What are the 2 points on the line: y= 1/3x
Answer:
Points are (3, 1) and (6, 2) ....
Step-by-step explanation:
When y is 1
[tex]{ \tt{1 = \frac{1}{3}x }} \\ \\ { \tt{x = 3}}[/tex]
Point: (3, 1)
When y is 2;
[tex]{ \tt{2 = \frac{1}{3}x }} \\ \\ { \tt{x = 6}}[/tex]
Point: (6, 2)
VERY IMPORTANT DBA !!!
Table C represents a linear relationship that is also proportional. This is because the ratio of the output (y) to the input (x) is always constant, which is a characteristic of proportional relationships.
What is a linear equation?
A linear equation is an algebraic equation that represents a straight line on a graph. In general, a linear equation takes the form:
y = mx + b
A linear relationship between two variables, x and y, is one in which the change in y is directly proportional to the change in x. This means that for any change in x, there is a corresponding change in y that can be expressed as a constant multiple of x.
For example, if we have a linear relationship y = mx + b, where m is the slope of the line and b is the y-intercept, then if we increase x by 1 unit, y will increase by m units. This relationship holds true for all values of x and y along the line.
A proportional relationship, on the other hand, is a special case of a linear relationship in which the ratio of y to x is always the same constant. This means that if we increase x by 1 unit, y will increase by a constant multiple of x. In other words, the relationship between x and y is one of scaling, where the size of y is always a fixed multiple of the size of x.
Therefore, Table C represents a linear relationship that is also proportional. This is because the ratio of the output (y) to the input (x) is always constant, which is a characteristic of proportional relationships.
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Subtract 4 times the positive square root of z from thrice of x
The algebraic representation of the statement subtract 4 times the positive square root of z from thrice of x is 4√z - 3x.
What are coefficients?A mathematical expression is composed of term. A coefficient is an integer that is either multiplied by the variable it is associated with or put alongside the variable. A coefficient is, in other words, the numerical component of a phrase that contains both constants and variables. For instance, the coefficient in the expression 2x is 2.
The algebraic representation of the given statement is:
4 times the positive square root of z: 4√z
4 times the positive square root of z: 4√z - 3x
Hence, the algebraic representation of the statement subtract 4 times the positive square root of z from thrice of x is 4√z - 3x.
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6=a/4+2 two step equations
Answer:
12
Step-by-step explanation:
6=a/4+2
L.C.M=4
4(6)=a+6(2)
24=a+12
24-12=a
a=12
Stanley left his home at 8:00 a. M. And went to Peter's house. For the first 48 minutes of his trip, his average driving speed was 50 km/h. (a) Find the distance travelled (in km) for the first 48 minutes of his trip. (b) For the remaining journey, Stanley drove at a speed of 90 km/h for 1 minutes. It is given that his average driving speed for the whole journey is 70 km/h. Hours if he -Example 73) (i) Express, in terms of 1, the distance travelled (in km) for the remaining journey. (ii) Did Stanley arrive at Peter's house before 9:30 a. M. ? Explain your answer
(a) Distance travelled in first 48 minutes = 40 km. (b)(i) Distance travelled in remaining journey = 30 km. (ii) Yes, Stanley arrived at Peter's house before 9:30 a.m. as he reached there at 9:28 a.m.
(a) To find the distance travelled for the first 48 minutes, we can use the formula:
distance = speed x time
The speed is 50 km/h and the time is 48/60 hours (since 48 minutes is 0.8 hours). So,
distance = 50 x 0.8
distance = 40 km
Therefore, Stanley travelled 40 km for the first 48 minutes of his trip.
(b) Let's use the formula for average speed:
average speed = total distance ÷ total time
We know that the total distance is the distance travelled in the first 48 minutes plus the distance travelled for the remaining journey. Let's call the distance for the remaining journey "d". We also know that the total time is 1 hour (60 minutes).
So,
70 = (40 + d) ÷ 1
40 + d = 70
d = 30
Therefore, the distance for the remaining journey is 30 km.
