Quadrilateral Prism Shape

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same (source-Wikipedia). If the prism has four sides then it is called quadrilateral prism. in this article we shall discus about some quadrilateral prism.

Quadrilateral prim shapes:

Cube:

It is one type of quadrilateral prism. Its all dimensions are equal in measure (all side are equal).it has eight vertices. The shape of the cube is shown in below.
Formula for Volume of the cube V= a3 cubic units
Where,
            V = volume of the cube
 a =side length (or edge) of the cube.
Formula for surface area of the cube (A) = 6 a² square units

Rectangle prism:

It is one type of quadrilateral prism. The shape of the cube is shown in below.
Volume of rectangular solid (V) = length x width x height Cubic units
     = l x w x h cubic unit.
Total surface area of rectangular prism
                                    = 2 (length x breadth + breadth x width + length x height)
                                    =2(lb + bh+ lh) square units

Trapezoidal prism:

It is one type of quadrilateral prism.it has one set of parallel sides. The shape of the trapezoidal prism is shown in the figure.
Volume of trapezoidal prism = l x area of the base cubic units
                                                       l – Length of trapezoidal prism
                              Area of the base= 1/2 x (a + b) x h square units
                                                       a and b are length of parallel sides.
                                                       h – Height

Quadrilateral prism shape - Example problems:

1. Find the volume and surface area of the cube whose side length is 17 cm.
Solution:
            Given:
                        Side length (a) = 17 cm
Formula:
            Volume of the cube V= a³ cubic units
                                                = 17³
                                                = 17 x 17 x 17
                                                = 4913 cm³
            Surface area of the cube (A) = 6 a² square units
                                                          = 6 x 17²
                                                          = 6 x 289
                                                          = 1734 cm²

2. Find the lateral surface area and total surface area of a rectangular whose length is 6 cm, width 3cm and height 6cm.
Solution:
Given:
                              Length (l) = 6 cm
                         Width (w) = 3 cm
                           Height (h) = 6 cm
Total surface area of rectangular prism =2(wh + lw+ lh) square units
                                                               = 2(3 x 6+ 6 x 3 +6 x 6)
                                                               = 2 (18 + 18 + 36)
                                                               = 2 (72)
                                                               = 144
Total surface area of rectangular prism = 144 cm²
            Volume of the rectangular prism (V) = l x w x h cubic units
                                                                           = 6 x 3 x 6
                                                                           = 108
                                                                           = 108 cm³
3.figure out the volume trapezoidal prism whose length 14 cm, height 5cm, length of parallel sides a=7 cm and b=4cm.
Solution:
Given:
                  Length (l) = 20 cm
                  Height (h) = 5 cm
                  Parallel sides a=7 cm and b=4 cm
Formula:
Volume of trapezoidal prism = l x area of the base cubic units

Area of the base:
Area of the base= 1/2 x (a + b) x h
                                                   = 1/2x (7 + 4) x 5
                                                   =1/2 x 11 x 5
                                                     = 27.5 cm²
Volume of trapezoidal prism = 20 x 27.5
                                                  = 550

Volume of trapezoidal prism = 550 cm³

Figuring Volume of a Structure

Volume of structure
Volume is how much three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Volume is measured in "cubic" units.


Volume of Cylinder
A cylinder is a solid that has two parallel faces which are congruent circles. These faces form the bases of the cylinder. The cylinder has one curved surface. The height of the cylinder is the perpendicular distance between the two bases.
The volume of a cylinder is given by the formula:
Volume = Area of base × height

V =   r²h where r = radius of cylinder and h is the height or length of cylinder.

Volume of hollow cylinder

Volume of hollow cylinder
V=πR²h-πr²h
Where R is the radius of the outer surface and r is the radius of the inner surface.

Volume of Cone

A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex at the top. The height of the cone is the perpendicular distance from the base to the vertex.

Volume of cone = 1/3 Area of base × height
V = 1/3πr²h where r is the radius of the base and h is the height of the prism.

Volume of Pyramid

A pyramid is a solid with a polygonal base and several triangular lateral faces. The pyramid is named after the shape of its base. For example, rectangular pyramid, triangular pyramid.
The lateral faces meet at a common vertex. The height of the pyramid is the perpendicular distance from the base to the vertex.

The volume of a pyramid is given by the formula:

Volume of pyramid =  1/3Area of base × height

V = 1/3Axh where A is the area of the base and h is the height of the pyramid.

