find the slope for the graph of the equation

Before we know how to find the slope for the graph of the equation ,lets see what is slope.

Slope is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. It is also always the same thing as how many rises in one run.

Using calculus, one can calculate the slope of the tangent to a curve at a point.

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Question:-

Draw the graph of 2x-3y=12. Use the graph to find the slope of line and area between the axes.

Answer:-

This Online help with math shows us ,how to draw the graph for the above problem.

by simplifying the equation we get.

y=0, x=6
y=-2, x=3
y=-4, x=0

We also use the slope formula to get the slope on the line .

A problem on pythogoras theorm

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

The theorem can be written as an equation:

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.



Question:-

Calculate the value of cos θ in the above triangle.

Solution:-

Use Pythagoras theorem to evaluate the length of PR.

PR=√122+202

       adjacent
Cosθ= ----------
       hypotenuse

   RQ
= ----
   PR

    12
 = -----
   √544


 = 0.514 is the Answer

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How to find the factors for an equation

Topic:-Factors of an equation

what is factorization : the decomposition of an object a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into prime factors as 3 × 5, and the polynomial x² − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained. However, factors are not needed to divide evenly because they are still divisible by any number. For example, technically 3 and 8/3 are factors of 8. But, when factoring for tests teachers are looking for the even divisibility of the numbers.
Let's look an example on it.

Question:-

Solve x²+16x+69=6

Answer:-
x²+16x+69=6

x²+16x+69-6=0

x²+16x+63=0

For factoring it ,we find two factors of 63
which add up to 16,they are 9 and 7

x²+9x+7x+69=6

Pair these terms

x(x+9)+7(x+9)=0

(x+9)(x+7)=0

x+9=0 0r x+7=0

x= -9 or x= -7

So x= -9 or -7 is the answer.

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A word problem on Distance

Distance is a numerical description of how far apart objects are. In everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").

If we have a map which shows the distance between two cities and they have given distance as well.

Let's see how to find the actual distance ...

Question:-

On a road map ,if 10 miles is equal to 1 cm.Them find the value for 40 miles and find the actual distance as well.



Answer:-

10 miles=1 cm

40 miles=?

Let's take it as 'x'

We take the ratio and equate them.

10       40
----- = -------
 1        x

      40
 x = ------
      10


 x =4
So
40 miles = 4 cm on map

Now,using the same ratio


Distance on map = 45 cm

  we know 1 cm = 10 miles

Actual Distance = 45*10

               = 450 miles

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Probability problems

Topic:Probability

Probability, is a way of expressing knowledge or belief that an event will occur or has occurred.

Here is some probability problems ,which help you to understand the concept much better.

Questions & Answers

1- Julie was given two bags full of goodies. There were 10 chocolate bars and 5 mint
candies in bag A. There were 7 chocolate bars and 2 strawberry mini-pies in bag B.
She wanted to eat a chocolate bar. From which bag should she take the candy (so that
the probability of taking a chocolate bar was greater)?

Ans:-

She should take a candy from bag B, because P (A) = 0.67 < P (B) = 0.78.

2- Mike’s class made a project about pets. There were 21 students in his class. Nine had
dogs, five had only cats, three had cats and hamsters and the rest had no pets at all.
They randomly picked out a name.
a) What was the probability that that student had a dog?
b) It turned out the student didn’t have any pets. They randomly picked out a
second name (excluding the first chosen). What is the probability that that
student had at least a cat?

Ans:

a) The probability that the student had a dog is 9/21, so P = 0.43.

b) The probability that the student had at least a cat is 8/20, so P = 0.4.

I hope it helped you ,for more math help ,you can reply me.

Question to Find the Slope of given Straight line Equation

Here is a Linear Programming problem on Slope and y intercept of a Straight line, Straight line equation is a linear equation in two variables.
Slope is a the ratio of the altitude change to the horizontal distance between any two points on the line.
y-intercept is the y-value of the point where the graph of the function or a relation intercepts the y-axis of the rectangular coordinate system (x = 0)

Topic : Slope and y-intercept of a Straight Line

This the simple and direct question to find slope and y intercept.

