A senior accounting major at Midsouth State University has job offers from four CPA firms. To explore the offers further, she asked a sample of recent trainees how many months each worked for the firm before receiving a raise in salary. The sample information is submitted to Minitab with the following results: Analysis of Variance Source Factor df Ss MS F 3 28.17 9.39 5.37 0.133 15 26.26 1.75 18 54.43 Total What is the decision rule? (Round your answer to 2 decimal places.) eject H0 if F> At the 0.01 level of significance, is there a difference in the mean number of months before a raise was granted among the four CPA firms? Do not reject O Reject

Answer:

60 years old

Step-by-step explanation:

A=Ashu's age. M=Ashu's Mother's Age

M=A*3

M/3 = (A*3)/3  ==> divide 3 on each side to solve for A

A = M/3

A+5 = M/3 + 5  ==> A+5=25 since that'll be Ashu's age 5 years from now

25 = M/3 + 5  => plug in 25 for A+5

25-5 = M/3 + 5 - 5 ==> solve for M

20 = M/3

M = 20 * 3

M = 60 years old

The range of the starting salaries is 53,000

Given,

The box and whisker plot below shows the starting salaries

The number of graduates in a small college = 120

We have to find the range of the starting salaries;

Range of a data;

The difference between the highest and lowest values for a given data collection is the range in statistics. For instance, the range will be 10 - 2 = 8 if the given data set is 2, 5, 8, 10, and 3. As a result, the range may alternatively be thought of as the distance between the highest and lowest observation.

Here,

Lowest value = 19,000

Highest value = 72,000

Then,

Range of the set = 72,000 - 19,000 = 53,000

That is,

The range of the given data set is 53,000

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Thus, the required trig values corresponding to the cosФ = -3/5 have been shown,

These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.

Here,
cosФ = -3/5
cosФ = -3/5 =  base / hypotenuse,

perpendicular = √5² - 3² = 4

sinФ = perpendicular / hypotenus
       = 4/5

Simultaneously,
cosecФ = 5/4
secФ = -5/4
tanФ = -4/3
cotФ = -3/4

Here,
sin and cosec are positive because this function as a positive positive value between π/2 and π, while else trigonometric function as a negative value for π/2 to π.

Thus, the required trig values are shown above.

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A radius is a part of a circle that doubles to form its diameter. The radius of each sphere is 3.34 cm.

A circle is a shape bounded by curved line known as circumference. some of its parts of a circle are; diameter, radius, arc, chord etc.

A radius is that part of the circle that is half of its diameter. This implies that;

   radius = [tex]\frac{diameter}{2}[/tex]

Such that;

diameter = 2*radius

A sphere is an object that can be derived from the volume of a circle.

Given that the height of the cylindrical can is 20 cm, and three identical spherical balls would fit entirely within the can.

Then;

diameter of each spherical balls = [tex]\frac{20}{3}[/tex]

                                                      = 6.67 cm

Thus;

the radius of each of the spherical balls =  [tex]\frac{diameter}{2}[/tex]

                                                   =  [tex]\frac{6.67}{2}[/tex]

radius of each of the spherical balls = 3.335 cm

Therefore, the maximum radius of each ball is 3.34 cm.

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The nominal rate investment ranges from 4.90%, 4.92%, 4.95%, and 4.96%.

Part (c), or IV, III, II, and I, is therefore the right response to the question.

We must now determine the effective interest rate [tex]R_{e}[/tex] , we'll use the formula:-

[tex]R_{e} = (1 + i)^{k} - 1[/tex]

where, i = r / k

Here,

r = interest rate,

I = the nominal interest rate,

k = is the number of times that interest is compounded annually.

(i) 4.87% compounded quarterly

Then i = (4.87 / 100) * 4 =

i = 0.0487 * 4 = 0.012

So, [tex]R_{e}[/tex]  = (1 + 0.012)⁴ - 1

[tex]R_{e}[/tex] = 0.0496 = 4.96 %

Therefore, the effective Interest Rate = 4.96%

(ii) 4.85% compounded monthly

Then i = (4.85 / 100) * 12

i = 0.0485 * 12 = 0.0040

So, [tex]R_{e}[/tex]  = (1 + 0.0040)⁴ - 1

[tex]R_{e}[/tex] =0.04959 = 4.95 %

(iii) 4.81% compounded daily (365 days)

