The vector orthogonal to the plane through the points P, Q, and R is (8, -8, 12), and the area of triangle PQR is [tex]$\sqrt{301}$[/tex].
For part (a), we can find a nonzero vector orthogonal to the plane through the points P, Q, and R by constructing a normal vector to the plane using the cross product of two vectors in the plane. The two vectors in the plane are [tex]$\vec{PQ} = (1- (-2), 0-1, 1-3) = (3, -1, -2)$[/tex] and [tex]$\vec{PR} = (1-4, 0-2, 1-5) = (-3, -2, -4)$[/tex]. Taking the cross product of the two vectors gives us the normal vector to the plane, [tex]$\vec{n} = \vec{PQ} \times \vec{PR} = (8, -8, 12)$[/tex]. Therefore, the vector orthogonal to the plane is [tex]$\vec{n} = (8, -8, 12)$.[/tex]
For part (b), the area of triangle PQR can be found using the formula for Heron's Formula, which states that the area of a triangle with sides a, b,and c is given by
[tex]$A = \sqrt{s(s-a)(s-b)(s-c)}$[/tex]
where [tex]$s = \frac{a+b+c}{2}$[/tex] . In this case, the sides are [tex]$a = \|\vec{PQ}\| = \sqrt{3^2 + (-1)^2 + (-2)^2} = \sqrt{14}$, $b = \|\vec{QR}\| = \sqrt{(-2-4)^2 + (1-2)^2 + (3-5)^2} = \sqrt{14}$[/tex], and[tex]$c = \|\vec{PR}\| = \sqrt{(-3)^2 + (-2)^2 + (-4)^2} = \sqrt{29}$[/tex]. Plugging these values into the formula, we get
[tex]= \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{\frac{7(7-\sqrt{14})(7-\sqrt{14})(7-\sqrt{29})}{4}} = \sqrt{7(7-\sqrt{14})^2(7-\sqrt{29})} = \sqrt{7(7-\sqrt{14})^2(\sqrt{14}+\sqrt{29})} = \sqrt{7(7-\sqrt{14})^2\sqrt{43}} = \sqrt{301}$[/tex]Therefore, the area of triangle PQR is [tex]$\sqrt{301}$[/tex]
The vector orthogonal to the plane through the points P, Q, and R is (8, -8, 12), and the area of triangle PQR is [tex]$\sqrt{301}$[/tex].
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An air traffic controller is tracking two planes. To start, Plane A is at an altitude of 3329 feet and Plane 8 is at an altitude of 4000 feet. Hane A is gaining atitude
at 605 feet per second and Plane 8 is gaining atitude at 45.25 feet per second.
How many second will pass before the planes are at the
Time taken to have both planes at same altitude = 1.2 seconds.
What is relative speed?Relative speed can be defined as the speed of a moving body with respect to another. When two bodies move in the same direction, their relative speed is computed by their difference. But when two bodies are moving in the opposite direction, the relative speed is calculated by adding the speed of both the bodies.
Given,
Plane A is at altitude of 3329 feet.
Plane B is at altitude of 4000 feet
Plane A is gaining altitude at Distance between them
= 4000 - 3329 feet
= 671 feet
Speed of plane A = 605 feet per second.
Speed of plane B = 45.25 feet per second
Since both plane are moving upwards
Relative speed = 605 - 45.25 = 559.75 feet per second
Time of both planes to be on same altitude
= 671/559.75
= 1.198 seconds.
≈ 1.2 seconds.
Hence, it will take 1.2 seconds to have both planes at same altitude.
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A and B and C are three villages on a highway. The distance between A and B is 18km. A motorbiker moved from A at speed of 10km/hr. At the same time , another man at village B started moving at 4km/hr. How long does it take the motorbiker to catch up to the man ? If they both reach C at the same time then what is the distance between C and B.