(i) To express the distance travelled for the remaining journey in terms of 1, we can use the formula we found in part (b):
d = 30 km
(ii) To find out if Stanley arrived at Peter's house before 9:30 a.m., we need to know the total time of his journey. We know that he left his home at 8:00 a.m. and his average speed was 70 km/h. So, the total distance he travelled is:
distance = speed x time
distance = 70 x time
We also know that the total distance is the distance for the first 48 minutes plus the distance for the remaining journey:
distance = 40 + 30
distance = 70 km
Therefore, the total time of his journey is:
time = distance ÷ speed
time = 70 ÷ 70
time = 1 hour
Since he arrived at Peter's house at 9:00 a.m. (1 hour after he left his home), we can conclude that he arrived before 9:30 a.m.
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a school pays 1,852 for 150 shirts . this includes the 25$ flat-rate shipping costs. c. what are the initial value and rate of change of the function? what does each on represent
Therefore, the initial value of the function is $1,852 and the rate of change is $12.18 per shirt. The initial value represents the cost of the shirts before any were purchased,
What is function?In mathematics, a function is a rule that assigns to each element in a set called the domain, a unique element in another set called the range. In other words, a function is a mathematical object that takes an input and produces a specific output, according to a specific set of rules or operations.
by the question.
et the initial value be represented by a and the rate of change by r.
The given information can be represented by the following equation:
a + 150r = 1,852
Since the flat-rate shipping cost is $25, the cost of the 150 shirts alone would be:
a + 150r - 25 = 1,827
The initial value, a, represents the cost of the shirts before any shirts were purchased. In this case, it would be the cost of the shirts if no shirts were purchased plus the flat-rate shipping cost of $25.
So, a = 1,827 + 25 = 1,852.
The rate of change, r, represents the increase in cost for each additional shirt purchased. In this case, it would be the cost of one shirt.
So, r = (1,852 - 25)/150 = 12.18.
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Sam’s SnowBoards. Com offers 9/8/5 chain discounts to many of its customers. The Ski Hut ordered 21 snow boards with a total list price of $5,103
The net price is $4,060.75 and the trade discount amount is $ 1,042.25.
Chain discounts = 9/8/5
Total number of snow boards = 21
Calculate the trade discount:
Trade discount = List price x Discount rate
= 5,103 x 0.09
= 459.27
Calculating the net price after the first discount:
= List price - Trade discount
= 5,103 - 459.27
= 4,643.73
Calculating the net price after the second discount:
= Net price after first discount x (1 - Discount rate)
= 4,643.73 x (1 - 0.08)
= 4,274.47
Calculating net price after the third discount:
= Net price after second discount x (1 - Discount rate)
= 4,274.47 x (1 - 0.05)
= 4,060.75
Calculating the trade discount amount -
= 459.27 + 379.26 + 203.72
= 1,042.25
Complete Question:
Sam’s Ski Boards. com offers 9/8/5 chain discounts to many of its customers. The Ski Hut ordered 21 ski boards with a total list price of $5,103. What is the net price and trade discount amount?
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5.6 Enrichment and Extension
Please answer A, B, D, F, H, l, N, and P
Step-by-step explanation:
a. f(x) = s(q(x))
b. f(x) = q(s(q(x)))
d. f(x) = g(p(h(x)))
f. f(x) = q(s(x))
i. f(x) = h(r(x))
n. f(x) = h(g(s(x)))
not sure about h and p tho
hope this helps
Arun’s mother’s age is 6 years more than 4 times Arun’s age. If Arun’s age is m years, find
mother’s age
As per the unitary method, Arun's mother would be 36 years old if Arun is 3 years old.
Let Arun's age be m years.
Let Arun's mother's age be n years.
From the problem statement, we know that n = 4m + 6. This means that Arun's mother's age is directly proportional to Arun's age, with a constant ratio of 4 and a constant difference of 6.
To solve for n, we can use the unitary method. We can set up a proportionality between the two ages as follows:
n / m = (4m + 6) / m
To solve for n, we can cross-multiply to get:
n = m x (4m + 6)
Expanding the right-hand side of the equation, we get:
n = 4m² + 6m
Therefore, Arun's mother's age is 4m² + 6m years. We can simplify this expression by factoring out 2m:
n = 2m(2m + 3)
This gives us a simpler form of the equation for Arun's mother's age. To find her age, we simply substitute Arun's age (m) into this expression and simplify.