Then i = (4.81 / 100) * 365

i = 0.0481 * 365 = 0.00013

So, [tex]R_{e}[/tex]  = (1 + 0.00013)⁴ - 1

[tex]R_{e}[/tex] = 0.0492 = 4.92 %

(iv) 4.79% compounded continuously

Then i = (4.79 / 100)

i = 0.0479

So, [tex]R_{e}[/tex]  = (1 + 0.0479)⁴ - 1

[tex]R_{e}[/tex] =  0.0490 = 4.90 %

Therefore, the nominal rate investment is as follows: 4.90%, 4.92%, 4.95%, and 4.96%.

The right response to the question is section (c), which is IV, III, II, and I.

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This differential equation's general answer is

u(x,t)=Acos⁡(αx−ωt)+Bsin⁡(αx−ωt)+Cx+D

The wave equation with free-end conditions is a differential equation of the form

????2∂2????∂x2=∂2????∂????2,00a2∂2u∂x2=∂2u∂t2,00

where ???? is the longitudinal displacement of a vibrating elastic bar, and x and t are the spatial and temporal variables, respectively.

∂????∂x∣∣∣x=0=0,∂????∂x∣∣∣x=????=0,????>0∂u∂x|x=0=0,∂u∂x|x=L=0,t>0

and the initial condition is

????(x,0)=x∂????∂????∣∣∣????=0=−2,0

u(x,0)=x,∂u∂t|t=0=-2,0

To solve this differential equation, we use the method of separation of variables. We first rewrite the equation as

a2∂2u∂x2=∂2u∂t2,

and then we assume that u can be written as a product of two functions, one of x and one of t. That is,

u(x,t)=X(x)T(t).

Substituting this into the wave equation and rearranging, we obtain two equations for X and T:

a2X″(x)=−ω2T(t)

a2T″(t)=−ω2X(x).

X(x)=Acos⁡(αx)+Bsin⁡(αx).

T(t)=Ccos⁡(ωt)+Dsin⁡(ωt).

Therefore, the general solution of the wave equation with free-end conditions is

u(x,t)=Acos⁡(αx−ωt)+Bsin⁡(αx−ωt)+Cx+D.

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Answer:

$126,000

Step-by-step explanation:

The net cash flow for the year is $103, 000.

After all debts have been settled, net cash flow can represent either a gain or a loss in money over a time period. A company is said to have positive cash flow if, after paying all of its operational expenses, it still has cash left over.

We have,

Sales for the year = $ 218, 000

Cash expenses = $92, 000

Depreciation Expenses = $23, 000

So, the Net cash flow is

= 218,000 - (92, 000 + 23, 000)

= $ 103, 000.

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Answer:

A

Step-by-step explanation:

change 'its' to   it's    which is a contraction for 'it is'

The volume of the solid is 250r³ when  the base of s is a circular disk with radius 5r.

Given that,

We have to discover the described solid's volume, v. The parallel cross-sections perpendicular to the base of s are squares, and the base of s is a circular disk with radius 5r.

We know that,

A cylinder is a solid with square cross sections parallel to the bases and circular bases. Its height is equal to the base circle's diameter.

The radius = 5r,

So the height = diameter = 10r

Volume of the cone is base × height= πr²×h

=π(5r)²×(10r)

=250πr³

Therefore, The volume of the solid is 250r³ when  the base of s is a circular disk with radius 5r.

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A forumla for the general term an ( not the partial sum) of the infinte series (starting with a1) is [tex]a_n=\left(\frac{1}{2}\right)^n[/tex]

[tex]We rewrite $a_1, a_2, a_3, a_4$ in the sequence so that we can find a formula for the nth term of the sequence.$$\begin{aligned}& a_1=\frac{1}{2}=\left(\frac{1}{2}\right)^1 \\& a_2=\frac{1}{4}=\left(\frac{1}{2}\right)^2 \\& a_3=\frac{1}{8}=\left(\frac{1}{2}\right)^3 \\& a_4=\frac{1}{16}=\left(\frac{1}{2}\right)^4\end{aligned}[/tex]

Based on the pattern of the first four terms, we know this sequence is a geometric sequence, and the nth term of the sequence is given by the formula[tex]a_n=\left(\frac{1}{2}\right)^n[/tex]

Geometric Sequence

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant. We can write a formula for the nth term of a geometric sequence in the form

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C. The point estimate of the difference between the two population means is 2.