Answer:
Let A B = x miles = 18
B C = y miles
t1 = (18 + y) / 10 time for biker 1 to reach C
t2 = y / 4 = y / 4
t1 = t2 both bikers reach C at same time
(18 + y) / 10 = y / 4
72 + 4 y = 10 y
y = 12 distance from B to C
Check:
A C = 12 + 18 = 30 km
t1 = (18 + 12) / 10 = 3 hr
t2 = 12 / 4 = 3 hrs
So both meet at C in 3 hrs
a scientist is studying bacteria whose cell population doubles at constant intervals, at which times each cell in the population divides simultaneously. four hours from now, immediately after the population doubles, the scientist will destroy the entire sample. how many cells will the population contain when the bacteria is destroyed?
Since the cell population doubles at constant intervals, we can use the formula for exponential growth: N = N0 * 2^(t/k), where N0 is the initial population, t is the time elapsed, and k is the doubling time.
Since we are told that the population will double four hours from now, and since we are not given the initial population or the doubling time, we cannot determine the final population when the bacteria will be destroyed.
As the division two hours ago and the division just now, the population quadrupled that is two doubling periods, so each doubling period is about an hour.
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A= 60, a=9, c=10. what is the smallest value of angle B
Answer:
Step-by-step explanation:
assuming its a right triangle 30 deg with sides being 9 3 10 and angles being 30 60 90
Solve the following system by substitution
6. -2x + 2y = 4
x = y + 2
Answer:
Step-by-step explanation:
Answer: No Solution
1. Substitute the given value of x into the equation:
-2(y+2)+2y=4
2. Solve the equation for y
No Solution :-)
GIVE BRANLIEST
Infinitely many solutions
∞
x - y = 2
2x - 2y = 4
x = y + 2
2•(y +2) - 2y = 4
0 = 0 => Infinitely many solutions
a house is advertised as having 1640 square feet under roof. what is the area of this house in square meters?
The area of this house in square meters is 152.24.
To convert square feet to square meters, you can use the conversion factor of 1 square foot = 0.092903 square meters.
A house is advertised as having 1640 square feet under the roof.
So, the area of the house in square meters is:
1640 square feet * 0.092903 square meters/square foot = 152.24 square meters.
Unit Conversion: It is defined as the changing from one quantity unit to another quantity unit followed by the method of division, and multiplication by a conversion factor.
Thus, the area of this house in square meters is 152.24 if the house is advertised as having 1640 square feet under the roof.
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a vending machine contains $3.00 in nickels and dimes. if the number of dimes is 5 more than twice the number of nickels, how many coins of each type
Answer:
tmr afternoon on a thursday bc im getting a hamburger with no honey mustard.
Step-by-step explanation:
im doing it rn.
To solve this problem, we can set up an equation by considering the value of nickels and dimes. By solving the equation, we can find the number of nickels and the number of dimes.
Explanation:Let's assume the number of nickels as x.
The value of nickels is 0.05x dollars.
The number of dimes is stated as '5 more than twice the number of nickels,' which can be expressed as (2x + 5) dimes.
The value of dimes is 0.10(2x + 5) dollars.
Since the total value of the coins is $3.00, we can set up the equation:
0.05x + 0.10(2x + 5) = 3.00
Solving the equation will give us the number of nickels and dimes.
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Translate the following English phrases into algebraic equation and vice versa.
The product of a number t and eight is twelve. Answer: ?
Twice the sum of a number m and five is the difference of nine and n. Answer: ?
Twenty-one is the quotient of a number y and 4. Answer: ?
Thrice the difference of number b and one is the product of a number c and then. Answer: ?
Answer:The product of a number t and eight is twelve: t * 8 = 12
Twice the sum of a number m and five is the difference of nine and n: 2(m + 5) = 9 - n
Twenty-one is the quotient of a number y and 4: y / 4 = 21
Thrice the difference of number b and one is the product of a number c and ten: 3(b - 1) = c * 10
Step-by-step explanation:
Answer:
1. b-7= 10
2. T-(8)=12 or T X 8 =12
3. 2 (X+5)
4. 21= 4/y
5. 3 (b-1)= C (10)
The perimeter of a square is represented by the expression 4(4x−6). Which statement is true?
