Question on Finding the area of a Field

Topic : Area of Rectangle

Question : A farmer wants to use 1000 meters of fencing to enclose 3 sides of a rectangular field. What dimensions should be used to maximize the area of the field, and what is the maximum area of the field?

Solution :

Equation of Circle Passing Through Point (-1,-9)

Topic : Equation of Circle

Question : Write the Standard from of the equation of the circle that passes through the given point (-1,-9) and whose center is the origin.

Solution :
The Standard equation of the circle with center at origin (0,0) is x² + y² = r²
r being its radius as x² + y² = r² passes through (-1,-9) we have
(-1)² + (-9)² = r²
or
r² = 1 + 81 = 82
So the required circle is x² + y² = 82

Problem on Simultaneous Equation

Topic : Simultaneous Equation

Question : Solve and find the value of x and z
x + z = 22
x +6z = 50

Solution :
x + z = 22---------------(1)
x +6z = 50--------------(2)

Subtracting second equation from first equation
we get, -5z = -28
divide both sides by -5
So z = 5.6

Substitute the value of z in equation (1)
x+5.6 = 22
x=22-5.6
x= 16.4

So the value of x = 16.4 and z = 5.6

Problem on Square Roots

Topic : Square Roots

Question : Solve:
(x – 2)^2 = 4

Solution :
Taking square roots on both the sides, we get
(x – 2) = ± rad(4)
Case (i)
x – 2 = 2
Add 2 on both the sides of the equation
x – 2 + 2 = 2 + 2
x = 4
Case (ii)
x – 2 = -2
Add 2 on both the sides of the equation
x – 2 + 2 = -2 + 2
x = 0
The solutions are x = 0, 4

Question on Calculating the Value of a Property

Topic : Value of a Property

Question : John left 1/3 of his property to his son, ¼ to his daughter and remaining to his wife. If the wife’s share was $40,000. Find the total value of his property.
Solution : Let Total value of property = $ Y
Son’s share = Y x 1/3 = $ Y/3 (1/3 of the property)
Daughter’s share = Yx1/4 =$ Y/4 (1/4 of the property)
Property left for son and daughter = Y/3 + Y/4
= (4Y+3Y)/ 12 (by LCM)
= 7Y/12
Hence Remaining part of property = Y – 7Y/12 (Subtracting son’s & daughter’s
Share from total property)
= (12Y-7Y)/12
= 5Y/12
This remaining was given to wife. According to given information
Wife’s share = $40,000
5Y/12 = 40,000
Y = 40000 x 12/5 = $ 96000
Hence Total value of property= $96000

A Question From Probability

Topic : Probability Word Problem

Question : In transmitting dot and dash signals, a communication system changes ¼ of the dots to dashes and 1/3 of the dashes to dots. If 40% of the signals transmitted are dots and 60% are dashes, what is the probability that a dot received was actually a transmitted dot?

Solution :
[a]Dot Signals transmitted = 40%
[b]Dash signals transmitted = 60%
Let A be the event of receiving a dot.
And B the event of a transmitted dot received as dot.
So the required probability P(BA). = P(A and B)/P(A).
P(A) = 3/4 X 0.40 + 1/3 X 0.60 = 0.30 + 0.20 = 0.50
P(A and B) = 3/4 X 0.40 = 0.30
So the required Probability = P(BA) = 0.30/0.50 = 3/5.

Differences between Theoretical Probability and Experimental Probability

Topic : Theoretical Probability and Experimental Probability

Question : "Theoretical Probability" and the differences between "Theoretical Probability and Experimental Probability"

Answer :
Theoretical probability is the chance of an event happening as determined by calculating the mathematical result.
Experimental probability is the chance of an event happening based in repeated testing or observation
For example, if I flip a coin the theoretical probability of a Heads is 1 / 2
whereas the experimental probability is often quite different.
When the number of experiments increase(quite a big number) the experimental probability in the absence of any other bias tends to match the theoretical probability.

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