Pyramid of Biomass

Pyramid of Biomass is the graphic representation of biomass of the successive trophic levels of an Eco system. Biomass refers to the total weight of dry matter present in the ecosystem at anyone time. The pyramid of biomass indicates the gradual reduction in biomass at each trophic level from base to top. The total biomass of producers is much more than the total biomass of herbivores. Like wise the total biomass of the third level will be less than the herbivores and so on. The pyramid of biomass are generally upright, but in case of parasites, the pyramid of biomass is inverted. Similarly, if organisms of lower levels are much smaller than those of higher levels, biomass pyramid may be inverted.                                                            

Examples of Pyramid of Biomass.1

Pyramid of Biomass in a Marine ecosystem

In the marine Eco system the Phytoplankton occupy the basic trophic level and the next trophic level is occupied by the Zoo plankton, the third and fourth are occupied by carnivores such as worms, and molluscs and carnivores fishes respectively. Here too the biomass falls in a graded manner from the basic trophic level to higher trophic level.

Examples of Pyramid of Biomass.2

Pyramid of Biomass in  a  grass land 

In a grass land the biomass of grasses is the maximum and it gradually decreases towards the consumer level. That is  the biomass is decrease in this order. Grasses have higher biomass than that of snake  and snake have higher biomass than that of Hawk.

                                                 Grass ------------> Snake -------------> Hawk

How to Rewrite Radicals

Radicals are nothing but a root which we also called square root. Square root is indicated with the symbol (sqrt) and sqrt (). Normally, radicals are rewrite like, sqrt of (a) is rewrite as a ^(1/2). Likewise radicals are expressed in various forms. The expression sqrt (8) is read as “radical eight”, or “the square root of eight”. Thus, we are going to see how to rewrite radicals in different ways.

Formula for how to rewrite radicals:

Formula for how to rewrite radicals:
General expression with exponent and radical:
                                          `( ^nsqrt (a) ) ^m`   = `( ^nsqrt (a) ) ^m`   = `(a1/n) ^m ` = `am/n`                         
Multiplication property for radical expression:  (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
Division property for radical expression:  ( `^nsqrt (a/b)` ) = (  `^nsqrt (a)` ) / (  `^nsqrt (b)`   )
Different forms of radicals is,
                    A square (second) root is written as (  ` sqrt(x)` )

     A cube (third) root is written as (  `^3sqrt(x)`  ),

     A fourth root is written as (  `^4sqrt(x)`   ),

     A fifth root is written as:  (  ` ^5sqrt(x)` ).

Example for how to rewrite radicals:

Example for how to rewrite radicals: Rewrite radical(`sqrt (1225)` )
Given: (`sqrt (1225)` )
Solution: Given question says, radical (1225),
      When, we take radical for 1225, we obtain 25*49.
Because,                 (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
                                       `sqrt (1225)` = `sqrt (25)` `sqrt (49)`
                                                        = `sqrt(5)`  * `sqrt(7)` 
                                       `sqrt (1225)` = `35`
Thus, we can do how to rewrite radicals in the prefered way.

Example for how to rewrite radicals: Rewrite radical( `^3sqrt (512)` )
Given: (  `^3sqrt (512)` )
Solution: Given question says, cubic root of (512),
      When, we take cubic root for 512, we obtain 8.
Because,                           `8*8*8 = 512`
                                          `^3sqrt(512)`` =` `8^3`
  Therefore, cubic root for (512) = 8^3
Thus,we can do how to rewrite radicals in the prefered way

Example for how to rewrite radicals: Rewrite radical `^3sqrt (729)`
Given: (`^3sqrt (729)` )
Solution: Given question says, cubic root of (729),
      When, we take cubic root for 729, we obtain 9.
   Because,                         `9*9*9 = 729`
                                         `^3sqrt(729) = 9^3`
     Therefore, cubic root of (729) = 9^3