Problem : Given straight line is 4x -8y - 32 = 0, find the slope and y-intercept of the straight line.
Solution :
Slope is always denoted by 'm'
and Slope = - Co-efficient of x / Co-efficient of y
So from the given straight line 4x - 8y - 32 = 0
=> - 4/- 8 = 1/2
Slope = 1/2

Now to find y-intercept
put x = 0 in the given straight line linear equation,
So 4x - 8y - 32 = 0
-8y - 32 = 0
=> - y = 32/8
=> y = - 4
=> y-intercept is (0,-4)

You can find lot more problems related to the linear programming problems in geometry help and algebra help.

Word Problem on Equation of Circular Orbit of a Satellite

Satellites revolving around the earth forms a circular orbit, hence below word problem describes how to derive equation of that circular orbit which is same as finding equation of circle.

Topic : Circular Orbit of a Satellite

Circular Orbit is considered from the center of the earth.

Question : Weather Satellites. Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation. A weather satellite circle 0.6unit above earth with the center of its circular orbit at the center of earth. Find the equation of the orbit of the satellite on this map.

Solution :

Equation is x² + y² + 2x + 4y - 4091 = 0

adding and subtracting 1 and 4 ;

x² + 2x + 1 + y² + 4y + 4 = 4091 + 1 + 4

(x+1)² + (y+2)² = 4096

(x+1)² + (y+2)² = 64²

So the earth is represented by (x+1)² + (y+2)² = 64²

given a weather satellite circle 0.6unit above earth with the center of its circular orbit at the center of earth.

So radius of this orbit would be 64.6

Thus the equation of orbit is

(x + 1)² + (y + 2)² = (64.6)² = 4173.16

x² + 2x + 1 + y² + 4y + 4 = 4173.16

x² + y² + 2x + 4y + 4 = 4173.16

x² + y² + 2x + 4y = 4173.16

x² + y² + 2x + 4y = 4168.6

So orbit is represented by the equation x² + y² + 2x + 4y -4168.6 = 0


Please leave back your comments or queries in the space provided.

Question to List Formulas for Volume of Geometrical Figures

Topic : Volume Formulas

Question : List the Formula for all kind of Geometrical figures.

Solution :

Problem to Find Unknown Values of Angles of Triangles

Topic : Angles of Triangle

Question to Indicate Order of Place Value with with an Example

Topic : Place Value

Question : Indicate Order of Place Value in a Number.

Solution :

A Simple Integration Problem

Topic : Integration

Problem : Integrate x^3 ln x



Solution :

Solved Differential Equation

Topic : Differential Equation

Problem : If y = 4 Sin 3x prove y" + 9y = 0

Solution :
y = 4 Sin 3x
differentiate with respect to x
y' = 4 Cos 3x . 3
y' = 12 Cos 3x
Again differentiate with respect to x
y" = 12 (- Sin 3x . 3)
y" = - 36 Sin 3x

Now consider y" + 9y = 0
Substitute the value of y and y"
-36 Sin 3x + 9(4 Sin 3x)
-36 Sin 3x + 36 Sin 3x = 0
Hence proved

Topic : Integration

Question : Solve by Integration using Trignometric Values
∫Cos³ 2x Sin² 2x dx

Solution :
∫Cos³ 2x Sin² 2x dx
∫Cos² 2x Sin² 2x(Cos 2x) dx

Now Sin 2x = 2 Sin x Cos x
Sin 4x = 2 Sin 2x Cos 2x
Sin² 4x = 4 Sin² 2x Cos² 2x
Multiplying and dividing by 4

¼ ∫4 Sin² 2x Cos² 2x(Cos 2x) dx
¼ ∫ Sin² 4x Cos 2x dx
Now Cos 2x = 1 – 2 Sin² x
2 Sin² x = Cos 2x – 1
Sin² x = (Cos 2x – 1) / 2
Sin² 4x = (Cos 8x -1) / 2

