Answer:
(y+1)³
Step-by-step explanation:
(y+1)⁵/(y+1)²
=(y+1)(y+1)(y+1)(y+1)(y+1)/(y+1)(y+1)
=(y+1)(y+1)(y+1)
=(y+1)³✓✓
Match each system of equations with the number of solutions the system has.
A. y = - (4/3)x + 4.
y = - (4/3)x - 1.
No solution.
B. y = 4x - 5.
y = - 2x + 7.
one solution.
C. 2x + 3y = 8
4x + 6y = 17.
No solution.
D. y = 5x - 15.
y = 5(x - 3).
Infinitely many solutions.
What is the graphical solution of a system of equations?Graphical solutions of a system of equations are a way of finding the intersection points of all the equations given.
A system with two or more than two equations has a solution only when they have a common intersection point.
We'll graph each system of equations, If they intersect at one point they have one solution, if they are parallel to each other they have no solution and if they are coinciding with each other they have infinitely many solutions.
y = - (4/3)x + 4.
y = - (4/3)x - 1.
These are parallel lines no solution.
y = 4x - 5.
y = - 2x + 7.
These lines intersect at (2, 3) one solution.
2x + 3y = 8
4x + 6y = 17.
They are parallel to each other.
y = 5x - 15.
y = 5(x - 3).
These are coinciding lines if we distribute the terms hence infinitely many solutions.
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The enrollment for a school in 2010 was 1510 students. In 2020, the enrollment was 860 students. Find and interpret the rate of change in enrollment from 2010 to 2020.
Then the enrollment change rate from 2010 to 2020 is -52.2%.
What is rate of change?A rate of change is the rate at which one quantity changes in relation to another. For example, Change in y = change in x = change in distance = change in time.Here ,
Formulate 860 - 1510 / 1510 based on the given condition.
We have calculate the difference or sum = -650/1245
By converting a fraction to a decimal we get, = -0.52208
we have to multiply a number by its numerator and denominator by 100,
We get,
= -0.52208*100/100
-52.208/100 is the product or quotient.
Hence ,
In 2010, a school had 1510 students enrolled. In 2020, there will be 860 students enrolled. Then the enrollment change rate from 2010 to 2020 is -52.2%.
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Gonga's health insurance deductible is $5,500. her medical bills for the year total $50,000. How much will she be required to pay out of pocket on her annual medical bills?
42,500
0
5,500
50,000
Answer:
If Gonga's health insurance deductible is $5,500 and her medical bills for the year total $50,000, she will be required to pay the first $5,500 of her medical bills out of pocket.
Step-by-step explanation:
This means that she will be required to pay a total of $5,500 out of pocket on her annual medical bills. After she has paid this amount, her health insurance will cover the remaining costs of her medical bills up to the limit of her policy.
Answer:
$5,500
Step-by-step explanation:
On the blueprint of a house, 44 millimeters represents 8 meters. The length of the living room is 33 millimeters on the blueprint. What is
the actual length of the living room?
Answer:
Step-by-step explanation:
The actual length of the explanation is 6 meters.
Where is Located World of Masha and the Bear?
Answer:
Russia
Step-by-step explanation:
The original language for the show is Russian
A certain statistic dˆ is being used to estimate a population parameter d. The expected value of dˆ is not equal to d. What property does dˆ exhibit?.
The correct option for the given problem is the statistic is biased which is (e) option.
What is Central Limit Theorem?The central limit theorem states that in the event that you have a populace with mean μ and standard deviation σ and take adequately enormous irregular examples from the populace with substitution , then, at that point, the conveyance of the example means will be roughly regularly circulated.
According to question:By using above central limit theorem establishes that for a proportion p in a sample of size n.
The expected value is μ = p
The standard error is s = [tex]\sqrt{\frac{p(1-p)}{n} }[/tex]
Now, in this problem the expected d value is different from the expected of μ = p.
Thus, the statistic is biased, and the correct option is E.
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Correct question:
A certain statistic d hat is being used to estimate a population parameter D. The expected value of d hat is not equal to D. What property does d hat exhibit?
