is focusing what ? Pass word cracking is rather easy subject  as banking system do not upgrade their security programming . Who want to spend extra for the p
oor customers when it is very easy to deny responsibility and passing the blame to other agencies.   This is the case of mass scale theft , Now the whole system has been made such, it is nearly impossible to find   out actual culprit.
What to do , going to bank counter and asking for money. Let them think happily you are a fool.

On online transactions, there is an agency or mediator with their computer program . More mediator means you do not know where your secret data is getting stored for future selling and for transaction.

The most dangerous part in totality no one or no agency is assuming responsibility and accountability and it is near to impossible to find the culprit as at every case  it is beyond our country .

SOLVED MATHEMATICS for STUDENTS

.Know mathematics.

Height and distance. Mensuration, GEOMETRY, TRIGONOMETRY
 FOR CLASS  IX/ X
CBSC/ICSC EXAMS.

IT IS EASY . POST MATHS IN E MAIL. GET SOLUTION IN THIS BLOG.



 Use Euclid's Division Algorithm to find the HCF of 255 and 867 (CBSE Sept 2012)


            Since  867> 255 taking  867 as dividend and 255 as divisor


                                    867= 255 X 3 +102


Now 102 is the remainder, which is not zero
 We have    255= 102 X 2 +51

Now 51 is the remainder and is not zero, so once again we have to apply Euclid's division algorithm

We have     102= 51 X 2+0
Now the remainder has become zero and 51 is the divisor.

Hence HCF of  867 and 255 is 51 (ANS)


Euclid Division Lemma 

Let there are two positive integers  p and q
There will exist two unique positive integers a and b which can be expressed as 

q= p*a + b 



where b is in between 0 and q, which is mathematically expressed 

We have taken two positive integers  p and q ,supposing p

         p | q |a
               x          let x =a*p
______________
let remainder =b      b= q-x





Let us take a real integer example, let two digits are   17 and  3

let us divide  17 by 3

              3  | 17 | 5
                     15

                    _____
                       2
This can be expressed as  17= 3*5 + 2

In the above example there is a remainder which is 2

Now let us take another example  of  15 and 3

         3| 15 | 5
              15
           _____        the equation becomes  15 = 3*5 +0
               0
                             So any two integer p and q , where supposing p< q 

can be expressed in a equation     q= a* p + b 






Now an important sum 

Prove that  squre root of any integer is irrational

Let us assume it is  square root of 3 

To solve a mathematical problem sometimes we are to help of inversion

So let us assume that  square root of 3 is rational

So  square root of 3 can be expressed in its simplest and lowest form of a/b ,

b is not equal to zero. a/b has no common factor other than  1

Square root of 3 = a/b 
then squaring  3 = a2/b2
                     or a2/b2= 3
                     or a2= 3 b2 

now  as 3b2 is divided by 3  so a2 is also divisible by 3 

as  a2 is divisible by 3 , a is also divisible by 3 ( 3 is a prime no.)

Now  let us assume that   a = 3c  ( for solving such mathematics it is general to assume to get to the solution)

Now assumption   a = 3c
                       or  a2= 9c2
                        or, 3b2= 9c2
                         or,  b2= 3 c2

Now with the same logic as above  3c2  can be divided by 3 so b2 can also be divided by 3 and if b2 is divided by 3 , b is also divided by 3 . ( 3 is also a prime no)

Now initially we have assumed that a/b is not divisible by any no other than 1 , now it is appearing that a and b are both can be divided by 3 .

Both logic are contrary to each other. So square root of 3 is irrational.

SOLVED MATHEMATICS for STUDENTS

.Know mathematics.

Height and distance. Mensuration, GEOMETRY, TRIGONOMETRY
 FOR CLASS  IX/ X
CBSC/ICSC EXAMS.

IT IS EASY . POST MATHS IN E MAIL. GET SOLUTION IN THIS BLOG.

SOLVED MATHEMATICS for STUDENTS

.Know mathematics.

Height and distance. Mensuration, GEOMETRY, TRIGONOMETRY
 FOR CLASS  IX/ X
CBSC/ICSC EXAMS.

IT IS EASY . POST MATHS IN E MAIL. GET SOLUTION IN THIS BLOG.