If Arun is 3 years old (m = 15), then his mother's age would be:
n = 2m(2m + 3) = 2(3)(2(3) + 3) = 2(3)(6) = 36
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An arcade deducts 3.5
points from your 50
-point game card for each game you play.
The linear equation y=−3.5x+50
represents the number of points y
on your card after x
games are played.
Use the Line Tool to graph the equation using the slope and y-intercept.
Answer: graph is attached
Step-by-step explanation:
your equation is y = -3.5x + 50
your slope is - 3.5, your y intercept is 50
to graph it yourself using a line tool
start your first point at the y intercept of 50
put your first point
next, your slope is - 3.5 this means y/x for slope is -3.5 / 1
rise is -3.5 down and run is over right +1
because we want a straight line and the slope is a fraction, make a table and you can use points that are on exact numbers instead of fractions.
at x = 0, y = 50
at x = 1, y = 46.5 ( 50 - 3.5)
at x = 2, y = 43 (46.5 - 3.5)
at x = 3, y = 39.5 ( 43 - 3.5)
at x = 4, y = 36 ( 39.5 - 3.5)
you can graph the even numbers and draw your line.
see attached graph and table.
Zahid is paid a set wage of $774.72 for a 36-hour week, plus time-and-a-half for overtime. In one particular week, he worked 43 hours. What were Zahid’s earnings?
Samantha is paid a set wage of $962.50 for a 35-hour week, plus double time for overtime. In one particular week, she worked 40 hours. What were Samantha’searnings?
Therefore, Zahid earned $1,001.16 in that particular week in hour and Samantha earned $1,237.50 in that particular week.
What do you mean by hour?An hour is a unit of time measurement that represents a duration of 60 minutes or 3,600 seconds. It is commonly used to measure time in various contexts, such as work schedules, meetings, and appointments. The abbreviation for hour is "hr" or "h".
Given by the question.
For Zahid:
Regular wage per hour = $774.72 / 36 hours = $21.52/hour
Overtime wage per hour = 1.5 x $21.52 = $32.28/hour
Zahid worked 7 hours of overtime (43 - 36 = 7)
Zahid's earnings for the week would be:
Regular earnings = 36 hours x $21.52/hour = $775.20
Overtime earnings = 7 hours x $32.28/hour = $225.96
Total earnings = Regular earnings + Overtime earnings = $1,001.16
For Samantha:
Regular wage per hour = $962.50 / 35 hours = $27.50/hour
Overtime wage per hour = 2 x $27.50 = $55/hour
Samantha worked 5 hours of overtime (40 - 35 = 5)
Samantha's earnings for the week would be:
Regular earnings = 35 hours x $27.50/hour = $962.50
Overtime earnings = 5 hours x $55/hour = $275
Total earnings = Regular earnings + Overtime earnings = $1,237.50
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(x+2)^2=-16 This equation had no solution, why not?
Answer:
It has no solution within Real numbers. Its solutions are Complex/imaginary, since its discriminant is negative (-64).
Step-by-step explanation:
The normalized form of the equation
(x+2)^2= - 16
x^2 + 4x + 4 = - 16
x^2 + 4x + 20 = 0
We have in the form ax^2 + abx + c = 0
a = 1
b = 4
c = 20
Discriminant is b^2 - 4ac = 4^2 - 4*1*20 = 16 - 80 = - 64
Discriminant is negative, therefore, no Real solutions.
Seven bags of cement weighs 3kg 52g what Is the weight of the each?
Answer:
436g
Step-by-step explanation:
1kg=1000g
3kg=3000g
3000+52=3052
3052÷7=436
What are the solutions to the following system of equations?
x − y = 6
y = x^2 −6
A) (0, −6) and (7, 1)
B) (0, −6) and (1, −5)
C) (0, −6) and (9, 3)
D) (0, −6) and (−9, 3)
Answer:
B
Step-by-step explanation:
1. Notice all the LHS answers are (0,-6) so that's definitely an answer
2. Substitute out the y so 2nd equation becomes x-6=x^2-6; so x^2-x=0; therefore x=0 or x=1
3. x=1 so y=-5 and that's the second answer hence B
Answer:
The answer is B.