A. The sampling distribution of d hat is normal.
B. The sampling distribution of d hat is binomial.
C. The sampling distribution of d hat is uniform.
D. d hat is unbiased.
E. d hat is biased.
Ellie and Mary went to the Stef bakery. Ellie bought 3 cakes and 6 filled donuts for $10.14. Mary bought 4 of each for $8.32. What is the price of each type of pastry?
On solving the provided question, by substitution method we got that y=1.43 and x=o.65
What is substitution method?To solve systems of linear equations, use the algebraic technique of substitution. The name of the technique implies that it involves substituting the values of variables from one equation into another.
x=cakes , y=donuts
3x+6y=$10.14 ------ eqt.1
4x+4y=$8.32 ------ eqt.2
solve eqt.1 for "y"
y=1.43
x=o.65
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The correct question is
Ellie and Mary went to the Stef bakery. Ellie bought 3 cakes and 6 filled donuts for $10.14. Mary bought 4 of each for $8.32. What is the price of each type The price of a is The price
kevin opened a savings account with texas national bank. his account has an apr of 1.75% compounded quarterly. if kevin opens his account with $2500, how long will it take for the account to earn $7500?
As per the concept of compound interest, it take 62.915 years for the account to earn $7500
Compound interest:
The general formula to calculate the compound interest is written as,
A = P(1 + r/n)^nt
where
A = Accrued amount (principal + interest)
P = Principal amount
r = Annual nominal interest rate as a decimal
n = number of compounding periods per unit of time
t = time in decimal years;
Given,
Compounding frequency = Quarterly
Interest rate = 1.75%
Opening balance = $2500
Final amount = $7500
Now, we have to calculate the time time for the person to achieve the final amount,
First, convert R as a percent to r as a decimal
r = R/100
r = 1.75/100
r = 0.0175 per year,
Then, solve the equation for t
t = ln(A/P) / n[ln(1 + r/n)]
t = ln(7,500.00/2,500.00) / ( 4 × [ln(1 + 0.0175/4)] )
t = ln(7,500.00/2,500.00) / ( 4 × [ln(1 + 0.004375)] )
t = 62.915 years
Therefore, the time required to get a total amount of $7,500.00 with compounded interest on a principal of $2,500.00 at an interest rate of 1.75% per year and compounded 4 times per year is 62.915 years.
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pls help due now!!!!!!!!!!!!!!!!
The solution to the following system using substitution is (2,1).
What is system of equations ?
Simultaneous equations, system of equations Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a singular solution.
[tex]$$\left[\begin{array}{c}y=-4 x+9 \\y=3 x-5\end{array}\right]$$[/tex]
Substitute y=3 x-5
[3 x-5=-4 x+9]
Isolate x for 3 x-5=-4 x+9: x=2
For y=3 x-5
Substitute x=2
[tex]y=3 \cdot 2-5[/tex]
Simplify
y=1
The solutions to the system of equations are:
y=1, x=2
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A groundskeeper needs
grass seed to cover a circular field, 290 feet in diameter.
A store sells 50-pound bags of grass seed.
One pound of grass seed covers about 400 square feet
of field
What is the smallest number of bags the groundskeeper must buy to cover the circular field?
Explain or show your reasoning
The smallest number of bags the groundskeeper must buy to cover the field is 165 bags.
What is Area of a circle?Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula, A = πr²
A groundskeeper needs grass seed to cover a circular field, 290 feet in diameter. A store sells 50 pound bags of grass seed. One pound of grass seed covers about 400 square feet of field.
The smallest number of bags the groundskeeper must buy to cover the field.
Diameter of the field = 290 feet
Radius = 145 feet
Area of the circular field = πr²
= 3.14 ×145²
= 66018.5ft²
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What is 7x +3-2x - 3x simplified?
Answer:
2x+3
Step-by-step explanation:
Combine like terms.