Responses
A. The perimeter can also be represented by the expression 8x−2 because 4(4x−6) can be simplified to 8x−2.The perimeter can also be represented by the expression 8 x minus 2 because 4 times open paren 4 x minus 6 close paren can be simplified to 8 x minus 2 point
B. The perimeter can also be represented by the expression 8x−6 because 4(4x−6) can be simplified to 8x−6.The perimeter can also be represented by the expression 8 x minus 6 because 4 times open paren 4 x minus 6 close paren can be simplified to 8 x minus 6 point
C. The perimeter can also be represented by the expression 16x−6 because 4(4x−6) can be simplified to 16x−6.The perimeter can also be represented by the expression 16 x minus 6 because 4 times open paren 4 x minus 6 close paren can be simplified to 16 x minus 6 point
D. The perimeter can also be represented by the expression 16x−24 because 4(4x−6) can be simplified to 16x−24.
Answer:
A. The perimeter can also be represented by the expression 8x−2 because 4(4x−6) can be simplified to 8x−2.
what is 4-2y+(-8y)+7.5 in a simplified answer
Answer:
The expression 4 - 2y - 8y + 7.5 can be simplified as follows:
4 - 2y - 8y + 7.5
= 4 - (2y + 8y) + 7.5
= 4 - 10y + 7.5
= 4 + 7.5 - 10y
= 11.5 - 10y
So, the simplified answer is 11.5 - 10y.
Answer: -10y+11.5
Step-by-step explanation:
PLS SOMEONE HELPP ME IT'S URGENT
The complex number in trigonometric form is given as follows:
8.49 with an angle of 45º.
How to model the complex number?A complex number has a real part and an imaginary part, and is defined as follows:
z = a + bi.
In which:
a is the real part.b is the imaginary part.Considering that the x-axis gives the real part, and the y-axis gives the imaginary part, the number in this problem is given as follows:
z = -6 - 6i.
For the trigonometric form, we have that:
The norm is given as follows: sqrt(a² + b²) = sqrt(72) = 6 x sqrt(2) = 8.49.The angle is given as follows: arctan(b/a) = arctan(1) = 45º.More can be learned about complex numbers at https://brainly.com/question/10662770
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Please help!!! A Square pyramid is shown, with dimensions in centimeters (cm).
What is the volume, in cubic centimeters, of the pyramid? Show all work solving this problem.
- formula used and all work for the area of the base
- formula used and all work for the volume of the pyramid
The volume of the pyramid is given by the equation V = 400 cm³
What is the volume of a pyramid?The volume of a pyramid, is the product of the area of its base with the height of the pyramid, and divide by 3. We express this product with the formula V = ( 1/3) ×B×h, where B is the area of the base of the pyramid, and h is its height.
Volume of Pyramid = ( 1/3 )a²H
Given data ,
Let the volume of the pyramid be represented as V
Now , the equation will be
The height of the pyramid H = 12 cm
The base side length of the pyramid a = 10 cm
So , Volume of Pyramid = ( 1/3 )a²H
Substituting the values in the equation , we get
Volume of Pyramid V = ( 1/3 ) x 10² x 12
Volume of Pyramid V = 4 x 100
Volume of Pyramid V = 400 cm³
Hence , the volume of the pyramid is 400 cm³
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Describe the likelihood of the event given its probability. The probability that the sun rises tomorrow is 1.
Answer
Because 1 = 100%, it is certain that the sun will rise tomorrow
Step-by-step explanation:
The jacobs family is planning to buy a home. They have some money for a down payment already. They see a home they like and compute that they would need to borrow $213,000 from a bank over a 30-year period. The apr is 6.75%
a. What js the monthry payment to the nearest cent?
b. What's the total of all monthly payments over the 30 years?
c. What's her total interest for the 30 years?
