Automatic Math Solutions

Introduction to automatic math solutions:
Automatic math solution deals with solving math problems in online. Only through online we get automatic math solution for the given problem. We get solution for any kind of math problems automatically. The following are some of the example problems which give the solution for the given question automatically. Here mathematics solution includes calculus problem, linear algebra problem and algebra problem. Some of the solved problems are given below. I like to share this Combinations Calculator with you all through my article.


Automatic math solutions examples:

Example 1:

Find the y value from the equation

(y - 4)(y + 8) = - y + 4

Solution:

Given equation with slight changes in the right hand side

(y - 4) (y + 8) = - (y - 4)

Make the above equation with right side equal to zero.

(y - 4)(y + 8) + (y - 4) = 0

Take (y – 4) as common factor.

(y - 4)(y + 8 + 1) = 0

Simplify the above equation
y - 4 = 0

y + 9 = 0

To obtain
y = 4
or
y = 9

Example 2:

Solve the algebraic expression

4(x -1) + 2y - 5(x -y -4) + 5

Solution:

Given algebraic expression is

4(x -1) + 2y - 5(x -y -4) + 5

Multiplying the integer terms

= 4x - 4 + 2y -5x + 5y + 20 + 5

Grouping the above terms

= -x + 7y + 21

Example 3:

Solve the Expression     -2(c - 3) – 6c - 1 = 8(c +2) – 2c

Solution:

Given equation is
-2(c - 3) – 6c - 1 = 8(c + 2) – 2c

Multiplying the integer term
-2c + 6 – 6c - 1 = 8c + 16 – 2c

Grouping the above terms
-8c + 5 = 6c + 16

Subtract 6c + 5 on both sides
-8c + 5 – 6c -16 = 6c +16 -6c-16

Grouping the above terms
-14c = 11

C = -11 / 14

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Automatic math solutions practice problems:

1) Find the a value from the equation

(a - 3)(a + 4) = - a + 3

Answer: a = 3 or a = -4

2) Find the x value from the given function using differentiation.

f(x) = x 4 - 108x + 100

Answer: x = 3 or x = -3

Related Rates Calculus Problems

Introduction to related rates calculus problems:
The study of rates of change is called as calculus. Calculus related rates is dealing with finding the rates of the changes of the two variables, the two variables are related to each other in a certain way. Using the rate of change of one variable we can find the rate of change of another variable usually with respect to time.

Examples for Related Rates Calculus Problems

Q :     The given problem is to solve the calculus related rates. A man and a woman start from the same point. The man walks S60o E at 1.3 m/sec. The women walks south at 1.6 m/sec. At what rate is the distance between the man and women changing after 35 minutes?

Sol :          Let x be the distance the man has traveled, y the distance the women has traveled and z the distance between the man and the women.

Speed of the man, (dx/dt)=1.3 m/sec and the speed of the women, (dy/dt)=1.6 m/sec. The rate of change of the distance between them, (dz/dt) after 35 minutes have passed. After t seconds, the distance the man has traveled is 1.3t while the women’s distance is 1.6t. So after 35 minutes, or 2100 seconds, the man’s distance is x=1.3(2100)=2730m and the women’s distance y=1.6(2100)=3360m.

Use the cosine law to equate the variables.

z^2=x^2+y^2-2xycos?

The angle between the paths of the man and the women will always be 60o. Since it is a constant, substitute it into the equation before differentiating.

z^2=x^2+y^2- 2xycos(60o)

z^2=x^2+y^2 -xy

Using the chain rule, differentiate the equation with respect to time.

2z(`dz/dt` )=2x(`dx/dt` )+2y(`dy/dt` )-(y(`dx/dt` )+x(`dy/dt` ))

2z(`dz/dt` )=2x(`dx/dt` )+2y(`dy/dt` )-y(`dy/dt` )-x(`dy/dt` )

2z(`dz/dt` )=(2x-y)(`dx/dt` )+(2y-x)(`dy/dt` ) 

` dz/dt` = ((2x-y)(`dx/dt` )+(2y-x)(`dy/dt` ))/2z

Before substitute the information into the equation, must find the value of z after 35 minutes. x=2730 and y=3360 after 35 minutes, substitutes these values into the Cosine Law formula to find the value of z at t=35 minutes.

z^2=x^2+y^2-xy

z=sqrt(x^2+y^2-xy)

=sqrt((2730)2+(3360)2-(2730)(3360))

˜ 3093.5 m

Substitute the information given in the equation into the equation for dz/dt.

`dz/dt` = (2(2730)-(3360)(1.3))+(2(2360)-(2730)(1.6))/2(3093.5)

= ((5460-3360)(1.3)+(6720-2730)(1.6))/6187

= ((2100)(1.3)+(3990)(1.6))/6187

= (2730+6384)/6187

= `9114/6187`

˜1.47 m/sec

The calculus related rates problem are therefore 35 minutes, the distance between the man and the women is increasing at 1.47 m/sec.

Practice Problem for Related Rates Calculus Problems:

Q :        The given problem is to find the calculus related rates a car leaves a gas station at an intersection at 4.30 p.m and travels east at 80 km/h. A truck traveling north at 100 km/h arrives at the same gas station at6.30 p.m. At what time are the two vehicles closest together?

Answer: Approximately 5.43.p.m