Example for how to rewrite radicals: Rewrite radical `^4sqrt (1296)`
Given: (`^4sqrt (1296)` )
Solution: Given question says,  fourth root of (1296),
      When, we take fourth radical for 1296, we obtain 6.
Because,                             `6*6*6*6 = 1296`
                                             `^4sqrt(1296 )= 6^4`
     Therefore,   fourth root of (1296) = 6^4

Example for how to rewrite radicals: Rewrite radical `^5sqrt (3125)`
 Given: (`^5sqrt (3125)` )
Solution: Given question says, fifth root of (3125),
      When, we take Fifth radical for 3125, we obtain 8.
Because,                       `5*5*5*5*5 = 3125`
                                             `^5sqrt(3125) = 5^5`
     Therefore,    fifth root of (3125) = 5

           

Math Scientific Expressions:

In this article we shall discuss about math scientific expressions. Scientific notation expressions are meant by transform very big numbers into simplest form of a number. That is called as scientific notation expressions to integer. Scientific notation expressions is nothing but exponential of the base integer 10 have expressed in the number scientific notation expressions. Example for scientific notation expressions is 785000000 this can be note down as 7.85 x 10⁸. Here, 8 is the power of 10.   

Example problems based on math scientific expressions:

The example problems based on math scientific expressions to integer is given below that,

• Example 1:

How do you write 67000 in math scientific expressions?
Solution:
Step 1:

The given number is 67000.

Step 2:

The given number is not a big number, but it is simple to convert scientific notation.

First rounding 6.7 x 10⁴ = 67000 [10⁴ = 10000]  

Step 3:

The concluding answer for math scientific expressions is 6.7 x 10⁴.

• Example 2:  

How do you write 2800000 in math scientific expressions?
Solution:
Step 1:

The given number is 9100000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 9.1 this is not equivalent to the given number.

Step 3:

Here, 6 digits subsequent the decimal point.

So, 9.1 × (1000000)

Therefore the rounding will be 10⁶.

Step 4:

The concluding answer for math scientific expressions is 9.1 x 10⁶

• Example 3:

How do you write 843000 in math scientific expressions?
Solution:
Step 1:

The given number is 843000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 8.43 this is not equivalent to the given number.

Step 3:

Here, 5 digits following the number are there so we can write 10⁵.

Step 4:

The concluding answer for math scientific expressions is 8.43 x 10⁵.

.

Practice problems based on math scientific expressions:

The practice problems based on math scientific expressions is given below that,

• Problem 1:

How do you write 9510000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 9.51 x 10⁶.

• Problem 2:

How do you write 852000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 8.52 x 10⁵.

• Problem 3:

How do you write 75000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 7.5 x 10⁴.

Just Math Answer

Some of the main branches of mathematics are algebra, geometry and calculus. Various number of problems exists in these branches.
In this article of just math answer, several problems related to various branches of mathematics are given with their respective answer. In addition, several practice problems with answer key are given for better understanding.


Math Example problems with answers:

Example 1:

Find the Mean of the given data set: { 2, 6, 19, 29 }

Solution:

Mean   =  ( 2 + 6 + 19 + 29) / 4

=  56 / 4

= 14

Answer: Mean  = 14

Example 2:

Find the volume of cylinder given the radius is 13 cm and 32 cm.

Solution:

Volume of cylinder = `pi` r2 h cubic units.

= (3.14) * 132 * 32

=  3.14 * 169 * 32

= 16981.12 cm3

Answer: Volume of cylinder = 16981.12 cm3

Example 3:

Find the area of the square with the side length of 9 cm.

Solution:

Area of square  =  ( a ) 2

= ( 9 )2

= 9 * 9

= 81 cm2

Answer: Area of square =  81 cm2

Example 4:

Find the area of a triangle through base of 15 m and a height of 8 m.

