Volume Divided by Area

Volume is how much three-dimensional space a substance (solid, liquid, gas, or plasma) or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic meter.

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The surface area of a 3-dimensional solid is the total area of the exposed surface. (Source: From Wikipedia).

Use of Volume Divided by Area:

The term volume divided by area is used to find the a dimension of a solid.

For example,

In a sphere, volume divided by area gives the radius of the sphere.

Volume of sphere = `4/3 pi r^3`

Surface area of sphere = `4 pi r^2`

Volume divided by area = `(4/3 pi r^3)/(4 pi r^2)`

= `r/3`

In a cube, volume divided by area gives the side of the cube,

Volume of cube = a^3

Surface area of cube = 6a^2

Volume divide by area = `(a^3)/(6a^2)`

= `a/6`

In a cylinder, volume divided by area gives the radius of the cylinder

Volume of cylinder = `pi r^2 h`

Surface area of cylinder = `2 pi r h`

Volume divided by area = `(pi r^2 h)/(2 pi r h)`

= `r/2`

Example Problems for Volume Divided by Area:

Here, we are going to see some example problems to find the dimensions of solid shapes using volume divided by area.

Example 1

Find the side of a cube whose volume is 125 cubic meter and the surface area is 150 square meter.

Solution

For a cube, Volume divide by area = `a/6` = `125/150`

`a/6` = `5/6`

a = 6

So, the side of the given cube is 5 meters.

Example 2

Find the radius of a sphere whose volume is 2393.88 cubic centimeter, and the surface area is 865.26 square centimeter.

Solution

We know that, the volume divided by area in a sphere gives `r/3`

Here, volume of the sphere = 2393.88 cubic centimeter

Surface area of the sphere = 865.26 square centimeter

Volume divided by area; `r/3` = `(2393.88)/(865.26)`

r = `(2393.88)/(865.26)` * 3

r = 8.3

So, the radius of the given sphere is 8.3 centimeter.

Example 3

The volume and lateral surface area of a cylinder are 565.2 cubic feet and 188.4 square feet respectively. Find the radius of the cylinder.

Solution

We know that, the volume divided by area in a sphere gives `r/2`

Volume of the cylinder = 565.2 cubic feet

Lateral area of cylinder = 188.4 square feet

Volume divided by area; `r/2` = `(565.2)/(188.4)`

r = `(565.2)/(188.4)` * 2

r = 6

So, the radius of the cylinder is 6 feet.

Solving Trigonometric Examples

Answering the trigonometric examples is nothing but we are solving the trigonometric functions and trigonometric identities. Here we will take trigonometric functions and equations. Trigonometric examples contain trigonometric functions. In trigonometric model we will solve the Sin, Cos, Tan Identities. We can find the angles from these identities. We will solve some trigonometric examples.

Explanation for Solving Trigonometric Examples:

Ex 1:     Solve the following trigonometric equation Cos4A – Sinn 2A =0

Sol :      Cos 4A – sin2A =0

2Sin2 (2A) + Sin (2A) – 1 = 0

Here we can use quadratic formula to find A value. Let us take any variable equal to Sin 2A

Let us take y = Sin 2A

2y2 + y – 1 =0

2y2+2y – y – 1 = 0

2y(y + 1) – (y + 1) = 0

If we factor this we will get two values for y.

(y + 1)(2y – 1) = 0

Now y + 1 = 0 2y – 1 = 0

Now plug y = Sin2A

Sin 2A + 1 = 0                                      2Sin2A – 1 =0

Sin 2A = -1                                          2Sin 2A = 1

2A = Sin-1 (-1)                                       Sin 2A =

2A = 270                                             2A = Sin-1

2A = 30

A = 135                                                      A = 15

From this we will get two value for A.