So,
¼ ∫ [(Cos 8x -1) / 2] Cos 2x dx
1/8 ∫ (Cos 8x Cos 2x - Cos 2x) dx
1/8 [ ∫ (Cos 8x Cos 2x ) dx - ∫ Cos 2x dx ]

Now 2 Cos A Cos B = Cos (A+B) + Cos (A-B)
1/8 [ ∫ Cos 10x + Cos 6x dx - ∫ Cos 2x dx ]
1/8 {[ (Sin 10x)/10 + (Sin 6x)/6] – (Sin 2x)/2}
1/8 [(3 Sin 10x + 5 Sin 6x – 15 Sin 2x) / 30]
3 Sin 10x + 5 Sin 6x – 15 Sin 2x) / 240

Problem on Multistep Equation

Topic : Multistep Equation

Problem : Solve 7x + 2 = -54

Solution :
7x + 2 = -54
7x = -54-2
7x = -56
dividing both sides by 7
7x/7 = -56/7
x=-8
So the solution is x = -8

Question on Parabola

Topic : Parabola

Question : Natural bridge is a stone arch formed over a river or stream. The longest natural bridge in the world is rainbow bridge in Utah. if the origin is placed at one end of the arch, the cure of the arch can be modeled by the equation h=-0.042d^2+3.57d where h meters represents the height and d meters represents the horizontal distance.
a. what is the max height of the arch, to the nearest meter?
b. at a horizontal distance of 10 m from the vertex, what is the height of the arch, to the nearest meter

Answer :

H(d) = -0.042d^2 + 3.57d
where h(d) represents the height of the arch and d represents the horizontal distance from one end of the base of the arch in meters. Find the maximum height of the arch.
-------------------
Max and min problems use the 1st derivative of the function..
h(d) = -0.042d^2 + 3.57d
Set the 1st derivative to zero.
0 = -0.084d + 3.57
0.084d = 3d = 42.5 meters
------------
h(42.5) = -0.042*(42.5^2 )+ 3.57*(42.5)= -75 .8625+ 151.725h = 75.8625 meters = 76 appx..
At a horizontal distance of 10 m from the vertex, what is the height of the arch, to the nearest meter
h(d) = -0.042d^2 + 3.57d
h(10) = -0.042*10^2 + 3.57*10
=-4.2+35.7=31.5 meters

Question on Factorisation using identities:

Topic : Factorisation using identities:

Question : Factorise 4y^2 – 12y + 9

Solution :
We know that (a + b)^ 2 = a^2 + 2ab + b^2 …………… (I)
(a – b)^2= a^2 – 2ab + b^2………….. (II)
(a + b) (a – b) = a^2 – b^2………… (III)
let us use these identities for factorisation

Observe 4 y^2 = (2y) 2, 9 = 3^2 and 12y = 2 × 3 × (2y)
Therefore, 4 y^2 – 12y + 9 = (2y)^ 2 – 2 × 3 × (2y) + (3)^2
= (2y – 3)2 (required factorisation)

Factorising a Quadratic Equations

Topic : Quadratic Equations

Questions : Explain the procedure involved during Factorisation of Quadratic Equation.

Solution :

The general form of a quadratic equation is :
ax² + bx + c = 0
In order to factor this we need to find two numbers such that, the sum of those two numbers is b, which is the co-efficient of the middle term and the product of those two numbers is ac, where a is the co-efficient of leading term and c is called the constant term.

Suppose, the two numbers are p and q
then p + q = b
and pq = ac

ax² + bx + c = ax² + (p + q)x + pq/a
= (ax² + px) + (qx + pq/a)
= ax(x + p/a) + q (x + p/a)
= (ax + q) + (x + p/a)

Hope the method of how to solve a quadratic equation helps you to solve number of questions like this.

Question to Find the Area of Region

Topic : Area of the Region

Question : Find the area of the region obtained by the intersection of a plane perpendicular to x-axis and passing through(2,0,0) with 2x^2+3y^2+4z^2=24.

Answer :

Putting x=2 in the equation
gives us 3y^2+4z^2=16
get this in to ellipse standard form
and use Pi*a*b
to get 8pi/sqrt(3).

Hope the hint will help you to solve the problem easily.