D. The 90% confidence interval estimate of the difference between the two population means is (1.748, 0.867)

Given that:

Sample 1 mean (x1) = 9

Sample 2 mean (x2) = 7

Sample 1 standard deviation (σ1^2) = 2.28

Sample 2 standard deviation (σ2^2) = 1.79

C. To find point estimate of the difference between the two population   means.

sample mean(x1) - sample mean(x 2)

 = 9 - 7

= 2

Therefore, point estimate = 2

D. To find 90% confidence interval estimate of the difference between the two population means

90% confidence for 't'

df = (n1 + n2) - 2

   = 12-2

   = 10

90%  confidence with df = 10 is t

t = 1.812

point estimate + 1 - t * [tex]\sqrt{\frac{s^2_{1} }{n_{1} } + \frac{s^2_{2} }{n_{2} } }[/tex]

2 + 1 - 1.812*[tex]\sqrt{\frac{2.28^2}{6} + \frac{1.79^2}{6} }[/tex]

3 - 1.812 *[tex]\sqrt{0.866 + 0.534}[/tex]

1.188 * 0.9305 + 0.7307

(1.748, 0.867)

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(a) Distribution of X is a bell curve with mean of  27 min

(b) The probability of a randomly selected LA worker has a commute that is longer than 35 minutes is 79.81%

(c) 70th percentile for the commute time of LA workers is 34.866

Given,

Mean = 27 minutes

Standard Deviation = 15 minutes

a. The distribution is a bell curve, with the peak of the curve corresponding to the mean = 27 minutes.

b. We have to find the probability that a randomly selected LA worker will have a commute of over 35 minutes. This value will fall on the right hand side of the bell curve, i.e, on the right side of the mean. For that, we need to find the z-score for 35.

z score, z = [tex]\frac{35-27}{15} = 0.53[/tex]

Using the z-tables, we get the z-value to the left of 0.53 as 0.2019, and therefore, the z-value to the right will be 1 - 0.2019 = 0.7981, which gives us a probability percentage of around 79.81%.

c. Here, we have to find the 70th percentile for the commute time for the LA workers. Let us take the marker as 0.70 on the bell-curve, so a z-score that gives us a value of 0.70 should be the right answer.

The answer to that is 0.5244.

Now, a little calculation:

0.5244 x 15 = 7.866 = x - 27

x = 34.866

Final answers:

a. A bell curve

b. 0.7981

c. 34.866

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a. The distribution is a bell curve, with the peak of the curve corresponding to the mean = 27 minutes.

b. Using the z-tables, we get the z-value to the left of 0.53 as 0.2019, and therefore, the z-value to the right will be 1 - 0.2019 = 0.7981, which gives us a probability percentage of around 79.81%.

c. Here, we have to find the 70th percentile for the commute time for the LA workers. Let us take the marker as 0.70 on the bell-curve, so a z-score that gives us a value of 0.70 should be the right answer.

The answer to that is 0.5244.

0.5244 x 15 = 7.866 = x - 27

x = 34.866

The meaning when LA Lakers had a PCT of 58.3 in the 2020-21 season is that the percentage of winning is 58.3%.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.

From the information given, the LA Lakers had a PCT of 58.3 in the 2020-21 season. It should be noted that PCT simply means percentage. Therefore, it denotes 58.3 percent.

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Complete question

The LA Lakers had a PCT of 58.3 in the 2020-21 season. What does this mean in practical terms

The van der Pol equation which is a second-order differential equation is given by

d²x/dt² - μ(1 - x²)dx/dt + x = 0

Here we are asked to convert it to a two-dimensional system.

Here we need to set y = dx/dt, where y represents the system's velocity.

Hence we will rewrite it as

d²x/dt² = μ(1 - x²)y + x

Now if we set the derivatives of x and y with respect to time to be equal to different variables we get

dx/dt = y

dy/dt = μ(1 - x²)y + x

Hence we get the above systems of equations as the 2-dimensional representation of the Van der Pol equation where x and y represent the system's position and velocity respectively.

If we set μ = 1 and x = 0, then we get

dx/dt = y

dy/dt = y

Hence we get the system of differential equations representing the behavior of the van der Pol equation with μ = 1 and x = 0 in two dimensions.