7x-2x-3x=2x
Then add 3 to expression
Answer:
2x+3
Step-by-step explanation:
we know that we would have to add up all the commen variables first to give the most simplified answer
first take the -2x and add it with the -3x. you get -5x.
than take the -5x and add it to the 7x. you will grt 2x
lastly add them up and you get 2x+3
A machine that drills holes for wells drilled to a depth of
72 feet in one day.
At this rate, how many hours will it take until the drill
reaches its final depth of -132 feet?
It takes 43 hours to reach final depth of -132 feet with rate of change 72 feet per day.
How to find rate of change?Depending on the context, several phrases may be used to describe rate of change. For instance, acceleration/deceleration is the rate of change with respect to speed or velocity. The slope of the line of best fit determines the pace of change in statistics and regression modeling. It is the population growth rate. Rate of change is frequently referred to as momentum in the financial markets.
The formula [tex]R=\frac{D}{T}[/tex], or rate of change equals the distance traveled divided by the time required to do so, can be used to approach rate of change problems in general. The term "distance" can be changed to another depending on the context of the issue, such as change in worth or price.
Calculation:Given,
Rate of change=72 feet per day =3 feet per hour.
Final depth=132feet;
Time taken= [tex]\frac{Final}{Rate}=\frac{132}{3} =43hours\\[/tex].
It takes 43 hours to reach final depth of -132 feet with rate of change 72 feet per day.
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an economist wants to estimate the mean per capita income (in thousands of dollars) for a major city in california. he believes that the mean income is $18.6, and the standard deviation is known to be $12.1. how large of a sample would be required in order to estimate the mean per capita income at the 80% level of confidence with an error of at most $0.25? round your answer up to the next integer.
Therefore , large of a sample would be required in order to estimate the mean per capita income will be $3,838.
Test statistics – what are they?You can assess the degree to which the distribution of your data conforms to that predicted underneath the null hypothesis of the statistical test you are just using by utilizing a test statistic. The data distribution, which is determined by the confidence interval and deviation from the central tendency of each observation, represents the frequency of each observation.
Here,
First step is to find z 80% confidence level
At 80% confidence level the z will be:
α/2 = 1 - 80% = 1 - 0.80 = 0.20
α/2= 0.20 / 2 = 0.01
Zα/2= Z0.01= 1.28 from z table
Now let calculate the Sample size using this formula
Sample size = n= [( Z[tex]\alpha[/tex]/2* σ ) / E]^2
Let plug in the formula
n = (1.28*12.1/0.25)^2
n= 3,838.05
n=3,838.(Approximately)
Therefore , large of a sample would be required in order to estimate the mean per capita income will be $3,838.
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.
The correct representations of the inequality expression are (a) 6x – 15 > -10 + 5x and (c) 3(2x – 5) > -5(2 – x)
How to determine the correct representationsFrom the question, we have the following expression that can be used in our computation:
–3(2x – 5) < 5(2 – x)
Divide bot sides of the inequality by -1
So, we have the following representation
3(2x – 5) > -5(2 – x)
Next, we open the brackets
This gives
6x – 15 > -10 + 5x
Hence, the representation is 6x – 15 > -10 + 5x
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Complete question
Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.
(a) 6x – 15 > -10 + 5x
(b) 6x – 15 < -10 + 5x
(c) 3(2x – 5) > -5(2 – x)
(d) 3(2x – 5) < -5(2 – x)
A rectangle is shown on the coordinate plane below
Select all of the transformations that carry the rectangle back onto itself.
a reflection across the line x=32
a reflection across the line y=23x
a clockwise rotation of 360 degrees about the origin
a reflection across the x-axis followed by a reflection across the y-axis
a clockwise rotation of 180 degrees about the point (6,−2), followed by
a translation of 9 units in a negative x-direction
Answer:
Step-by-step explanation:
Looks like the clockwise rotation of 180 degrees about the point (6,−2), followed by a translation of 9 units in a negative x-direction. That is because once it's rotated it will be at x=15 and then if you subtract -9x you will land exactly where you started
A food truck vendor averages 150 tacos sold when the price per taco is $1. A study shows that the average number of tacos sold will increase by 25 tacos for every $1 that the price per taco is
lowered
Which equation models the food truck's average sales in dollars, S, in terms of the price for taco, x?