Solution:

Area of a triangle = ½ b h

= ½ ( 15 ) ( 8 )

= 0.5 * 15 * 8

= 60 m2

Answer: Area of triangle = 60 m2

Example 5:

Solve :  6 ( x + 3 ) = 72

Solution:

6( x + 6)  =  72

6x + ( 6 x 6 )  =  72

6x + 36  =  72

6x  =  36

x  =  6

Answer:  x =  6

Example 6:

Find lateral surface area of cone with radius 5 cm and slant height 16 cm.

Solution:

Given:  Radius r = 5 cm

Slant height l =16 cm

Lateral surface area  = `pi` r l   square units.

= 3.14 x 5 x 16

= 251.2 cm2

Answer: Lateral surface area = 251.2 cm2



Example 7:

Find the range of the data set { 25, 42, 36, 21, 48 }

Solution:

Here, The maximum value is 48

The minimum value is 21

Range  =  Maximum value – Minimum value

=  48 – 21

= 27

Answer: Range  = 27


Math Practice problems with answer:

1) Find the range of the data set { 15, 43, 29, 21, 14 }

Answer: 29

2) Find the area of the square with the side length of 11 cm.

Answer: 121 cm2

3) Find the Mean of the given data set: { 8, 16, 20, 26 }

Answer: 17.5

4) Find lateral surface area of cone with radius 8 cm and slant height 18 cm.

Answer: 452.16 cm2

Practice Divide Monomials

A polynomial is an algebraic expression with literal. Mostly we use x, y literal in polynomials. Literals are also called as variables. In polynomial, the variables have only positive integral exponents.

For example,  15 + 2x + x2 , 7x3 + 5xy2 + 12 y3 .

The first example is polynomials in one variable x.

The second example is polynomials in two variables x and y.

Polynomials having only one term are known as monomials. Monomial is one of the type of polynomials.  For example,

x3 , x2  x , y5, 5x5, 6y3 .

Let us practice division of a monomial problems.



Divide monomial problems for Practice:

There are two rules for dividing a monomial by a monomial. They are following,

Rule 1: The coefficient of the quotient of two monomials is equal to the quotient of the coefficients of the monomials in question.

Rule 2: In the quotient of two monomials, the variable part is equal to the quotient of the variable parts in the monomials in question.

Problem 1:

Divide by x5 by x3 monomials.

Solution:

x5 ÷ x3 = `x^5/x^3 `

= x2 .

Problem 2:

Divide by 15x5 by 5x4 monomials.

Solution:

15x5 ÷ 5x4 = `(15x^5)/(5x^4) `

= 5x .

Problem 3:

Divide by -20x4 by 10x monomials.

Solution:

- 20x4 ÷ 10x = `(-20x^4)/(10x) `

= `(-20/10)`` (x^4/x)`

= - 2 x3 .

Problem 4:

Divide by 3y3 by `sqrt(3)` y monomials.

Solution:

3y3 ÷ `sqrt(3)` y = `(3y^3)/(sqrt(3)y) `

= `(3/sqrt(3))`` (y^3/y)`

= `sqrt(3)` y2.

Problem 5:

Divide by 2x2 by 2x monomials.

Solution:

2x2 ÷ 2x      = `(2x^2)/(2x) `

= `(2/2)`` (x^2/x)`

=  x .

Problem 6:

Divide by -3x3 by x2 monomials.

Solution:

- 3x3 ÷ x2 = `(-3x^3)/(x^2) `

= `(-3/1)`` (x^3/x^2)`

= - 3 x .

Practice Problems on monomials:

Practice Problem 1:

Divide by `2/3` x2 by x monomials.

Answer:

`2/3` x .

Practice Problem 2:

Divide by `sqrt(5)` x4 by 5x3 monomials.

Answer:

`x/sqrt(5)` .

Practice Problem 3:

Divide by `sqrt(3)` a3 by 2a monomials.

Answer:

`(sqrt(3)a^2)/2` .

Practice Problem 4:

Divide by 4a4 by - 2 `sqrt(2)` a2 monomials.

Answer:

`-sqrt(2)a^2` .