Practice Problem for Solving Trigonometric Examples:

Ex 2:          Solve the assessment of the following trigonometric identity Sin 75 - Cos 15

Sol :           Sin 75 – Cos 15

Here we have to use sum and variation formula to find the value os Sin 75 - Cos 15

Sin (45 + 30) – Cos (45 - 30)

Sin (A + B) = Sin A Cos B + Cos A Sin B

Cos (A - B) = Cos A Cos B + Sin A Sin B

Here A = 45

B = 30

Sin (45 + 30) = Sin45.Cos30 + Cos45Sin30

= 0.7071 * 0.8660 + 0.7071 * (0.5)

= 0.6123 + 0.3536

Sin 75 = 0.9659

Cos (45 – 30) = Cos45Cos30 + Sin45Sin30

=0.7071 * 0.8660 + 0.7071 * (0.5)

=0.6123 + 0.3536

Cos 15 = 0.9659

Now plug the values in the equation is

Sin 75 – Cos15 = 0.9659 – 0.9659

Sin 75 – Cos15 = 0

Ex 3:               Solve for x Sin x = 0.5, Cos x = 0.8660

Sol :             (I)  Given    Sin x = 0.5

x = Sin-1 (0.5)

x = 30o

(II)    Cos x = 0.8660

x = Cos-1 (0.8660)

x = 300

So from this angle x =30o. Here we use the opposite trigonometric functions to find the value of x.

Solving Simple Linear Equations

Solving simple linear equation involves the process of the solving basic linear equations in simple method. Linear equations come under the category of linear algebra whereas linear algebra is defined as the process of calculating system of unknown variables with the help of known things. Simple linear equations have relations with the families of vectors called vector or linear spaces. The following are the simple linear equations examples for solving.

Simple Linear Equations Examples for Solving:

Example 1:

Solve the simple linear equation to find the variable value.

-2(h - 1) – 4h - 1 = 3(h + 2) – 4h

Solution:

Given expression is
-2(h - 1) – 4h - 1 = 3(h + 2) – 4h

Multiplying the integer terms
-2h + 2 – 4h - 1 = 3h + 6 – 4h

Grouping the above terms
-6h + 1 = -h + 6

Subtract 1 on both sides
-6h + 1 - 1 = -h + 6 -1

Group the above terms
-6h = -h + 5

Add h on both sides
-6h + h = h + 5 -h

Group the above terms
-5h = 5

Multiply -1/5 on both sides
h = - 5/5

h = - 1 is the solution.

Example 2:

Solve the simple linear equation to find the variable value.

-4(h + 3) = h + 9

Solution:

Given expression is
-4(h + 3) = h + 9

Multiplying the integer terms
-4h - 12 = h + 9

Add 12 on both sides
-4h - 12 + 12 = h + 9 + 12

Grouping the above terms
-4h = h + 21

Subtract h on both sides
-4h - h = h + 21 -h

Grouping the above perms
-5h = 21

Multiply -1/5 on both sides
h = -21/5

h = -21/5 is the solution.

Simple Linear Equations Practice Problems for Solving:

1) Solve the simple linear equation to find the variable value.

-3(h - 2) – 2h - 3 = 2(h + 1) – 4h

Answer:  h = -1/3 is the solution.

2) Solve the simple linear equation to find the variable value.

-2(h + 3) = h - 1

Answer:   h = - 5/3 is the solution.

Explanation for Multiples and Factors of an Integers

Introduction to multiples and factors of an integers:
                        In math, the natural numbers are form the integer and another name of number is an integer. The factor is divisor of a given number. This divisor is divides the given integer without any remainder. The factors may be two or more in an integer. The multiple is a one quantity in multiplication. Now we are going to see about sum of factors of a number.

Explanation for Multiples and Factors of an Integers

Multiples of an integers:

                          If we multiply the one integer with another integer means that first integer is called as multiples. For example, x. y = x is multiple of y that is 2 x 3 = integer 2 is a multiple of integer 3.

Properties of multiple:

• The integer zero is common multiple of all integers.
• If we multiply any integer with 1 means that any integer is multiple of 1.

Factors of integers:

                             The factors are referred as divisors and each integer contains more than one factors. Types of factors are,

• Prime factors – It define the prime numbers (only two divisors).
• Composite factors – It define the composite number (two or more factors).