A S = 25x + 250
B
S=-25x + 250
C S = x(25x + 250)
D
S = x(-25x + 250)
S = 25x + 250 is the equation models the food truck's average sales in dollars, S, in terms of the price for taco, x.
Define equation.Equation is a declaration that two expressions with variables or integers are equal. In essence, equations are questions, and attempts to systematically identify the solutions to these questions have been the driving forces behind the creation of mathematics. Simple algebraic equations with merely addition or multiplication to differential equations, exponential equations with exponential expressions, and integral equations are examples of different types of equations. They are employed to represent a number of physics laws. also see equation system.
Given
A food truck vendor averages 150 tacos sold when the price per taco is $1.
A study shows that the average number of tacos sold will increase by 25
tacos for every $1 that the price per taco is lowered.
The calculation needed to determine which equation best represents the food truck's average sales in dollars, s, in terms of the price per taco, x, is as follows:
S = 25x + 250
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USPS offers the flat-rate shipping option below small box : 5 3/8 by 8 5/8 inches by 1 5/8 inches cost:$ 6.80 medium box: 11 7/8 inches by 3 3/8 inches by 13 5/8 cost : $13.45 large box 12 inches by 12 inches by 5 1/2 cost: $18.75 An artist makes necklaces and packs each necklace in a small jewelry box that is inches by 1 3 /4 by 2 1/4 inches by inch. A department store ordered 270 necklaces. The artist plans to ship the necklaces 3 /4 to the department store using USPS flat-rate shipping boxes. Which flat-rate box should this artist use to minimize his shipping costs?
You can mail small things weighing 70 pounds or less to any location in the US for a set price using USPS Small Flat Rate boxes. The cost of a USPS Small Flat Rate package is only $8.45.
What is USPS?The United States Postal Service (USPS), often known as the Post Office, U.S. Mail, or Postal Service, is a separate executive branch agency that is in charge of delivering postal service across the United States, including its associated states and insular territories.
It is one of the few governmental organizations that the US Constitution expressly grants authority to. 516,636 career employees and 136,531 non-career employees work for the USPS as of 2021.
The first postmaster general of the United States Postal Service, Benjamin Franklin, was appointed in 1775 during the Second Continental Congress. He also held a comparable role for the colonies of the Kingdom of Great Britain.
With the passing of the Postal Service Act in 1792, the Post Office Department was established.
Hence, A USPS Small Flat Rate box only costs $8.45.
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A cyclist rode 3.75 miles in 0.3 hours.
a. How fast was she going in miles per hour?
b. At that rate, how long will it take her to go 4.5 miles?
The speed of a cyclist rode 3.75 miles in 0.3 hours is 0.36 hours.
Given that, a cyclist rode 3.75 miles in 0.3 hours.
What is the speed?The speed formula can be defined as the rate at which an object covers some distance. Speed can be measured as the distance travelled by a body in a given period of time. The SI unit of speed is m/s.
A) We know that, speed =Distance/Time
Now, speed =3.75/0.3
= 12.5 miles per hour
B) Here, distance =4.5 miles and rate =12.5 miles per hour
Now, 12.5 =4.5/Time
⇒ Time =4.5/12.5
⇒ Time =0.36 hour
Therefore, the speed of a cyclist rode 3.75 miles in 0.3 hours is 0.36 hours.
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Paige is filling her tub at a rate of 3/4 gallons every 2/3 minute how many gallons would be in the tub after 3 minutes?
If Paige is filling her tub at a rate of 3/4 gallons every 2/3 minute, 3.375 gallons would be in the tub after 3 minutes.
What is an arithmetic operation?It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.
It is given that, Paige is filling her tub at a rate of 3/4 gallons every 2/3 minute.
We have to find gallons that would be in the tub after 3 minutes
We can utilize multiplication for the given task,
2/3 minutes = 3/4 gallons
1 minute = 3/4 × 3/2
1 minute =1.125
3 minutes = 3 × 1.125
3 minutes =3.375 gallons
Thus, if Paige is filling her tub at a rate of 3/4 gallons every 2/3 minute, 3.375 gallons would be in the tub after 3 minutes.