The monthly payment to the nearest cent is $1381.51,
The total of all monthly payments over the 30 years is $497,343.6,
and her total interest for the 30 years is $284,343.6.
What is the interest rate?The amount that the lender charges the borrower over and beyond the principal amount is referred to as the interest rate. A person who deposits money in a bank or other financial institution also earns additional income in terms of the recipient, known as interest, taking into account the time value of money.
Given amount borrowed = $213,000
time period = 30 years
interest rate = 6.75% = 0.0675
to find the monthly payment,
so M = [tex]\frac{p(\frac{r}{n} )(1 + \frac{r}{n} )^{nt}}{(1 + \frac{r}{n} )^{nt} - 1}[/tex]
here n = 12 for the monthly payment
M = monthly payment
p = principal
t = time = 30 years
r = trate = 0.0675
p = $213000
so 1 + r/n = 1 + 0.0675/12
1 + r/n = 1.005625
M = [213000(0.005625)(1.00625)³⁶⁰] ÷ [(1.00625)³⁶⁰ - 1]
M = $1381.51
B: What's the total of all monthly payments over the 30 years,
there will be 360 payments in 30 years,
a total amount of 360 payments = $1381.51*360
a total amount of 360 payments = $497,343.6
C: What's her total interest for the 30 years
total interest in 30 years = $497,343.6 - $213000
total interest in 30 years = $284,343.6.
Hence monthly payment is $1381.51,
total of all monthly payments is $497,343.6,
and total interest for the 30 years is $284,343.6.
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assume z is a standard normal random variable. then p(1.41 < z < 2.85) equals . a. .4772 b. .3413 c. .8285 d. .0771
The value of P(1.41 < Z < 2.85) is 0.0771.
Hence, the correct answer is d.
A normally distributed random variable with mean μ= 0 and standard deviation σ= 1 is referred to as a standard normal random variable. The letter Z will always be used to represent it.
Because the Standard Normal Distribution is a probability distribution, the area under the curve between two points indicates the likelihood that variables will take on a range of values.
The whole area under the curve is one, or one hundred percent.
The mean and variance of a normal distribution are governed by two factors.
A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
The probability that a standard normal random variable Z is between 1.41 and 2.85 can be found using a standard normal table with a standard normal cumulative distribution function.
The answer is approximate:
P(1.41 < Z < 2.85)
= P(Z < 2.85) - P(Z < 1.41)
= 0.9927 - 0.9185
= 0.0742
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please solve for x.....
The values of x in the circles and secant lines are 2.3, 4 and 1
How to determine the values of xTo calculate x, we make use of the tangent-secant line theorem which stated that
c² = b * d
Using the above as a guide, we have the following:
Shape (10)
(14 + x + 2) * 14 = 16^2
This gives
16 + x = 18.3
Collect and evaluate the like terms
x = 2.3
Shape (11)
(4x + 9) * 9 = 15^2
This gives
4x + 9 = 25
Collect and evaluate the like terms
x = 4
Shape (12)
x * (x + 8) = 3^2
This gives
x^2 + 8x - 9 = 0
Factorize
(x + 9)(x - 1) = 0
So, we have
x = -9 and x = 1
x = -9 is invalid
Hence, the value is 1
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how many \textbf{nm} are there in 4.6 x 10^{-4} \textbf{m}? enter the appropriate exponent as a number and not a word.
There are 4.6×10⁵ nanometer present in "4.6×10⁻⁴ meter " .
The Unit nanometer (nm) is a very small unit of length measurement, used to measure the size of particles, proteins, and other small objects.
One meter is equivalent to 10⁹ nanometers, so converting from meters to nanometers is done by multiplying by 10⁹ .
To convert 4.6 × 10⁻⁴ meters to nanometers, we simply need to multiply the value by 10⁹ :
⇒ (4.6 × 10⁻⁴ meters)×(10⁹ nanometers/meter)
⇒ 46,000 nanometers = 4.6×10⁵ nanometers .
