Right Circular Cylinder

Right Circular Cylinder :- If the number of circular sheets is arranged in a stack, if the stack is kept vertically up then it is called the right circular cyl. If the base is circular and the other sheets are kept at ninety degrees at the base, then it is called the right circular cyl.  The right circular cyl can be made by a rectangular sheet by folding it in round shape. The area of the sheet gives us the curved surface area of the cyl because the length of the sheet is equal to the circumference of the circular base which is equal to 2 π r . Therefore the curved surface area of the cyl is equal to the area of the rectangular sheet. The area of the rectangular sheet is equal to the product of the length and the breadth. The curved surface area of the cylinders is also equal to the product of the perimeter of the base and the height of the cylinders.  If the top and bottom of the cylinders is also consider because it is needed to make the circular tin, If the area of the bottom and top of the cylinders is included in the curved surface area we get the total surface area. It means total surface area of the right circular cylinder is equal to the area of the base plus the area of the top of the cylinder plus the curved surface area of the cylinder. The cylinder has two circular base at the bottom and at the top.The diameter of the base of the cylinder can be measured directly with the help of the scale. 

What is the Formula for Volume of a Cylinder :- As we know that the volume of cuboids is equal to the product of the length, breadth and the height. The volume is three dimensional. The cuboids are built up with the rectangle of the same size sheets. In the same way the right circular cylinders can be made by using the same principle. So, by using the same argument as for cuboids, we can see that the volume of a cylinders can be obtained by the product of the base and height of the cylinders.  Therefore, the volume of a cylinder = the base area of the cylinder X height of the cylinder. As we know that the base of the cylinder is the circle. The area of the circle is equal toπ r². The height of the cylinder can be assumed as h. Therefore the volume of the cylinder = π r²h, where r is the base radius and h is the height of the cylinder.

Example :- Let us find the volume of the cylinder which has the radius of the base is twenty one centimeter and its height is thirty five centimeter.

Solution :- The formula for the volume of the cylinder, V = π r²h
Radius r = 21 centimeter, height h = 35 centimeter
Therefore the volume of the cylinder V = π ×21² ×35 = 48490.48 cubic centimeters.

Trend Line

The trend lines are drawn for the prediction of the data. These lines are associated with the data series. The trend lines are used for any type of improvements from the available data. The trends lines can be drawn to follow the equation of a line, i.e y = mx + c, where m is the slope of a lines. The value of c is constant which decides the equation of a lines. The trends line is used for exponential or logarithmic formulae. We can choose the right  type of it for our data as per our requirements. Linear trend lines, logarithmic trend lines, Polynomial trend lines, power trend lines, exponential and moving average trend lines can be drawn as per our requirements. We can add, remove or modify a trend lines as required.

Now a days, online tools or calculators are available to draw a trend line. We can display the equation of any trend line on the chart. The R squared value can be displayed. We can format the present structure to draw a fresher one. The online tools are available to draw any type or desired line quickly to take the faster decisions. Prediction of whether forecasting, draught, can be made by the data available. These  are the key of the success of our business, trade, education and research.

Equation of a Line Calculator: - Line calculators are the online tool which can be used to find the slope of a lines. Slope of a line is the tangent of the line. Slope is the ratio of the rise to run.  A line can be written in the standard form, slope intercept form, intercept form, general form, point slope form, two point form.
The equation of line calculators can also be used to find the equations of the lines when the coordinates are given. This calculator has four inputs and one output. The input to the calculators is the value or the coordinates of two points. The output terminal shows the value of the slope which is ratio of the difference between the y coordinates to the difference of x coordinates. The four inputs to the line calculators are x- coordinate one, x_ coordinate two, y- coordinate one and y- coordinate two. After giving all the inputs we have to press the output button. In the output, we can get a fraction or the degree of the tangent. If we want to check the equation of a line then press the output button for equation of the line. The slope of the lines can be used to find the height and the base when we making the bridge or when a road is planned in the hill area. The tangent angle is required to hit a moving or stationary target.

In the hanging bridge technology the line calculators are used to find the slope of the tension wire to hold the bridges. All the computers which are used to find the moving target and to fire guided missiles use the line calculators to calculate the tangent angles.

Mixed Fractions

Mixed Fraction: - The mixed numbers are like 2 1/2 which is two and one half or 35 3/32 which is equal to thirty five and three thirty two seconds.  To express the mixed fractions we have to keep exactly one blank between the whole numbers and the fractions. The mixed fraction are in the form of 3 2/3 ( three and two third) , 7 1/5 ( seven and one fifth) , 13 1/7 ( thirteen and one seventh ) , 113 7/100 ( one hundred thirteen and seven hundredth) and so on.

The rules for adding the mixed numbers: - To add the mixed number, convert them into the fraction. The algebraic formulae can be used for the addition of these, for example a/b + c/d = (ad + bc) / bd.

Example :- Let us add the two mixed fractions a b/c and d b/c

Solution :- The given fraction are a b/c and d b/c. The first step which is to be used is to convert the mixed fraction to the fraction. The fraction of the mixed term a b/c is equal to (ac + b)/c. The procedure is to convert the mixed fraction to the fraction is to multiply the whole number to the denominator of the fraction and then add the fraction. The total value is divided by the denominator of the fraction. The fraction of the mixed term d b/c is equal to (dc + b)/c. Now we have to add the two fractions as below.

(ac + b )/c + (dc + b)/c. As the denominators of both the terms are same, the numerators can be added directly as below
(ac + b + dc + b ) /c =   (ac +2 b + dc) /c

The rules for subtracting the mixed numbers: - To subtract the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the subtraction of the fractions, for example a/b - c/d = (ad - bc) / bd.