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Answer:

------------------------------

The function f(x) is horizontally stretched when the function f(bx) has the value of b between 0 and 1.

We have a function y = - x - 4.

If we need to find the equation of horizontally stretch of this function by a factor of 5, then our b = 1/5.

So the function is:

The frequency table with the classes is:

Category                        Frequency

85 99                                     7

100-114                                   8

115-129                                   13

130-144                                   7

145-159                                   5

In the given question,

The daily total number of students who used the state university swimming pool on 40 days during the summer is given.

A frequency table with the classes is 85 99, 100-114, 115-129, 130-144, and 145-159.

First we make table of given frequency and write the given number who lies in between that numbers.

Category                                         Numbers in interval

85-99                                          90, 93, 98, 89, 86, 97, 98

100-114                                 110, 107, 108, 110, 102, 105, 109, 104

115-129                  121, 119, 128, 129, 128, 115, 120, 129, 118, 122, 126, 118, 116

130-144                                      142, 132, 139, 137, 131, 132, 131

145-159                                            145, 149, 149, 152, 159

As we know that the frequency is the total numbers that cones between given interval.

Now the frequency is:

Category                        Frequency

85 99                                     7

100-114                                   8

115-129                                   13

130-144                                   7

145-159                                   5

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The right question is:

The daily total number of students who used the state university swimming pool on 40 days during the summer is as follows:

90 98 137 108 128 115 152 122 110 132 149 131 102 109 118 126 121 145 89 149 86 120 97 118 142 139 128 110 105 104 131 159 93 119 107 129 132 129 98 116

For a frequency table with the classes 85 99, 100-114, 115-129, 130-144, and 145-159.

Answer:

x=-6.-3.0.3.6

y=6.3.0.-3.-6

Step-by-step explanation:

So in this problem you're given the table at the top for your values of X is going across their corresponding P of X values and their corresponding Q of X values. So in part A we want to find P fq of X. So first off we're gonna start when X is equal to zero. So that would mean we want to find p. of Q. of zero. So remember you start with your inside function. Well Q of zero is equal to zero. So now we need to find P. Of zero which is equal to three. Alright so next we want to find when X is one. So we have P of Q of one. Well Q of one is equal to one and I'm sorry that would leave us with PS one and PS one happens to also be one. So that'll be value for one. Next for one. X. is to so api of Q of two. Well Q two is equal to four and then P. Of four. That would leave us a pr four and pr four is equal to five. So that will be our answer for the next one. It will be five. Alright, next for three. So we have p. of Q. of three. So we have two of three which is equal to two. So now sorry that should be a P. Then we find P. F two which is also to So that'll be the answer to the 3rd 1 Next we need it for four. So we're looking for p of Q of four. Well looking according to our thing. Q4 is equal to five and then we have to find P. Of five which is zero. And lastly for five. So first we'll start with p. of Q. of five. So Q. A. Five we're told is three. So then we have to find pf three which is equal to four. So that's how you would find your first values. Okay, So now part B. We want to find sfx but no, this is the opposite way this time it's Q. A. P. Of X. So when X zero, we're going to be looking for Q. Of P. Of zero. Well no this P. Of zero is equal to three. So then we have to find Q. of three which is equal to two. It'll be our first answer. Next we'll do one. So we're gonna have Q. Of P. Of one. Well p. of one is equal to one. So then we find Q. Of one which is also one. So bring that down. All right, so now for two. First we're gonna start with Q. A. P. F. Two. Well P. F two is two. So that means we're essentially just finding Q. Of two which is four. Alright, next for three. So you're gonna have cute Of p. of 3? Well p of three is equal to four And Q four is equal to five. So I'll be your answer there. I'll do a new color because I think we're running out of room here. So next we're doing for, so I'm gonna go right here. So we're gonna have Q. A. P. Of four. Well if you four is equal to five, so that means we have to find Q. Of five which is three. And lastly for five. So we're gonna have Q. A. P. Of five. So first we'll find PFE which is zero. So that means now we need Q. Of zero which is zero. And now we filled out both of your tables.

On solving the provided question we can say that - correlation coefficient of the question is r = [tex]\frac{S_{xy} }{\sqrt{S_{xx} S_{yy} } }[/tex] =  -0.91

The Pearson's correlation coefficient, also known as the Pearson's r, Pearson's product-moment correlation coefficient, bivariate correlation, or simply correlation coefficient, is a statistical indicator of the linear relationship between two sets of data.

