Triangular Prism Image

Introduction to triangular prism image

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. Equivalently, it is a pentahedron of which two faces are parallel, while the surface normals of the other three are in the same plane. These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle. (Source: Wikipedia)



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Triangular Prism Image

Triangular prism

Volume of the prism

The method utilized for computing the volume of triangular prism, is identified as,

Volume of prism = Base area x Length of prism

Base area of prism = `(bh)/2`

Wherever, b = base of the triangle

h = height of the prism.

First we can find the bottom area of the triangular prism is to be computed then we have found the volume of the prism.

Examples for Triangular Prism Image

Example 1 for triangular prism image

Compute the volume of the triangular prism, whose base is 10 cm, height is 12 cm and length is 16 cm.

Solution

The method used for computing the volume of prism, is known as,

Volume of triangular prism = Base area x Length of prism

base = 10 cm, height = 12 cm and length = 16 cm

Base area = `(bh)/2`

= `(10xx12)/2`

= `(120)/2`

= 60 cm2

Volume     = 60 x 16

= 960 cm3.

Therefore the volume is 960 cm3

Example 2 for triangular prism image

Compute the volume of the triangular prism, whose base is 15 cm, height is 16 cm and length is 20 cm.

Solution

The method used for computing the volume of prism, is known as,

Volume of triangular prism = Base area x Length of prism

base = 15 cm, height = 16 cm and length = 20 cm

Base area = `(bh)/2`

= `(15xx16)/2`

= `(240)/2`

= 120 cm2

Volume     = 120 x 20

= 2400 cm3.

Therefore the volume is 2400 cm3

Multiplication with Decimals

Introduction : 

Decimal values are used to symbolize fraction numbers. In the multiplication with decimals are writing exclusive of a fraction having denominator and denominator. Decimal values are might be greater than 1 or less than 1. For the model, the fraction 4/10 might be written as the decimal value as 0.4 and marked as four tenths or zero point four. The decimal of 45/10 is 4.5 which can be marked as four and five tenths. In this article we shall discuss about multiplication with decimals.

Sample Problem for Multiplication with Decimals:

To Multiplying Decimals, we require the following steps.

• Multiplying the two values with null decimal point.

• Once we obtain the final answer, make sure to design the decimal point.

• Number of digits at the back the decimal point in the outcome will be equivalent to the total number of digits after the decimal point in the two numbers being multiplied.

Pro 1:  Solving the decimal multiply for the given values 0.04 by 1.6    

Sol :    Start with: 0.04 x 1.6

Multiply without decimal points: 4x 16 = 72

0.04 has 2 decimal places, and 1.6 has 1 decimal place.

So the answer has 3 decimal places: 0.072

Pro 2:  Solving the decimals multiply for the given values 0.08 by 3.2   

Sol :    Start with: 0.08 x 3.2

Multiply without decimal points: 8 x 32 = 256

0.08 has 2 decimal places, and 3.2 have 1 decimal place.

So the answer has 3 decimal places: 0.256

Pro 3:  Solving the decimals multiply for the given values 0.03 by 2.8   

Sol :    Start with: 0.03 x 2.8

Multiply without decimal points: 3 x 28 = 84

0.03 has 2 decimal places, and 2.8 have the 1 decimal place.

So the answer has 3 decimal places: 0.084

Pro 4 :  Solving the decimals multiply for the given values 0.06 by 4.8   

Sol :   Start with: 0.06 x 4.8

Multiply without decimal points: 4 x 48 = 288

0.06 has 2 decimal places, and 4.8 have the 1 decimal place.

So the answer has 3 decimal places: 0.288


Practice Problem for Multiplication with Decimals:

Solving the decimals multiply for the given values 0.12 by 5.3

Ans : 0.636

Solving the decimals multiply for the given values 0.16 by 7.8

Ans : 1.248

Solving Absolute Value Inequalities

Introduction to solving absolute value inequalities:

Solving absolute value inequalities is nothing but the value without considering its sign. Here we are going to learn how we are solving absolute value inequalities. The main difference between the equalities and inequalities is less than and greater than symbol. We will see some example problem for solving absolute value inequalities. For graphing the absolute value inequalities we will use the number lines.

Example Problems for Solving Absolute Value Inequalities:

Solving absolute value inequalities example 1:

Solve the following |x + 5| `gt=` 5

Solution:

Here we are having the inequality |x + 5| `gt= ` 5

If we write these inequalities in absolute value form we will get

x + 5 `gt=` 5 ………… (1) And - (x + 5) `gt=` 5 ………….. (2)

Equation 1:

x + 5 `gt=` 5

Add -5 on both sides we will get

x + 5 – 5 `gt=` 5 – 5

x `gt=` 0

Equation 2:

-(X + 5) `gt=` 5

Add+5 on both sides we will get

- x - 5 + 5 `gt=` 5 + 5

-x > = 10

Divide by -1 so

x `lt= ` -10

So the x lies between 0 and -10

-10 `gt=` x `gt=` 0

We can mark these inequalities using the number line.