Step-by-step explanation:
hope this helps
(a) The number of automobiles sold weekly at a certain dealership is a random variable with expected value 16. (i) Give an upper bound to the probability that next week's sales exceed 18. (ii) Suppose now that the variance of the number of automobiles sold weekly is 9. Give a lower bound to the probability that next week's sales are between 10 and 22, inclusively.
Markov's inequality says that for a positive random variable X and any positive real number a, the probability that X is greater than or equal to a is less than or equal to the expected value of X divided by a.
(i) Let X be the number of automobiles sold next week. We have E(X) = 16, and we want to find an upper bound to P(X > 18).
Using Markov's inequality, we have:
P(X > 18) <= E(X)/18 = 16/18 = 8/9
Therefore, an upper bound to the probability that next week's sales exceed 18 is 8/9.
(ii) Let X be the number of automobiles sold next week. We have E(X) = 16 and Var(X) = 9.
By Chebyshev's inequality, we have:
P(|X-E(X)| < k*sqrt(Var(X))) >= 1 - 1/k^2
We want to find a lower bound to P(10 ≤ X ≤ 22).
First, we standardize X by subtracting the mean and dividing by the standard deviation:
Z = (X - E(X))/sqrt(Var(X)) = (X - 16)/3
We want to find P(10 ≤ X ≤ 22), which is equivalent to finding P(-2 ≤ Z ≤ 2). Using Chebyshev's inequality with k=2, we have:
P(-2 ≤ Z ≤ 2) >= 1 - 1/2^2 = 3/4
Substituting back for X, we get:
P(10 ≤ X ≤ 22) = P(-2 ≤ (X - 16)/3 ≤ 2) >= 3/4
Therefore, a lower bound to the probability that next week's sales are between 10 and 22, inclusively, is 3/4.
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determine whether the series is convergent or divergent. 1/2 3/4 1/8 3/16 1/32 3/64..... convergent or divergent correct?. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)
Answer: The geometric series is convergent and the value is 16.
Step-by-step explanation:
How to illustrate the information?
Recall that the sum of an infinite geometric series, S, given first term, t_1, and common ratio, r, is given by:
S = t_1/(1 - r)
Note that:
3 = (4)(3/4)
9/4 = (3)(3/4
27/16 = (9/4)(3/4)
So this a geometric series with t_1 = 4 and r = 3/4. Therefore:
4 + 3 + 9/4 + 27/16 + ... = 4/(1 - 3/4) = 4/(1/4) = 16
The correct option is 16.
Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum 4+3+ 9/4 +27/16 +???
choices are
1. 3/4
2. 12
3. 4
4. divergent
5. 16
give three examples of contracts you are currently a part of or have been a part of in the past. identify whether they are unilateral or bilateral; express or implied; executed or executory.
The three examples of contracts are:
Employment ContractRental AgreementPurchase AgreementContracts are legal agreements between two or more parties that involve the exchange of goods, services, or money. They can be classified as unilateral or bilateral, express or implied, executed or executory.
Here are three examples of contracts that a person can be a part of:
Employment Contract: An employment contract is a bilateral, express contract between an employer and an employee. It defines the terms and conditions of employment, including salary, benefits, and job responsibilities. An employment contract is executed when both parties have agreed to the terms of the agreement and have signed the contract.Rental Agreement: A rental agreement is a unilateral or bilateral, express or implied, executory contract between a landlord and a tenant. It outlines the terms of the lease, such as the duration of the tenancy, rent, security deposit, and maintenance responsibilities. A rental agreement can be either oral or written. It is considered executed when the tenant moves in and starts paying rent.Purchase Agreement: A purchase agreement is a bilateral, express contract between a buyer and a seller. It outlines the terms of the sale, including the price, payment terms, delivery method, and warranty. A purchase agreement is executed when the buyer pays the agreed-upon amount and the seller delivers the product or service.To know more about the "contracts":https://brainly.com/question/5746834
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you walk 1 1.5 miles to the gym and then another 1 1/10 miles to a basketball court. How many yards did you walk in all?
You walked a total of 4576 yards to get to the basketball court.
What is unit conversion?In order to represent amounts in a more practical or acceptable unit of measurement, unit conversions are crucial for addressing mathematical issues. In this task, for instance, we were given distances in miles but had to translate them into yards to get the overall distance travelled. We wouldn't be able to compare or combine values that are stated in various units without unit conversions. When working with formulae or equations that contain physical quantities with multiple units, unit conversions are also crucial.