Example :- Let us subtract the two mixed fractions a b/c and d b/c
(ac + b )/c - (dc + b)/c

As the denominators of both the terms are same, the numerators can be subtracted directly as below
{(ac + b) – (dc + b)} /c = (ac + b – dc - b) /c = (ac – dc)/ c = (a – d) [by canceling the common term in the numerator and denominator]. 

The rules for multiplying the mixed numbers: - To multiply the mixed numbers or the fractions we have to convert them into the fractions. The algebraic formulae can be used for multiplication of the fractions, for example a/b *c/d = a c /bd

The rules for dividing the mixed numbers :- To divide the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the division of the fractions, for example a/b ÷ c/d = a/b ×d/c = ad/bc.

Fraction Calculator Online: - The fraction calculator online is a tool which can be used to add, subtract, to multiply and to divide the fractions. We have to enter the fractions to be calculated, enter the function is to be carried out (i.e add, subtract, multiply, divide). The output in the form of the fraction will be displayed in the output window.

Direct and Inversely Proportional

Proportionality:

A quantity is said to be proportional to another quantity if change of one of the quantities is always accompanied by the change of the other. This property is known as proportionality.

Proportionality is of two types:

(i)                  Direct proportionality

(ii)                Inverse proportionality

Direct Proportionality:

A quantity is said to be directly proportional to another quantity if the change in both of them is in the same direction. This means that if one of the quantities increases then the other also increases. If one of the quantities decreases then the other also decreases.

It is denotes by the symbol a. If ‘a’ is directly proportional to ‘b’ then:

We write as a a b ==> a = k * b where k is the proportionality constant.

i.e.  a / b = k = constant à a1 / b1 = a2 / b2

Graph:

Let us consider that x a y. If we plot the values of x and y on a graph sheet we obtain the graph showing the relation between these two quantities. Generally, the graph of directly proportional quantities is a straight line. Thus by seeing the graph we can conclude the relation and proportionality between two quantities.

Inversely proportionality:

A quantity is said to be inversely proportional to another quantity if the change in both of them is in the opposite direction. This means that if one of the quantities increases then the other quantity decreases. If one decreases then the other increases.

Inverse proportionality is also uses the symbol a but the reciprocal of the second quantity is written.

If ‘a’ is inversely proportional to b then:

We write as a a 1 / b

You can see that the reciprocal of b is written to indicate inverse proportionality. We can also say that ‘a’ is directly proportional to the reciprocal of ‘b’.

If a a 1 / b à a = k / b where k is the proportionality constant.

i.e. a * b = k = constant à a1 * b1 = a2 * b2

Graph:

Let us consider x is inversely proportional to y i.e. x a 1 / y

Now plot the values of x and y on the on a graph sheet we obtain the graph showing the relation between the two quantities. Generally the graph is not a straight line but a curve. On seeing the graph we can analyze the relation and proportionality between the two quantities.

Problem:

If the volume of a gas at a given temperature is 2 liters when its pressure is 1 bar, then what will its volume when the pressure increases to 3 bars? (Volume in inversely proportional to pressure)

Sol: Given, initial volume v1 = 2 l

Initial pressure p1 = 1 bar

Final pressure p2 = 3 bars

As volume is inversely proportional to pressure we have, p1 v1 = p2 v2

Now v2 = p1 v1 / p2 = (2 * 1) / 3 = 0.66 liters (approx.)

Thus final volume the gas is 0.66l

Properties and Area of a Rectangle

Rectangles
For a normal 4th grader, a rectangle would mean a plane figure that has four sides. However, more precisely in geometry, a branch of math, a rectangle is a special type of a quadrilateral that has 4 right angles. It would look as shown in the picture below:

Properties of a rectangle:
1. It has four sides.
2. It has four angle and all the angles are right angles.
3. It has four vertices.
4. Each pair of opposite sides are congruent.
5. Opposite sides are parallel.
Examples of rectangles:
1. Top of a book.
2. Face of a cuboid.
3. Top of a table.
4. Front of a cupboard.
5. Etc.

The Area of the Rectangle:
The Formula for the Area of a Rectangle is as follows:
A = l * w
Where,
A = area of the rectangle,
L = length of the rectangle
W = width of the rectangle.

It is customary to denote the longer side of the rectangle as length and the shorter side as width. Another custom is to denote the horizontal sides as the length and the vertical sides as the width of the rectangle. It is shown in the picture below.

Let us now try to understand how to calculate the area of a rectangle with the help of a sample problem question.

Example 1: Find area rectangle from the figure shown below:

Solution:
From our formula for area of a rectangle we know that,
Area = A = L * W
For this problem,
L = length = 5 units and
W = width = 3 units
Therefore substituting these values of L and W into the above formula for area of the rectangle we have,
A = 5 * 3 = 15 sq units <- answer="" p="">
If instead of being given the measures of length and width, we are given the co ordinates of the vertices of the rectangle then its area can be found out as follows:

Consider a rectangle with the vertices at A (x1,y1),B  (x2,y2), C (x3,y3) and D (x4,y4) taken in clock wise direction. Therefore we know that if AB is the length of the rectangle then BC would be the width of the rectangle. The distance AB can be found using the distance formula as follows:

Length = L = AB = √[(x2-x1)^2 + (y2-y1)^2]

Similarly the distance BC can also be found using the distance formula as follows:

Width = W = BC = √[(x3-x2)^2 + (y3-y2)^2]

Both the above can be now used to find the area of the rectangle as follows:

A = L * W.

Sample problem:
Find the area of a rectangle having vertices at (3,7), (0,7), (3,-2) and (0,-2)

Solution:
First let us sketch a graph of the said rectangle.

From the picture we see that
Length = L = 3-0 = 3 and
Width = W = 7 – (-2) = 7+2 = 9

Therefore,
Area of rectangle = 3 * 9 = 27 sq units.