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In 2012 there were approximately 275 students
Answer:
is there a graph available????
Step-by-step explanation:
Amy ued her firt 2 token at gimmer arcade to play a game of roll and core then he played her favorite game balloon bouncer over and over until he ran out of token balloon bouncer cot 3 toke per game and amyl tart with a bucket of 35 token game token. Which equation can Amy ue to find how many game of balloon bouncer, g, he player?
The Number of Ballon bouncer games played is 11.
What is number ?
A number is a numerical unit of measurement and labelling in mathematics. The natural numbers 1, 2, 3, 4, and so on are the first examples. Number words can be used to represent numbers in language.
The expression for the number of ballon bouncer games played at the arcade is : 2 + 3b = 35
Let :
Glimmer arcade, g = $2 per game
Ballon bouncer, b = $3 per game
Total game tokens = 35
Her spending could be represented using the expression :
2 + 3b = 35 ; b = number of ballon bouncer games played
2 + 3b = 35
3b = 35 - 2
3b = 33
b = 33/3
b = 11
Therefore, the Number of Ballon bouncer games played is 11.
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Example 3. Find each measure.
The values of all the angles are shown:
(A) m∠1: 62° (B) m∠2: 39°
(C) m∠3: 26° (D) m∠4: 45°
(E) m∠5: 55° (F) m∠6: 45°
What are angles?Two rays that have a common terminal and are referred to as the angle's sides and vertices, respectively, make up an angle in Euclidean geometry.
Two rays can form angles in the plane where they are positioned.
Angles are also produced when two planes intersect.
These are what are known as dihedral angles.
So, the following values of the angles are:
(A) m∠1:
28 + 90 + ∠1 = 180
118 + ∠1 = 180
∠1 = 180 - 118
∠1 = 62°
(B) m∠2:
51 + 90 + ∠1 = 180
141 + ∠1 = 180
∠1 = 180 - 141
∠2 = 39°
(C) m∠3:
25 + (180-51) + ∠3 = 180
25 + 129 + ∠3 = 180
154 + ∠3 = 180
∠3 = 180 - 154
∠3 = 26°
(D) m∠4:
∠4 + ∠6 + 90 = 180
∠4 + ∠6 = 180 - 90
∠4 + ∠4 = 90 (∠4 and ∠6 are equal: Isosceles triangle)
2∠4 = 90
∠4 = 90/2
∠4 = 45°
(E) m∠5:
35 + 90 + ∠5 = 180
125 + ∠5 = 180
∠5 = 180 - 125
∠5 = 55°
(F) m∠6:
∠4 = ∠6 = 45°
Therefore, the values of all the angles are shown:
(A) m∠1: 62° (B) m∠2: 39°
(C) m∠3: 26° (D) m∠4: 45°
(E) m∠5: 55° (F) m∠6: 45°
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solve elimination method
3x+2y-15=0
5x-3y -25=0
Answer:
x=5
y=0
Step-by-step explanation:
3x+2y−15=0
5x−3y−25=0
multiply the first equation by 3, and multiply the second equation by 2.
3(3x+2y−15=0)
2(5x−3y−25=0)
becomes:
9x+6y−45=0
10x−6y−50=0
add these equations to eliminate y:
19x−95=0
Then solve 19x−95=0 for x:
19x−95=0
19x−95+95=0+95 (add 95 to both sides)
19x=95
19x
19
=
95
19
(divide both sides by 19)
x=5
now that we've found x let's plug it back in to solve for y.
write down an original equation:
3x+2y−15=0
substitute 5for x in 3x+2y−15=0:
(3)(5)+2y−15=0
2y=0 (simplify both sides of the equation)
2y
2
=
0
2
(divide both sides by 2)
y=0
Four of the interior angles of a pentagon are 2xº.
The fifth angle is xº.
Work out the value of xº.