Given that, the distance walked is 1.5 miles and 1 1/10 miles.
Coverting into yards we have:
1.5 miles is equal to 1.5 x 1760 = 2640 yards
1 1/10 miles is equal to (1 + 1/10) x 1760 = 1936 yards
Total distance is:
2640 + 1936 = 4576 yards
Hence, you walked a total of 4576 yards to get to the basketball court.
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Darnel is studying the movement of glaciers, which are bodies of dense ice. The median
annual movement of the Blue Valley Glacier is about 300.2 feet, and the interquartile range is
14 feet. The median annual movement of the Silver Lake Glacier is about 300.4 feet, and the
interquartile range is about 14 feet.
4) What can you conclude from these statistics? Complete the sentence.
Over a year, the Blue Valley Glacier typically moves about
the Silver Lake Glacier, and Blue Valley has
its annual movement compared to Silver Lake.
as
▾ variability in its annual movement compared to silver lake
Over a year, the Blue Valley Glacier typically moves about the same distance as the Silver Lake Glacier, and Blue Valley has the same variability in its annual movement compared to Silver Lake.
How to interpret the statisticsThe median annual movement of the Blue Valley Glacier is 300.2 feet, and the interquartile range is 14 feet.
The interquartile range indicates the spread of the data within the middle 50% of the data
So we know that the annual movement of the Blue Valley Glacier falls within a range of 300.2 ± 7 feet (i.e. 293.2 to 307.2 feet)
Similarly, the median annual movement of the Silver Lake Glacier is 300.4 feet, and the interquartile range is also 14 feet
So the annual movement of the Silver Lake Glacier also falls within a range of 300.4 ± 7 feet (i.e. 293.4 to 307.4 feet)
Since the ranges for both glaciers overlap and have the same size, we can conclude that they typically move about the same distance over a year, and that the variability in the annual movement of Blue Valley is comparable to that of Silver Lake.
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Sean pays £10 for 24 chocolate bars.
He sells all 24 chocolate bars for 50p each.
Work out Sean's percentage profit
so he got them for £10 all 24 bars, then he went and sold each for 50c, so hmmm 24 at £0.50 that's (24)(0.5) = 12, so he bought them for £10 and sold them for £12, he's got a surplus or profit of £2.
now, if we take 10(origin amount) as the 100%, what's 2 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 10 & 100\\ 2& x \end{array} \implies \cfrac{10}{2}~~=~~\cfrac{100}{x} \\\\\\ 5 ~~=~~ \cfrac{ 100 }{ x }\implies x=\cfrac{100}{5}\implies x=20[/tex]
Find the circumference of both circles to the nearest hundredth. A green circle has a radius of 22 meters. A blue circle is inside the green circle. A point on the circumference of the blue circle is on the circumference of the green circle. The radius of the small blue circle is half the radius of the green circle. The circumference of the green circle is about
meters
Green Circle's circumference is 138.16 metres.
Blue Circle's circumference is 69.08 metres.
The green circle has a radius of 22 metres.
The blue circle's diameter is 1/2. the green circle's radius is 12*22, or 11 metres.
The formula for circumference is C = 2r.
The Green circle's circumference is 2*3.14*22, or 138.16 metres.
The Blue Circle's circumference is 2 * 3.14 * 11 = 69.08 metres.
What is its diameter?
The circumference of a circle or other curved geometric object refers to its length. It is the boundary across any two-dimensional circular surface measured linearly in one dimension.
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numeric prefixes are used to name select one compounds. they are not used for select one compounds, but the charges must select one .
Greek number prefixes are employed in the naming of molecular (covalent) substances.
Greek number prefixes are employed in the naming of molecular (covalent) substances. Prefixes are always combined in order of size (left-to-right in the table below). For instance, "heptadecatetracta" (7 + 10 + 400) would be the prefix for the number 417.
We utilise prefixes (mono-, di-, tri-, etc.) at the beginning of each element in molecular compounds to denote the element's subscript, but we only use the prefix mono- for the compound's second element.
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The perimeter of a rectangle is 32 3232 centimeters. The width is 7 centimeters
Hello!