[tex]S_{xx} =[/tex]∑[tex]x^2[/tex] - (∑x[tex])^2[/tex] /n = 19300- ((420)^2 /10)= 1660

[tex]S_{yy} =[/tex] ∑[tex]y^2[/tex] (∑y[tex])^2[/tex]/n = 7454- ((270)^2 /10) = 164

[tex]S_{xy} =[/tex]∑[tex](xy)^2[/tex]/n -475

The correlation coefficient is:

r = [tex]\frac{S_{xy} }{\sqrt{S_{xx} S_{yy} } }[/tex] =  -0.91

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div( curl( grad f )) - scalar field; the divergence of a vector field is a scalar field

A quantity with both direction and magnitude, particularly when used to depict the distance between two points in space.

In a scalar field, each point in a space—possibly actual space—is assigned a single integer.

According to the given information

(a) You can only determine the curl of a vector field; curl f is meaningless.

(b) grad f - vector field; when you take a gradient of something , it results in a vector field

(c) div F - scalar field; take the divergence of a vector field, it results in a scalar field

(d) A vector field is produced by the operation curl(grad f) on a vector field.

(e) grad F - meaningless; gradients are only used for scalar fields

(f) grad( div F ) - vector field; the gradient of a scalar field is a vector field

(g) div( grad f ) - scalar field; if you take divergence of a vector field, the result is a scalar field

(h) Grad (div f) is useless; one cannot take a scalar field's divergence into account.

(i) vector field; a vector field is a vector field's curl (curl F).

(j) div(div F) is useless; it is impossible to calculate a scalar field's divergence.

(k) ( grad f ) x ( div F ) - meaningless; you cant cross a scalar field by a vector field !

(l) The divergence of a vector field is a scalar field, and its formula is div(curl(grad f))

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Answer:

$130.84

Step-by-step explanation:

If a 7% sales tax is applied, the cost of the item (including tax) will be 107% of the original price.

To calculate how much Emma can spend if she has $140 cash to spend, divide $140 by 107%:

[tex]\implies \dfrac{140}{107\%}[/tex]

[tex]\implies \dfrac{140}{\frac{107}{100}}[/tex]

[tex]\implies 140 \times \dfrac{100}{107}[/tex]

[tex]\implies \dfrac{14000}{107}[/tex]

[tex]\implies 130.84[/tex]

Therefore, Emma can spend $130.84 on items if there is a 7% sales tax.

The probability that Luke will touch the target's inner ring fewer times than the mean of X out of five attempts is 0.40951.

Probability refers to possibility.

A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.

The degree to which something is likely to happen is basically what probability means.

You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.

According to our question-

P(target) equals 0.90

Let the random variable X stand in for the number of times the arrow is fired at the target's ring.

The probability distribution of X is

  x:         0              1                2               3               4               5

P(x):    0.00001   0.00045  0.00810   0.07290   0.32805   0.59049

Hence, The probability that Luke will touch the target's inner ring fewer times than the mean of X out of five attempts is 0.40951.

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A.

[tex]7 \leqslant 3(1) + 4 \\ 7 \leqslant 3 + 4 \\ 7 \leqslant 7[/tex]

B.

[tex]0 \leqslant 3(0) + 4 \\ 0 \leqslant 0 + 4 \\ 0 \leqslant 4[/tex]

BOTH ARE TRUE

Answer:

  A.  yes

  B.  yes

Step-by-step explanation:

You want a graph of y ≤ 3x +4 and an indication whether (1, 7) and (0, 0) are solutions.

The graph is attached. The boundary line is solid because points on that line are in the solution set. Points in the shaded area are also in the solution set.

This point is on the boundary line. It is a solution.

This point is in the shaded area of the graph. It is a solution.

<95141404393>

The graph of the given proportional relation shows that the line passes through the point (0, 0) and continues through the point (15, 5). The relation is represented by the equation y = -0.33x. So, option A is correct.

The given table shows the x and y coordinates of a line.