More about Multiples and Factors of an Integers

Example problems for multiples and factors of an integers:

Problem 1: Find out the multiples of given number.

8

Answer:

The given integer is 8.

The multiples of given integer is 8 x 1 = 8

                                                     8 x 2 = 16

                                                     8 x 3 = 24

                                                     8 x 4 = 32

Therefore, the multiples of an integer 8 are 8, 16, 24, 32…….

Problem 2: Find out the factors of given integer.

42

Answer:

The given integer is 42.

Factors of an integer 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

Example problems for multiples and factors of an integers:

1. Determine the factors of an integer 20.

Answer: Factors are 1, 2, 4, 5, 10 and 20.

2. List the multiples of an integer 11.

Answer: The multiples of an integer are 11, 22, 33, 44….

Fraction least to Greatest

Introduction to fraction least to greatest:

   A fraction number value is one part of the whole number value in decimals.  A Fraction number value consisting of a two division in the number. The one part is top place of a number value is called as a numerator value. Another part is bottom place of the number value is called as a denominator value. That is numerator value of the fraction number divided by a denominator value of the fraction number. In this article we shall discuss about fraction least to greatest.

Sample Problem for Fraction least to Greatest:

This problem shows proper fraction for least to greatest:

Problem 1:

Find the value of given fraction numbers `(2)/(5)` + `(3)/(5)`

Solution:

   In the proper fraction a denominator values are same. So we directly add or subtract the numerator value.

Step 1: In the denominator values are same.

Step 2: Add the numerator values and put over the same denominator values.

       `(2)/(5)` + `(3)/(5)` = `(2 + 3)/(5)`

                 = `(5)/(5)`

Step 3: Now we simplify the fraction values.

               = 1

Problem 2:

Find the value of given fraction numbers `(2)/(9)` + `(7)/(9)`

Solution:

   In the proper fraction a denominator values are same. So we directly add or subtract the numerator value.

Step 1 Here  denominator values are same.

Step 2: Add the numerator values and put over the same denominator values.

      `(2)/(9)` + `(7)/(9)`   = `(2 + 7)/(9)`

                   = `(9)/(9)`  

Step 3: Now we simplify the fraction values.

               = 1

Improper fraction problem for least to greatest:

Problem 1:

Find the value of given fraction numbers `(11)/(7)` + `(13)/(5)`

Solution:

   In the improper fraction a denominator values are not same. So we don’t directly add or subtract the numerator values. So, we take least common multiplier for the denominator values.

Step 1: In the denominator values are not same. So, we take LCM for denominator

Step 2: multiply the numerator and denominator values for common multiplier.

The LCM value of the given fraction denominator value is 33

       `(11)/(7)` * `(5)/(5)` = `(55)/(35)`

        `(13)/(5)` * `(7)/(7)`   = `(91)/(35)`

 Step 3: Now we add the numerator values.

     `(11)/(7)` + `(13)/(5)` = `(11 + 13)/(35)`

                    = `(24)/(35)`

Practice Problem for Fraction least to Greatest:

Problem 1:

Find the value of given fraction numbers `(5)/(8)` + `(7)/(8)`

               Answer: `(4)/(3)`

Problem 2:

Find the value of given fraction numbers `(5)/(6)` - `(11)/(6)`

             Answer: -1

Mixed Word Problems

Introduction:

                A mixed word problems contain different types of word problems. The types of word problems are age problem, percent problems, quadratic problems, etc, to find the solution for the word problems first step is to analyze the word problem. After analyzing the problem use the appropriate method to find the answer.

Example on Mixed Word Problems

Example 1- Mixed word problems:

Kevin is two times old as john. The sum of their age is 30. Find the age of Kevin and john.

Solution:

Given Kevin is two times old as john.

So K = 2J `=>` 1

The sum of their age is 30

So, K+J = 30 `=>` 2

Now substitute the equation 1 in equation 2

2J + J = 30.