Answer:
20°
Step-by-step explanation:
2x°+2x°+2x°+2x°+x°=180°
8x°+x°=180°
180°=9x°
Dbs by 9
x°=20°
Use a ruler and protractor to make an accurate drawing of the triangle below. Measure length x in your drawing to 1 d.p 40 20
Answer:15
Step-by-step explanation:
explain why a 2 x 2 matrix can have at most two distinct eigenvalues. explain why an n x 11 matrix can have at most n distinct eigenvalues.
An n x 11 matrix can have a maximum of n different eigenvalues, whereas a 2 x 2 matrix can only have a maximum of two. Below, there is a detailed explanation of this.
What is a matrix simple definition?a collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics.
What's the matrix theory?The study of matrices is the main focus of the mathematical field known as matrix theory. It started out as a branch of linear algebra but quickly expanded to cover topics in graph theory, algebra, combinatorics, and statistics.
solution of a 2 x 2 matrix is in given form:
|A - λI| = 0
here,
A = eigenvalue
I = identity matrix
now
|A - λI| = (a11 - λ)
(a22 - λ) - a12a21 = 0
the solutions of above equation are 2 x 2 matrix's eigenvalues.
just like this, a matrix of size n by n may have up to n different eigenvalues as its characteristic equation is a polynomial of nth degree with up to n roots (i.e. n eigenvalues). all due to the equation form: |A - I| = 0, where A is an eigen and I is an identity matrix.
by expanding the determinant:
|A - I| = (-1)
having coefficients c1, c2,..., cn, having sum = zero.
which is nth degree polynomial equation consisting of n possible solutions, that are basically n x n matrix's eigenvalues .
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3X-y=7
4X+2Y=3
please explain each step, thank you.
Answer:
x= 1.7
y= -1.9
Step-by-step explanation:
I will be using substitution method
in the first equation i will isolate y
3x-y=7
+y +y
3x=7+y
-7. -7
3x-7=y
now since y is equal to 3x-7 we can put 3x-7 where y is in the second equation
4x+2(3x-7)=3
4x+6x-14=3
combine like terms
10x-14=3
+14 +14
10x=17
/10. /10
x= 1.7
now we fill in x in any equation to find y
i will put it into the first one
3(1.7)-y=7
5.1-y=7
-5.1. -5.1
-y=1.9
/-1. /-1
y= -1.9
hopes this helps please mark brainliest
PLEASEE HELP
At a local college, only 60% of students live off campus. Of those who live off campus, 58% of those students get a part-time job. Of those who live on campus, 67% work part-time. The tree diagram shows how the college students are divided into subgroups.
The tree diagram shows college students branching off into two categories, off campus and on campus students. Off campus students branches off into two sub-categories, work part-time and do not work. On campus branches off into two subcategories, work part-time and do not work.
What is the percentage of students who live off campus and do not have a part-time job?
42.6%
34.8%
25.2%
24.3%
The percentage of students who live off campus and do not have a part-time job will be 25.2%. Then the correct option is C.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
At a neighborhood school, just 60% of understudies live off the grounds. Of the people who live off-grounds, 58% of those understudies find a part-time line of work. Of the people who live nearby, 67% work part-time. The tree chart shows how the undergrads are separated into subgroups.
The number of students who live off campus and do not have a part-time job is given as,
⇒ 0.60 x (1 - 0.58) x 100
⇒ 0.252 x 100
⇒ 25.2%
The percentage of students who live off campus and do not have a part-time job will be 25.2%. Then the correct option is C.
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for fixed population standard deviation and level of significance, the minimum sample size needed to guarantee a given margin of error ......... as the margin of error increases.
The right response is (b), as it increases the minimum sample size required to ensure a given margin of error.
What is margin of error?When a tiny sample of data from a relatively large population is estimated, this is what is meant by the term "margin of error" . The standard deviation, sample size, and desired confidence level are often the factors that control the margin of error.
calculation
Let's take a look at the values in the supplied statement to discover the missing term.
the population's standard deviation is where
m stands for "Margin of Error," and Z stands for "Empirical Value of Z-Score" at a specific confidence level.