Therefore, 4.6×10⁻⁴ meter is equivalent to 4.6×10⁵ nanometers .
The given question is incomplete , the complete question is
How many "nm" are there in 4.6×10⁻⁴ m ?
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A builder is using the drawing below to build a house. If the owner decides to increase the living room dimensions by 20%, what is the new length and width of the living room floor?
A.14.4 feet x 9.6 feet
B.144 feet x 96 feet
C.13.2 feet x 8.2 feet
D.10 feet x 8 feet
Percentage increment means the part of original value is added to the value. The new length of the living room is 14.4 feet long and new width of the living room is 1.6 feet long. The option A is the correct option.
What is the percentage of a number?
Percentage of a number or ratio is fraction of hundred. It is expressed with the sign "%".
Given information-
The length of the living room is 12 feet.
The width of the living room is 8 feet.
The owner increases the 20 percent of length and 20% width.
Suppose new length is feet long. As the owner increases the 20 percent of length. Thus the new length will be sum of original length and the 20 percent of it. Hence,
Thus the new length of the living room is 14.4 feet long.
Suppose new width is feet long. As the owner increases the 20 percent of width. Thus the new width will be sum of original width and the 20 percent of it. Hence,
Thus the new width of the living room is 1.6 feet long.
Hence the new length of the living room is 14.4 feet long and new width of the living room is 1.6 feet long. The option A is the correct option.
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are the following examples; the return on a mutual fund, time to completion of a task, or the volume of beer sold as 16 ounces, examples of continuous or discrete random variables?
The return on a mutual fund, time to completion of a task and the volume of beer sold as 16 ounces are the examples of continuous variables.
Discrete and continuous variables are the two categories into which they fall in mathematics.
A continuous variable is a variable that can take either an infinite or an uncountable number of values. For example, if a variable is continuous throughout a non-empty range of real numbers, it can take on any value within that range. As a result, it is argued that the set of real numbers between x and y, where x, y R, and x y, is uncountable and infinite.
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Which value is the solution to the equation 2b +5b - 7 = 14 when the replacement set is {2, 3, 4, 5}?
Answer:
b = 3
Step-by-step explanation:
2b + 5b - 7 = 14
7b - 7 = 14 ( add 7 to both sides )
7b = 21 ( divide both sides by 7 )
b = 3
the value from the replacement set is 3
Finding the Area of a Circle continued Find the missing measurements. Use 3.14 for T. Show your work.
Radius: 6 mm
Diameter: 12 mm
Area:.
Radius:
Diameter: 24 cm
Area:
Radius:
Diameter: 16 ft
Area:
Radius:
Diameter: 2 km
Area:
Answer:
Step-by-step explanation:
1. Area=113.1 mm^2
2. Radius=12mm
Area=452.4 mm^2
3. Radius=8 mm
Area=201.06 mm^2
4. Radius=1 km
Area=3.1416 km^2
Does the value of the standard deviation depend on the value of the mean?A)No, because the center of a distribution and the spread are not related.B)Yes, because the mean must be known in order to calculate the standard deviation.C)Yes, because if the mean gets larger, the standard deviation will also get larger.
The correct option is A) No, because the center of a distribution and the spread are not related
How standard deviation depend on the value of the mean?The standard deviation is a measure of the spread or dispersion of a set of data. It is not directly related to the mean or average of the data.
The standard deviation is calculated based on the deviations of each data point from the mean, taking into account how many data points there are.
The mean does not affect the calculation of the standard deviation, as the formula for standard deviation is based solely on the data itself and not on any summary statistics like the mean.
The value of the standard deviation can remain the same even if the mean changes, and conversely, the mean can change without affecting the standard deviation.
Thus, option A) No, because the center of a distribution and the spread are not related is correct.
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Q1.
There are a total of 96 children in Years 4, 5 and 6
37 of these children cannot swim.
