Math Concepts and Examples

Most of the students are don’t realize what is Universal Set U
is all about ,Let’s see an example problem which explains us more ….

Question:-

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

Answer:-

This online help with math gives the solution of the above problem.
Here the total number of students in class is what is Universal Set U ,which is given as 40 students

Here is the venn diagram template for the complete question

Two students are taking neither class.
There are 38 students in at least one of the classes.
There is a 24/40 = 0.6 = 60%  probability that a randomly-chosen student in this
group is taking Chemistry but not English.

Topic : Straight Line Equation

Question :  Find the equation of the straight line with slope 1/3 and passes through the point ( 3, -3)

Answer :

Given point is (3,-3)

and slope 1/3

THe slope intercept form of a line is y = mx + b;

m is slope and

b is y intercept

Given y=m/3 so we have y= (1/3)x + b

This line passess through (3,-3), So we substitute x=3 and y = -3

y=(1/3)x+b

-3=(1/3)*3+b

-3=1+b

-1=-1  subtracting 1 on both the sides

——-

-4=b

Hence substitute b = -4

So the required equstion is y = (1/3)x -4

we can simplify further as,

3y= 3[(1/3)x-4]

3y= 3*(1/3)x -3*4

3y=x-12 is the equation of straight line.

Topic : Properties of Triangle

Question : List and Explain the properties of Triangle.

Solution :

Vertex : The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices. 

Base : The base of a triangle can be any one of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle. In an isosceles triangle, the base is usually taken to be the unequal side. 

Altitude : The altitude of a triangle is the perpendicular from the base to the opposite vertex. The altitudes intersect at a single point, called the orthocenter of the triangle.

Median : The median of a triangle is a line from a vertex to the midpoint of the opposite side. 

Area : The number of square units it takes to exactly fill the interior of a triangle.

Perimeter : The distance around the triangle. The sum of its sides. Definition: The total distance around the outside of a triangle

perimeter = a+b+c, where a,b and c are the lengths of each side of the triangle

Topic : Word Problem on Determining the Revenue Function.

Question : A cable television firm presently serves 5000 households and charges $20 per month. A marketing survey indicates that each decrease of $1 in the monthly charge will result in 500 new customers. Let R(x) denote the total monthly revenue when the monthly charge is x dollars. Determine the revenue function R. find the value of x that results in maximum monthly revenue.

Answer :

Total monthly revenue = number of customers x monthly rent
for $20 rent : Total monthly revenue = 5000 x 20 = $100000
for every $1 decrease in the rent , there is an increase of 500 customers.
So , the relation is formulated as follows:

decrease of $1 : monthly rent = $(20-1) = $19
new customers = 500 = 500 x 1 ,
total number of customers = 5000 + 500 = 5500 = 5000 + 500 x 1
total monthly revenue = (5000 + 500 x 1) x (20 – 1)

decrease of $2 : monthly rent = $(20-2) = $18
new customers = 500 + 500 = 1000 = 500 x 2 ,
total number of customers = 5000 + 1000 = 6000 = 5000 + 500 x 2
total monthly revenue = (5000 + 500 x 2) x (20 – 1)

decrease of $3 : monthly rent = $(20-3) = $17
new customers = 500 + 500 + 500 = 1500 ,
total number of customers = 5000 + 1500 = 6500 = 5000 + 500 x 3
total monthly revenue = (5000 + 500 x 3) x (20 – 3)
……………………..
the relation is now : R(x) = (5000 + 500 x X) x ( 20 – x) = 500 (10 + X)(20-X)
to find the max. monthly revenue we have to find the first derivative of R(x) and equate it to 0.
d/dx ( R(x)) = 500 { (10 + X)(-1) + (1)(20 – X) } = 500 { 10 – 2X }
we should equate that to 0 .
so, 10 – 2x = 0 , so , X = 5
So , the total monthly revenue will be maximum if the monthly rent is decreased by $5.
Finally the answer is :
Revenue Function is : R(x) = (5000 + 500 x X) x ( 20 – x)
The total monthly revenue will be maximum if the monthly rent is decreased by $5 ; i.e if monthly rent is $15 the total monthly revenue will be maximum.

Topic : Area of Circle.

Question : A circle with area A1 is contained in the interior of larger circle with area A1+A2. If the radius of the larger circle is 3 and A1,A2,A1+A2 are in A.P,then the radius of smaller circle is__________.

Answer :

radius of the larger circle = 3
area of the larger circle = A1 + A2

A1+A2 = pi r^2 = 9 pi

A1+A2 = 9 pi —-(1)

A1, A2, A1+A2 are in A.P

hence the difference is common

A2 -A1 = A1 + A2 – A2 = A1

A2 = 2A1 —(2)
solving (1) and ( 2) ,

3A1 = 9 pi -> A1 = 3pi

Area of the smaller circle = 3 pi

radius of the smaller circle = square root of 3

(we have two concentric circles ; A1is the area of the inner circle , A2 is the area between the inner and outer circle)

Topic : parabola

Question : Growth rate y (in pounds/month)of an infant is related to the present weight x(in pounds)by formula y= cx(21-x)where c is a positive constant.0<x<21.At what rate does the maximum growth occur.

Answer :

The y coordinate of the vertex is the max of a quadratic if the parabola is downward and its the minimum if its upward.

the parabola is upward if the co efficient of x^2 is positive and downward if its negative.

here we have y= c x(21-x)= 21cx -c x^2

as the co efficient of x^2 is negative, we get a maximum value

formula: for y = ax^2+bx+c , the vertex is given by ( -b/2a , f( -b/2a) )

hence the maximum y value is -b/2a

here the max growth rate = – 21c / -2c = 21/2 =10.5pounds/month.