Multipling and Dividing

The result of multiplying is the total number (product) that would be obtained by combining several (multipling) groups of similar size (multiplicand). Division is clear as a mathematics connotation which is the opposite procedure of multiplication. From the method of division the amount or relation of 2 numbers can be calculate. Sign of multiplying is ‘*’.The same result can be obtain by recurring multiplying. Otherwise the process of find how many period of one number is including in an additional one. Sign of division is ‘/’ or ‘÷’.If we are combining 6 groups with 3 objects in each group, we could arrive at the same answer by addition. For example, 6+6+6=18 is equivalent to the multiplication equation 6*3=18.

Descriptions about multipling and dividing:-

Steps for multiplying and dividing problems,
Step1:-
Multiplying and division of two digits by the similar signs will be positive sign
• (+ sign) ÷ (+ sign) = (+ sign).
• (- sign) ÷ (- sign) = (- sign).
• (+ sign) * (+ sign) = (+ sign).
• (- sign) * (- sign) = (+ sign).
Step2:-
Multiplying and division of two digits by the dissimilar signs will be negative
• (+ sign) ÷ (- sign) = (- sign).
• (- sign) ÷ (+ sign) = (- sign).
• (+ sign) * (- sign) = (- sign).
• (- sign) * (+ sign) = (- sign).

Example problems for multiplying and dividing:-

Problem1:-
Solve 24 dividing by 8
Solution:-
24 ÷ 8
= 3.

Problem2:-
Solve (-12) dividing by (-2)
Solution:-
(-12) ÷ (-2)
= 6

Problem3:-
Solve 4 dividing by (-2)
Solution:-
4 ÷ (-2)
= -2

Problem4:-
Solve (-9) dividing by 9
Solution:-
(-9) ÷ 9 = -1

Problem5:-
Solve 12 multipling by 3
Solution:-
12*3
=36.

Problem6:-
Solve (-10) multiplying by (-2)
Solution:-
(-10)*(-2)
=20.



Problem7:-
Solve 14 multiplying by (-4)
Solution:-
(14)*(-4)
=-56.

Problem8:-
Solve (-12) multiplying by 3
Solution:-
(-12)*(3)
=-36. 

Frequency Distribution Data

A Frequency Distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data e.g. to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc. Some of the graphs that can be used with frequency distributions are histograms, line graphs, bar charts and pie charts. Frequency distributions are used for both qualitative and quantitative data.

(Source: Wikipedia)

Types of Frequency Distribution data:

Different types of frequency distribution data are,

• Univariate frequency tables
• Joint frequency distribution

Univariate distribution tables:

                It is a list of values that can be ordered by the quantity. It can show the values for each value appear for number of times.

Joint frequency distribution:

                 It is used as two-way tables. It is also called as bivariate joint frequency distribution.

Example problem for Univariate frequency distribution data:

Example 1:
Construct the univariate frequency distribution table for the given data. For the following students in a class have marks in the exam.
Students scored marks in between 31-40 are 5
                                                        41-50 are 12
                                                        51-60 are 9
                                                        61-70 are 15
                                                        71-80 are 7
                                                        81-90 are 4
                                                        91-100 are 2
Solution:
         Determine the range:
                      100 – 31 = 70
         Determine the intervals:
                   Choose the interval as 10
         Construct the univariate frequency distribution table.

Marks	No of students	Cumulative frequency
31-40	5	5
41-50	12	17
51-60	9	26
61-70	15	41
71-80	7	48
81-90	4	52
91-100	2	54

Joint frequency distribution example problem:

Example 2:
In a school, boys and girls are participated in different sports competition that can be given. Using that set of values construct the joint frequency distribution.
Boys and girls are participated in running, long jump, and volley ball.
In running – 10 boys and 8 girls
In long jump – 8 boys and 7 girls
In volley ball –12 boys and 12 girls are participated.
Solution:
          Construct a joint frequency distribution table for the given set of data.

Joint frequency	Boys	Girls	Total
Running	10	8	18
Long jump	8	7	15
Volley ball	12	12	24
Total	30	27	57