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Solving Absolute Value Inequalities Example 2:

Solve the following |x + 3| `gt=` 9

Solution:

Here we are having the inequality |x + 3|` gt= ` 9

If we write these inequalities in absolute value form we will get

x + 3 `gt=` 9 ………… (1) And -(x + 3)` gt= ` 9 ………….. (2)

Equation 1:

x + 3 `gt= ` 9

Add -3 on both sides we will get

x + 3 – 3` gt=` 9 – 3

x` gt=` 6

Equation 2:

- (X + 3) `gt=` 9

Add +3 on both sides we will get

-x - 3 + 3 `gt=` 9 + 3

-x `gt= ` 12

Divide by -1 so

x `lt=` -12

So the x lies between 0 and -12

-12 `gt=` x` gt=` 0

These are the some examples for solving absolute value inequalities. We can mark these absolute value inequalities in number line.

Steps to Graphing Inequalities

Steps to graphing inequalities:

The graph displays the nature of relationship between the variables. One of the most useful graph that we obtain quite often is the graph. ax + by > c and ax + by < c are called  linear inequalities. When we combine the equation ax + by = c with ax + by > c we get                 ax + by `>=` c. Similarly by combining ax + by = c and ax + by < c we get ax + by `<=` c. Here also ax + by`<=` c and ax + by `>=` c are called inequalities.

Step to Draw the Graph of Inequalities:

The following steps are used to draw the graph of inequalities ax + by > c or ax  +  by < c.

Step 1:

Treat the in equation as if it were an equation. That is ax + by = c.

Step 2:

Draw the graph of ax + by = c.

Step 3:

The line ax + by = c divides the plane into two regions. Take any point in one of the regions (usually the origin (0, 0) is taken). Substitute this point in the given in. equalities. If the inequalities are satisfied then the region containing the point is the desired region and it is shaded. If the point does not satisfy the given in equation then the region not containing the point is the desired region and it is shaded.

The shaded region represents the given in equation. All the points in this region satisfy the inequalities and they are the solutions of the in equalities.

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Steps to Graphing Inequalities-example Problem:
Draw the graph of the in equation 2x + y `>=` 10.

Sol:

Draw the graph of 2x + y = 10.

2x + y=10 ? y =10 - 2x

X   1  5  0

y   8  0  10

Plot the points (1,8), (5,0) and (0,10) in the graph and join them to get the straight line 2x + y = 10. Take the point (0,0) and substitute in the in equation 2x + y `>=` 10.0 + 0 > 10 this is false. So the region not containing the point (0,0) is the desired region and it is shaded. The shaded region in the graph represents the in equation 2x + y `>=` 10. All the points in the shaded region satisfy 2x + y `>=` 10 and they are the solutions of 2x + y `>=` 10.

Vertical Line Equation

Introduction of vertical line equation:
On a graph sheet , we can draw infinite number of lines,  for each line we draw, they make an angle θ with x axis. if that angle is 90 degrees then that line will be vertical line, on a vertical line the x coordinate is always same, so the vertical line equation is X=C , where C is the point through which the line passes x-axis. the vertical line test is always parallel to y axis, since the equation of y axis is X=0 , the line parallel to y axis the vertical line equation is X=c.

in the grpah below, there are some vertcal lines which passes throuh 1,3,-2,-5 points on x axis, the vertical line equations are x=1, x=3, x=-2 and x=-5 respectively.

slope of vertical line equation:

slope of a vertical line equation is arc tan(90) which is infinity.and there is no y intercept for the verical line since it does not cut y axis as it is parallel to y axis always.

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Examples on Vertical Line Equations
Q 1:  Find equation of line which passes through (5,0) and parallel to y axis

sol : the vertical line equation is of the form X=c, here c=5

so, th equation of vertical line is x=5.

Q 2 :  Find vertical line equation which passes through (-7,0)

sol :  the vertical line equation is of the form X=c, here c=-7

so, th equation of vertical line is x=-7

Some more Examples on Vertical Line Equation
Q 3:   Find vertical line equation which passes through (7,0)

sol :  the vertical line equation is of the form X=c, here c=7

so, th equation of vertical line is x=7

Q 4:  Find vertical line equation which passes through (-6,0)

sol :  the vertical line equation is of the form X=c, here c=-6

so, th equation of vertical line is x=-6