Algebra Data Analysis Help

he Algebra data analysis is one of the important branch of algebra concerning the learn of rules of operations and dealings. The constructions and concepts arising from algebra data analysis. Algebra data analysis including terms, polynomials, expressions, quadratic formula and algebraic structures or pre-algebra structure. We will solve examples in polynomial equations and quadratic formula. This article has information about algebra data analysis help as well as examples.

Polynomial equations – Algebra Data Analysis Help:

In algebra data analysis, the polynomial equations are one of the most significant factors.
Now we will solve the polynomial problem in algebra data analysis with help of example problem.

Example Problem 1:
Solve given polynomial equations.
6x² + 4 + 9x + 3x² + 3x + 9 + 7x
Solution:
Step 1:
First we have to mingle terms x²
6x²+3x²=9x²
Step 2:
Now combine the terms x
9x + 3x + 7x = 19x
Step 3:
Then join the constants terms
4 +9 =13
Step 4:
Finally, combine all the terms
9x² + 19x +13
So, the final answer is 9x² + 19x +13

Example Problem 2:
Solve following polynomial expression.
11x² + 2 + 9x + 13x² + 5x + 4 + 3x
Solution:
Step 1:
First we have to mingle terms x²
11x²+13x²=24x²
Step 2:
Now combine the terms x
9x + 5x + 3x = 17x
Step 3:
Then join the constants terms
2 + 4 =6
Step 4:
Finally, combine all the terms
24x² + 17x +6
So, the final answer is 24x² + 17x +6

Quadratic formula - Algebra Data Analysis Help:

In algebra data analysis, quadratic equation with real or complex coefficients contains two solutions, is known as roots.
The roots are given by the quadratic formula

Example of quadratic formula:
Example Problem1:
Solve 7x² + 5x = -3.
Solution:
Now we will find the standard form for the quadratic equation and calculate a, b, and c.
7x² + 5x + 3 = 0
a = 7
b = 5
c = 3

Now applying the formula,


= `(-5 +- sqrt(5^2 - 4 xx 7 xx 3))/(2 xx 7)`
=`(-5+-(sqrt(25-84)))/(2xx7)`
=`(-5 +-(sqrt(-59)))/(14)`
=`(-5+- i 7.68)/14`
=2.68 `+-i` -7.68

Answer:
2.68 `+-i` -7.68
These are examples in algebra data analysis help.

That's all about algebra data analysis help.

Measurement of Angle

• Line joins two positions and which are nonstop in two directions. Lines have one endpoint of two lines and infinite in one direction is called angle.
•  The grouping point of two lines is called the vertex.

               

•  Angles are denoted as degrees. In measurement of angles can diverge in between 0 and 180 angles.

Concept -measurement Angle:

Concept of angles:

      There are three major categories of angles: right angle, obtuse angle, and acute angle.

• A right angle – a measurement of an angle is 90 degrees.
• An acute angle - a measurement of an angle in between 0 to 90 degrees.
• An obtuse angle - a measurement of an angle in between 90 and 180 degrees.

More about angles and measurement of angle:

More about angles:

1)      Vertical angle measurements are two angles with a common vertex and with sides that are two pairs of opposite of opposite rays. (That is, the union of the two pairs of sides is two lines.)  

                              

2)       A right angle measurements has a measure of 90^o.    

     

                       

3)      An acute angle measurement is an angle whose measure is larger than 0^o and smaller than 90^o.  

          

                

4)      An obtuse angle measurement has a measure larger than 90^o and less than 180^o.  

           

               

5)     Straight angle measurement has a measure of 180^o. The rays of a straight angle lie on a line.

6)      Complementary angle measurements are two angles, the addition of whose measures is 90^o.

               

7)      Supplementary angle measurements are two angles, the addition of whose measures is 180^o.

               

Step to measurement of angle:

          

Step 1: To find the angle is obtuse or acute before using the protractor.

          

Step 2: Position the protractor on the center mark on the vertex of the angle

          

Step 3: position the protractor, line up with one ray of the angle and another ray of the angle cross the protractor’s scale

          

Step 4: to check the measure of the angle of the line that crosses the protractor scale.

          

How to calculate volume

Example problems:

Here are some solved examples on How to calculate volume:
Problem 1:

Problem 2:
Calculate the volume of the cone with base radius = 5cm and height at the apex is 4 cm.
Solution:
Volume of a cone: (13)(π)(r2)h
                                 = (13)(π)(52)4
                                = 104.666 cm^3
Problem 3:

Problem 4:
Calculate the length of the rectangular prism, when the volume is 36 cm^3 , width = 3 m, height is = 4m
Solution:
Volume of the rectangular prism = l * w * h.
                                                     36 =  l* 4*3
                                          Therefore,  l= 3m
Students can understand How to calculate volume from the above examples and learn to solve on their own.