Given,
Perimeter of rectangle = 32 cm
Width = 7cm
Perimeter of a rectangle = 2 (l+b)
32cm = 2l + 2 (7cm)
32 cm = 2l + 14cm
32cm -14cm =2l
2l= 18cm
l = 9cm
Area of a rectangle = length × breadth
= 7cm × 9cm
= 63 cm^2
In △PQR
how many degrees is m∠Q?
Answer:
105 degrees
Step-by-step explanation:
sum of angles in triangle is 180 degrees
11x-5+6x+5+x = 180
simplify this to get 18x=180
180/18 = 10 = x
plug in 10 for x
11(10) - 5
110-5
105
the area of the shaded region is $78$ square inches. all angles are right angles and all measurements are given in inches. what is the perimeter of the non-shaded region?
The perimeter of the non-shaded region in the figure can be calculated to be 14 inches.
Given a figure.
The middle portion of the figure is not shaded.
It is required to find the perimeter of the non-shaded region.
It is given that:
The whole area of the shaded region = 78 inches²
The area of the small square which is situated at the bottom is:
Area of small square = 2 × 4
= 8 inches²
So, the area of the rest of the shaded area = 78 - 8
= 70 inches²
Now, the area of the whole region without the small square is:
Area = 10 × (10 - 2)
= 80 inches²
So, the area of the non-shaded region = 80 - 70
= 10 inches²
The width of the non-shaded rectangle is 2 inches.
So, length = 5 inches.
So, the perimeter is:
P = 2(5 + 2)
= 14 inches
Hence, the perimeter is 14 inches.
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The figure is given below.
if belongs to the interval , at which values of does the curve have a tangent line parallel to the line ?
Answer:
you need to show the numbers
Step-by-step explanation:
In the diagram below what is the measure in angel x
Answer:
143°
When solving angle problems, make sure to label the other angles (even the ones that are not asked) and use them to relate to each other to find the answer.
show that v is an eigenvector of A and find the corresponding eigenvalue, λ.A= [\begin{ccc}-1&1\\6&0\end{array}\right], v = [\begin{ccc}1\\3\end{array}\right]λ=_____
The matrix A does not eigenvector v corresponding to the given eigen value.
λ.A = [ -11 60 ]
v = [ 1 3λ ]
v is an eigenvector of A calculate corresponding eigenvalue,
A × v = λ × v
where A is the given matrix.
v is the given vector.
λ is the corresponding eigenvalue.
× denotes matrix multiplication.
Let's first calculate A × v we have,
A × v
= [-11 60] × [ 1 3λ]
= [-11-33λ 60+ 180λ]
Check A× v is equal to λ × v ,
λ × v =
λ × [ 1 3λ]
= [λ 3λ^2]
Set these two vectors equal to each other and get the following system of equations ,
-11-33λ = λ __(1)
60+ 180λ = 3λ^2 ___(2)
From equation (1) we get,
⇒34λ = -11
⇒ λ = -11/34
Substituting this value of λ into the second equation, we have,
60+ 180λ = 3λ^2
⇒ 60 + 180(-11/34) = 3(-11/34)^2
⇒ (2040 -1980)/ 34 = 3(-11/34)^2
⇒ 60/34 = 3( 11/34)^2
⇒ 60 × 34 = 3 × 11 × 11
⇒20× 34 = 11 × 11
Which is not true.
so the value of λ does not satisfies both equations.
Therefore, v is not an eigenvector of A with corresponding eigenvalue λ.
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The above question is incomplete, the complete question is:
Check whether v is an eigen vector of A if yes find the corresponding eigen value.
λ.A = [ -11 60 ]
v = [ 1 3λ ]
According to a recent poll by YouGov. Com, in the United States 78% of 18 to 29-year-old Citizens think that college should be free for all students. Select a random sample of SO U. S. Citizens ages 18 to 29 and let X = The number of individuals in the sample who think that college should be free for all students. (A) Show that the probability distribution of X is aproximately normal. (B) Calculate the mean and standard deviation of the aproximate normal distribution
(C) Use this normal distribution to calculate the probability that 35 or fewer of the individuals in the sample think that college should be free for all students
(D) An independent study at Waynesburg University found that 35 of a random sample of 50 Students think that college should be free for all students. Based on this result and your answer to part C does the data provide convincing evidence that less than 78% of Waynesburg University students think that college should be free for all students
(A). The probability distribution of X is approximately normal because the sample size is large enough (n = 50) and the population proportion (p = 0.78) is not too close to 0 or 1.