So, we can find the relationship among them by writing an equation from the given points as

y - y₁ = {(y₂ - y₁)/(x₂ - x₁)} × (x - x₁)

Here we have (x₁, y₁) = (15, 5); (x₂, y₂) = (9, 3)

⇒ y - 5 = (3 - 5)/(9 - 15) × (x - 15)

⇒ y - 5 = (-2)/(-6) × (x - 15)

⇒ y - 5 = 1/3 × (x - 15)

⇒ 3y - 15 = x - 15

⇒ 3y = x + 0

∴ y = 0.33x

Thus, the equation for the given proportions is y = 0.33x.

Since the line is in the form of y = mx, it passes through th point (0, 0) and continues through the point (15, 5).

So, option A is correct.

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The final distance is 31 km.

Time, t = Distance, d/Velocity, v

Let's assume the ambulance drives a distance of x along the road, which leaves us 50-x on the road, after which it has reached the road.

It forms a triangle, after which we can use Pythagoras Theorem.

Using it, we get distance down the hill, d1, and we can calculate distance on the road remaining till the hospital, d2.

t = d1/v1 + d2/v2

 = [tex]\frac{\sqrt{x^{2} + 30^{2} } }{30} + \frac{50 - x}{120}[/tex]

We need to minimise t. Therefore, we have to differentiate.

On differentiating and equating it to 0, we get the value of x as [tex]\sqrt{60} = 7.75[/tex]

How far down the road should the vehicle aim to reach the road to reach the hospital at the minimum time?

[tex]= \sqrt{900 + 7.75^{2} } = 31 km[/tex]

Thus, the final distance is 31 km.

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Answer:

2/3x

Step-by-step explanation:

Answer:

The width would be 5, and the length would be 15.

Step-by-step explanation:

To solve this problem, first look at the perimeter. If the perimeter of the rectangle is 40m, then we can now solve the problem.

Since we don't know the width or the length, first, say that the width is w.

Width=w

Now, since the length is three times the width, we can write 3w for the length.

Length=3w.

Now, combine like terms.

w+3w+w+3w=40

8w=40

Divide by 8 on both sides.

40/8=5

w=5.

Since now we know that the width is 5, plug in 5 for w.

3(5)=15

w=5

The width would be 5, and the length would be 15.

Hope this helps! Have a great day! :D

Answer:

Step-by-step explanation:

Let us assume that,

→ Perimeter = 40 m

→ Width = x

→ Length = 3x

Perimeter of rectangle formula,

→ P = 2(l + w)

Forming the equation,

→ 2(3x + x) = 40

Now the value of x will be,

→ 2(3x + x) = 40

→ 2(4x) = 40

→ 8x = 40

→ x = 40/8

→ [ x = 5 ]

Then the length and width is,

→ Width = x = 5 m

→ Length = 3x = 3(5) = 15 m

Hence, these are the answers.

A bond quote of 75.65 is equal to  $75.65 in dollars. The correct option is B.

A bond quote is the most recent price at which a bond traded, converted to a point scale and expressed as a percentage of par value. Par value is typically set at 100, or 100% of a bond's $1,000 face value. If a corporate bond is quoted at 99, for instance, it is currently trading at 99% of its face value. In this instance, each bond costs $990 to purchase.

The most recent price at which a bond traded is referred to as a bond quote.

Bond quotes are converted to a point scale and expressed as a percentage of par (face value). The standard par value is 100, which equals 100% of a bond's $1,000 face value. Bond quotes can include fractional values.

Therefore, based on the information given, the correct option is B.

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Relations in the options (a), (b) and (e) are the Equivalence relations as, the relations are reflexive, symmetric and transitive all.

Given, five relations as,

a) {(a, b) | a and b are the same age}

b) {(a, b) | a and b have the same parents}

c) {(a, b) | a and b share a common parent}

d) {(a, b) | a and b have met}

e) {(a, b) | a and b speak a common language}

we have to find which of them are equivalence relations

a) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b are the same age} is an equivalence relation.

b) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b have the same parents} is an equivalence relation.

c) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) need not to be in R for some a, b, c. Hence, Non - Transitive.

So, R = {(a, b) | a and b share a common parent} is not an equivalence relation.

d) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) need not to be in R for some a, b, c. Hence, Non - Transitive.

So, R = {(a, b) | a and b have met} is not an equivalence relation.

e) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b speak a common language} is an equivalence relation.

Hence, the relations in options (a) , (b) and (e) are the equivalence relations.

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