3J = 30

Now divide by 3 on both sides

3J/3 = 30/3

J = 10.

Substitute the value of J in equation 1 to find the value of K.

K= 2 (10)

K = 20.

Kevin’s age is 20 and John’s age is 10.

Example 2 - Mixed word problems:

What is 8 percent of 100?

Solution:

Given, what is 8 percent of 100

Let us take unknown as x, is is same as =, of refers ()

So what is 8 percent of 100

x = 8 %( 100)

8% = 8 /100

So the above expression would be

x = 8/100(100)

x= 8.

So 8 is 8 percent of 100.

More Example on Mixed Word Problems:

Example 3 - Mixed word problems:

The area of the rectangle is 24 cm² and the perimeter of the rectangle is 20cm

Solution:

Given, Area of the rectangle = 24cm² and the perimeter of the rectangle is 20cm.

The formula to find the area of the rectangle is length * width

The formula to find the perimeter of the rectangle is 2*(length + width)

Let us take l as length and w as width.

l*w =24

Divide by w on both sides

l= 24/w`=>` 1

2(l+w) =20

Divide by 2 on both sides

l+w =10`=>`

24/w + w =10

24+w² =10w

w² -10w +24 =0

w² – 6w-4w +24 =0

Take w as common from first two terms

w(w-6)-4w+24=0

Take -4 as common from last two terms

w (w-6)-4(w-6)=0

(w-6)(w-4)=0

w=6 or w=4

Now substitute w=6 and w=4 in equation 1 to find the corresponding length

When w=6`=>` l=24/6 `=>` l=4

When w=4`=>` l=24/4`=>` l=6

The length of the rectangle =6 and the width of the rectangle =4.

Exponential Functions Calculator

Introduction exponential functions  calculator:
Exponential function has the form of f(x) = b^x for a permanent base b which can be any positive real number. These exponential functions are defined by the fact that their rate of growth is proportional to their value.  Let us start with a population of cells so that its growth rate at any moment is proportional to its size. The number of cells after p years will then be a^p, an exponential function for some a>0.

How to Calculate Exponential Function:

The exponential function calculator is used to calculate the functions involving the exponential expressions.

Rules and examples:

These are the rules that should be followed by the exponential functions calculator.

 Rule1:  a^xa^y = a^{x + y}

           a^x , here the x is called exponential function Example: 2³2⁵ = 2³⁺⁵ = 2⁸

 Rule 2:  (a^x)^y = a^xy

            Ex: (4²)⁵ = 4¹⁰

 Rule3:  (ab) ^x = a^xb^x

           Ex: (3*7)³ = 3³7³

 Rule 4:  (`(a)/(b)` ) ^x = `(a^x)/(b^x)`

           Ex: (`(3)/(5)` )³ = `(3^3)/(5^3)`

 Rule 5: `(a^x)/(a^y)` = a^{x - y}

           Ex: `(5^7)/(5^4)`  = 5³

Ex 1:Simplify: 3^x - 3^{x + 1}

Solution:Step 1: Use rules:

Here apply rule1: 3^{x + 1} we can wriiten as 3^x3¹ in the given expression

so , 3^x - 3^{x + 1} written as 3^x - 3^x3¹

Step2: Factor 3^x out (take common term)

            = 3^x (1 - 3)

It provides the easy method of calculating the exponential functions using the exponential functions calculator.

Calculating Exponential Functions Using Calculator:

Ex1: Use this button in calulator for find exponential function.. Following points are showing how to use exponential function in a calculator.

Step 1: find the exponential value for (1.03E2)

Step 2: (1.03E2) the number is  1.03 x 10² so 

Step 3: press the following buttons.

                      1.03 EXP 2    

Step 4 : therefore answers is 103.

Ex 2: Step 1: find the exponential value for (12.3E5)

Step 2: (12.3E5) the number is  12.3 x 10⁵ so

Step 3: press the following buttons.

                      12.3 EXP 5  

Step4 : Therefore the answer is 123000