As a result, the recommended minimum sample size for a particular degree of confidence is
⇒ (Z×σ)²/m²
The minimal sample size is directly correlated with the population's standard deviation, as shown by the calculation above.
Therefore, when the population "standard deviation" increased, the minimal sample size needed would also "rise."
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Three numbers are in automatic sequence and there sum is 15 if 1,3,9 be added to them respectively they forms geometric sequence find the three numbers.
Answer:
There is no solution to this problem. It is not possible to find three numbers that form an arithmetic sequence and, when 1, 3, and 9 are added to them respectively, form a geometric sequence.
Step-by-step explanation:
To find the three numbers in the arithmetic sequence, let's call the first number in the sequence "a," the second number "a + d," and the third number "a + 2d," where "d" is the common difference between the numbers. The sum of these three numbers is:
a + (a + d) + (a + 2d) = 15
This simplifies to:
3a + 3d = 15
To find the three numbers in the geometric sequence, let's call the first number in the sequence "b," the second number "r * b," and the third number "r^2 * b," where "r" is the common ratio between the numbers. The sum of these three numbers is:
b + (r * b) + (r^2 * b) = 15 + b + (r * b) + (r^2 * b)
This simplifies to:
b(1 + r + r^2) = 15 + b + br + br^2
We are told that if 1, 3, and 9 are added to the three numbers in the arithmetic sequence, we get the three numbers in the geometric sequence. Therefore, we can set up the following system of equations:
3a + 3d = 15
b(1 + r + r^2) = 15 + b + br + br^2
Subtracting b from both sides of the second equation gives:
b(1 + r + r^2) - b = 15 + br + br^2
This simplifies to:
b(r^2 + r - 1) = 15 + br + br^2
Subtracting br and br^2 from both sides gives:
b(r^2 + r - 1) - br - br^2 = 15
This simplifies to:
b(r^2 + r - 1 - r - r^2) = 15
This simplifies to:
b(r - 1)(r + 1) = 15
Dividing both sides by (r - 1)(r + 1) gives:
b = 15 / (r - 1)(r + 1)
We are told that the three numbers in the arithmetic sequence add up to 15, so we can substitute 3a + 3d = 15 into the equation above to solve for "b":
3a + 3d = 15
b = 15 / (r - 1)(r + 1)
Substituting 3a + 3d = 15 into the equation above gives:
b = 15 / (r - 1)(r + 1)
Substituting b = 15 / (r - 1)(r + 1) into the second equation gives:
15 / (r - 1)(r + 1)(1 + r + r^2) = 15 + 15 / (r - 1)(r + 1) + 15 / (r - 1)(r + 1)*r + 15 / (r - 1)(r + 1)*r^2
This simplifies to:
15 / (r - 1)(r + 1)(1 + r + r^2) = 15 + 15r + 15r^2
Multiplying both sides by (r - 1)(r + 1) gives:
15 = 15 + 15r + 15r^2
Subtracting 15 from both sides gives:
0 = 15r + 15r^2
Subtracting 15r^2 from both sides gives:
0 - 15r^2 = 15r
This simplifies to:
-15r^2 = 15r
Dividing both sides by -15 gives:
r^2 = r
Solving for "r" gives:
r = 1 or r = 0
If "r" is equal to 1, then the common ratio between the numbers in the geometric sequence is 1, which means that the three numbers in the geometric sequence are the same. This does not satisfy the condition that the three numbers in the geometric sequence are formed by adding 1, 3, and 9 to the three numbers in the arithmetic sequence, so we can eliminate this solution.
If "r" is equal to 0, then the common ratio between the numbers in the geometric sequence is 0, which means that the three numbers in the geometric sequence are all 0. This does not satisfy the condition that the three numbers in the geometric sequence are formed by adding 1, 3, and 9 to the three numbers in the arithmetic sequence, so we can eliminate this solution as well.
Therefore, there is no solution to this problem. It is not possible to find three numbers that form an arithmetic sequence and, when 1, 3, and 9 are added to them respectively, form a geometric sequence.