11 children in Year 4 cannot swim.
21 children in Year 5 can swim.
There are 30 children in Year 6
18 of these 30 children can swim.
(1) Work out the number of children in Year 4 who can swim.
(ii) Work out the total number of children in Year 5
Answer:
Question 1:
Of the 37 children that can't swim, 11 are in Year 4.
Therefore, 26 children in Year 4 can swim.
Question 2:
There are 21 children in Year 5 who can swim.
In Year 6, there are 30 children, 18 of which can swim.
Therefore, a total of 21 + 18 = 39 children who can swim.
He gives away $800 each month. If he gives away 60% of his budget, then how much is it over budget? 
He is over budget by $1,333.33 - $800 = $533.33 per month.
What is budget?
Budget can be defined as a financial plan that outlines expected income and expenditures for a specific period of time, usually a month or a year. It is a tool used to manage personal to ensure that there is enough money coming in to cover all of the necessary expenses and to prioritize spending in a way that aligns with one's financial goals.
Budget helps to track and control spending, identify areas where expenses can be reduced and make adjustments as needed to stay on track.
If he gives away 60% of his budget, then his monthly budget is $800 / 0.6 = $1,333.33.
So, he is over budget by $1,333.33 - $800 = $533.33 per month.
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taking a cruise is a costly discretionary expense. in a recent year, the top five cruise lines in the world had this many passengers: 4,133,000 2,369,000 1,295,000 928,000 679,000 round your answers to the nearest integer. a. the computations will be easier to work if you view this problem in terms of thousands of passengers. represent each number in terms of thousands of passengers. place your answer in the same order as above using a semicolon in between the numbers. b. what is the mean number of passengers for these five cruise lines? (give the full number.) c. what is the range? (give the full number.) d. what is the standard deviation? (give the full number.)
a. The number of passengers for each cruise line in thousands:
4,133,000 passengers ÷ 1,000 = 4,133 passengers
2,369,000 passengers ÷ 1,000 = 2,369 passengers
1,295,000 passengers ÷ 1,000 = 1,295 passengers
928,000 passengers ÷ 1,000 = 928 passengers
679,000 passengers ÷ 1,000 = 679 passengers
So, the answer is 4,133; 2,369; 1,295; 928; 679.
b. The mean number of passengers can be found by adding the number of passengers for each cruise line and then dividing the total by the number of cruise lines, which is 5.
(4,133 + 2,369 + 1,295 + 928 + 679) ÷ 5 = 10,514 ÷ 5 = 2,102.8 passengers
So, the mean number of passengers is 2,102.8 passengers.
c. The range is the difference between the highest and lowest number of passengers, which is:
4,133 - 679 = 3,454 passengers
So, the range is 3,454 passengers.
d. The standard deviation can be calculated using the formula for the standard deviation of a sample. However, this calculation is quite complex and requires a significant amount of mathematical knowledge and computational resources. Without access to those resources, the standard deviation cannot be calculated
The standard deviation is a measure of the amount of variation or dispersion from the mean. It shows how spread out the values in a set of data are. To calculate the standard deviation, the difference of each value from the mean is squared and then divided by the number of items in the set. The standard deviation provides a way to quantify the degree of variation of the values around the mean. It is a useful tool for understanding how much each value in a set deviates from the average.
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the geometric series $a ar ar^2 \cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of $5.$ what is $r$?
The sum of an infinite geometric series is given by a+ar+ar^2 where a is the first term and r is the common ratio.