Parametric t Test

Conventional statistical procedures are also called parametric tests. In a parametric test a sample statistic is obtained to estimate the population parameter. Because this estimation process involves a sample, a sampling distribution, and a population, certain parametric assumptions are required to ensure all components are compatible with each other. 

For example, in Analysis of Variance (ANOVA) there are three assumptions:

• Observations are independent.
• The sample data have a normal distribution.
• Scores in different groups have homogeneous variances.

In a repeated measure design, it is assumed that the data structure conforms to the compound symmetry. A regression model assumes the absence of collinearity, the absence of auto correlation, random residuals, linearity...etc. In structural equation modeling, the data should be multivariate normal.

Why are they important? Take ANOVA as an example. ANOVA is a procedure of comparing means in terms of variance with reference to a normal distribution. The inventor of ANOVA, Sir R. A. Fisher (1935) clearly explained the relationship among the mean, the variance, and the normal distribution: "The normal distribution has only two characteristics, its mean and its variance. The mean determines the bias of our estimate, and the variance determines its precision." (p.42) It is generally known that the estimation is more precise as the variance becomes smaller and smaller.

Put it in another way: the purpose of ANOVA is to extract precise information out of bias, or to filter signal out of noise. When the data are skewed (non-normal), the means can no longer reflect the central location and thus the signal is biased. When the variances are unequal, not every group has the same level of noise and thus the comparison is invalid. More importantly, the purpose of parametric test is to make inferences from the sample statistic to the population parameter through sampling distributions.

When the assumptions are not met in the sample data, the statistic may not be a good estimation to the parameter. It is incorrect to say that the population is assumed to be normal and equal in variance, therefore the researcher demands the same properties in the sample. Actually, the population is infinite and unknown. It may or may not possess those attributes. The required assumptions are imposed on the data because those attributes are found in sampling distributions. However, very often the acquired data do not meet these assumptions. There are several alternatives to rectify this situation.

Free Sample Trinomials

Trinomials:

In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.(source : WIKIPEDIA)

The trinomial must be one of the following form .

Examples for free sample trinomial:

1. 9x + 3y + 5z , where x , y, z are variables.

2. xy + x + 2y, Where x, y are variables.

3. x2+x-8, where x is variable.

4. ax + by + c = 0 , Where a,b,c are constants and x,y are variables.

Our  tutor vista website provide opportunity to learn about sample trinomials with free of cost. In this article we are going to see some sample problems on solving and factoring trinomials.

Free sample problems on trinomials:

Problem 1:

Square the following trinomial,

x+3y-2

Solution:

Given, x+3y-2

We need to find the square for the given trinomial,

That is (x+3y-2)2

We can apply the following formukla to find the square for the above trinomial,

( a + b + c)2 = a2+b2+c2+2ab+2bc+2ca

(x+3y-2)2  = x2+(3y)2+22+2(x)(3y)+2(3y)(-2)+2(-2)(x)

= x2+9y2+4+6xy-12y-4x

= x2+ 9y2+ 6xy - 4x -12y + 4

Answer: (x+3y-2)2 = x2+ 9y2+ 6xy - 4x -12y + 4

Problem 2:

Factor the trinomail x2 + x – 156 .

Solution:

Given , x2 + x – 156 .

- 156 (product)

/     \

-12     13

\    /

1 (sum)

= x2 - 12x + 13x -156

= x ( x - 12 ) + 13 ( x -12 )

= (x+13) (x-12)

Answer: (x+13) and  (x-12) are the factors of the given trinomial.

Problem 3:

Solve the trinomial 2x2 - 2x = 12.

Solution:

Given, 2x2 - 2x = 12.

Subtract 12 on both sides,

2x2 - 2x - 12 = 12 - 12

2x2 - 2x - 12 = 0

2(x2 - x - 6 ) =0

Divide by 2 on both sides,

x2 - x - 6 = 0

-6

/  \

-3  2

\  /

-1

x2 -3x + 2x - 6 = 0

x(x - 3) + 2(x-3) = 0

(x+2) (x-3) = 0

(x+2) = 0

x = -2

(x-3) = 0

x =3

Answer: x = 3 , -2

Practice problems on free sample trinomials:

Problems:

1.Find the roots of the trinomial x2 - 5x = -6

2.Factor the trinomial 2x2 + 12 x - 14

Answer Key:

1. x = -1 , x = 6

2. ( x - 1) and (x + 7)