(B) The mean of the approximately normal distribution is μ = np = 50 x 0.78 = 39, and the standard deviation is σ = sqrt(np(1-p)) = sqrt(50 x 0.78 x 0.22) = 3.22.
(C) To calculate the probability that 35 or fewer individuals need to standardize the value of X using the formula z = (X - μ) / σ. Then, z = (35 - 39) / 3.22 = -1.24 in the standard normal distribution table or using a calculator. The probability is approximately 0.1093 or 10.93%.
(D). The null hypothesis is H0: p = 0.78, and the alternative hypothesis is Ha: p < 0.78. The test statistic is z = (35/50 - 0.78) / sqrt(0.78 x 0.22 / 50) = -1.97, which has a p-value of 0.0244 or 2.44%.
Probability is a branch of mathematics that deals with the study of random events. It is the measure of the likelihood of an event occurring, expressed as a number between 0 and 1. An event with probability 0 means it will never occur, while an event with probability 1 means it will always occur.
The calculation of probability involves analyzing the possible outcomes of an event and assigning a probability to each outcome based on its likelihood of occurring. The probability of an event is determined by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is used in various fields, such as statistics, finance, engineering, and science, to make predictions and informed decisions.
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Find the points on the ellipse that are farthest away from the point (1,0). (Round your answers to the nearest hundredth.) 4x^2 + y^2 = 4
(____ , ____) (largery-value )
(____ , ____) ( smaller y-value )
The required points are (0,2) and (0,-2) respectively.
The equation of the ellipse is 4x² + y² = 4. Find the points on the ellipse that are farthest away from the point (1,0). (Round your answers to the nearest hundredth.)Solution:The given equation of the ellipse is 4x² + y² = 4.Let (x, y) be a point on the ellipse.The square of the distance between (x, y) and (1,0) is given by:(x - 1)² + y²The maximum and minimum of (x - 1)² + y² occur at the same point where the maximum and minimum of [(x - 1)² + y²] is equal to the maximum and minimum of [4x² + y² - 4] and also where the gradient of both functions is perpendicular.Thus, we need to find the maximum and minimum of the function f(x, y) = (x - 1)² + y² subject to the constraint 4x² + y² = 4.Maximize and minimize the function f(x, y) subject to the constraint 4x² + y² = 4 using Lagrange multipliers.Let g(x, y) = 4x² + y² - 4 = 0 be the constraint equation.Let L be the Lagrange functionL(x, y, λ) = f(x, y) + λg(x, y)Therefore, L(x, y, λ) = (x - 1)² + y² + λ[4x² + y² - 4]Now differentiate L(x, y, λ) partially with respect to x, y and λ.Lx = 2(x - 1) + 8λxLy = 2y + 2λydL/dλ = 4x² + y² - 4 = 0From the equation dL/dλ = 0, we get4x² + y² = 4 ⇒ x²/1 + y²/4 = 1Thus, the ellipse can be represented as:x = cosθ; y = 2sinθwhere 0 ≤ θ ≤ 2πThus, Lx = 2(x - 1) + 8λx = 0⇒ 2cosθ - 2 + 8λcosθ = 0Ly = 2y + 2λy = 0⇒ 2sinθ + 2λ(2sinθ) = 0dL/dλ = 4x² + y² - 4 = 0⇒ 4cos²θ + 4sin²θ - 4 = 0⇒ cos²θ + sin²θ = 1∴ 4λsinθ + cosθ - 1 = 0Solving the equations obtained above, we get(1) θ = 0: (x, y) = (1,0)(2) θ = π: (x, y) = (-1,0)(3) θ = π/2: (x, y) = (0,2)(4) θ = 3π/2: (x, y) = (0,-2)The points on the ellipse that are farthest away from the point (1,0) are (0,2) and (0,-2).(0,2) has the largest y-value, and (0,-2) has the smallest y-value.Therefore, the required points on the ellipse are:(0, 2) (larger y-value)(0, -2) (smaller y-value)Hence, the required points are (0,2) and (0,-2) respectively.
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