The sum of the series can be represented by :
a / ( 1 - r) = 12
a = 12 ( 1- r ) (1)
So....note that the sum of the terms with odd powers of r can be represented by :
ar + ar^3 + ar^5 + ar^7 + ..... + ar^(2n - 1) = 5 (2)
And note that the sum of the terms with even powers of r can be represented by :
a + ar^2 + ar^4 + ar^6 + ar^8 + ..... + ar^(2n) = 7 (3)
Multiply (2) by r on both sides
ar^2 + ar^4 + ar^6 + ar^8 + ..... + ar^(2n ) = 5r (4)
Subtract (4) from (3)
a = 7 - 5r (5)
Sub (5) into (1)
7 - 5r = 12(1 - r)
7 - 5r = 12 - 12r
7r = 5 → r = 5/7
Check
a = 7 - 5(5/7) = 24/7
a / ( 1 - r) =
(24/7) / ( 1- 5/7) =
(24/7) / ( 2/7) =
24 / 2 =
12
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when each data value in one sample can be linked to a data value in another sample, the samples are said to be
When each value of the data in one of the sample linked to the data value in another given sample , then samples are known as paired samples.
Let us consider data value of sample one is A = { 1, 2, 3, 4, 5 }Now consider another sample as B = { a, b, c, d, e }1→a , 2→c , 3 →b , 4→e , 5→d.Each data value of one sample is linked with the data value of another sample.This relation of linking data values of two different samples is known as paired samples.All data values are uniquely associated with the other sample data values.Therefore, paired samples is representing the linking of data values of one sample to the data values of other sample.
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can someone help me do this graph for math
The one of many solutions of the system is x = 0 and y = 0.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
A system of equations,
x + 6y = 0
30y = -5x
Simplifying,
x + 6y = 0
Both are the same equation,
so there are infinitely many solution possibles.
Therefore, solution is x = 0 and y = 0.
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Use parallelogram ABCD. What are the values of x and y?
When the opposite sides of a parallelogram ABCD are equal, the values of x and y are 17 and 10, respectively.
What is parallelogram?A parallelogram is a basic quadrilateral with two pairs of parallel sides in Euclidean geometry. A parallelogram's opposing or facing sides are of equal length, and its opposite angles are of equal measure. A parallelogram is a sort of quadrilateral with opposing pairs of sides that are parallel and equal. Parallelograms are classified into four categories, three of which are exceptional. Parallelograms, squares, rectangles, and rhombuses are the four kinds. A parallelogram is a quadrilateral with opposing sides that are parallel and equal. It has equal interior opposite angles.
Here,
in parallelogram ABCD, opposite sides is equal.
4y-3=37
4y=40
y=10
3x-9=42
3x=51
x=17
The values of x and y is 17 and 10 in parallelogram ABCD when opposite sides is equal.
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magine a game of three players where exactly one player wins in the end and all players have equal chances of being the winner. the game is repeated four times. find the probability that there is at least one person who wins no games.
the probability that there is at least one person who wins no games.
P( A1 ∪ A2 U A3 ) = 16/81 + 16/81 + 16/81 - 1/81 - 1/81 - 1/81 - 0 = 5/9
Number of players = 3
number of times game is repeated = 4
P( any person wins a game ) = 1/3
P ( any person does not win a game ) = 1 - 1/3 = 2/3
P ( any person wins no game in 4 attempts ) = ( 2/3 )^4 = 16/81
Note : each player has equal chance of winning
Find the probability that there is at least one person who wins no games
lets represent the probability of each player not wining a game with alphabet A
A1 = player 1 wins no game
A2 = player 2 wins no game
A3 = players 3 wins no game
Applying the inclusion-exclusion formula
P( A1 ∪ A2 U A3 ) = P(A1 ) + P(A2) + P(A3) - P( A1 ∩ A2 ) - P( A2 ∩ A3 ) - P( A1 ∩ A3 ) + P( A1 ∩ A2 ∩ A3 )
where
P( A1 ∩ A2 ) = P( A1 wins all games )
P ( A1 wins all games in 4 attempts ) = ( 1/3 )^4 = 1/81
P( A1 ∩ A2 ∩ A3 ) = P ( no players wins any game in 4 attempts ) = 0
Hence
P( A1 ∪ A2 U A3 ) = 16/81 + 16/81 + 16/81 - 1/81 - 1/81 - 1/81 - 0 = 5/9
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