4. Find the general solution to 4y"+20y'+25y = 0

Answer:

So the y-intercept is 2 while the slope is -7/2.

Step-by-step explanation:

We are going to write this in slope-intercept form because it tells us the slope,m, and the y-intercept,b.

Slope-intercept form is y=mx+b.

So our goal is to solve for y.

-7x-2y=-4

Add 7x on both sides:

   -2y=7x-4

Divide both sides by -2:

   [tex]y=\frac{7x-4}{-2}[/tex]

Separate the fraction:

[tex]y=\frac{7x}{-2}+\frac{-4}{-2}[/tex]

Simplify:

[tex]y=\frac{-7}{2}x+2[/tex]

If we compare this to y=mx+b, we see m is -7/2 and b is 2.

So the y-intercept is 2 while the slope is -7/2.

Answer:

the slope m is:

[tex]m = -\frac{7}{2}[/tex]

The y-intersection is:

[tex]b = 2[/tex]

Step-by-step explanation:

For the equation of a line written in the form

[tex]y = mx + b[/tex]

m is the slope and b is the intersection with y-axis.

In this case we have the equation

[tex]-7x - 2y = -4[/tex]

So we rewrite the equation and we have to:

[tex]2y = -7x + 4[/tex]

[tex]y = -\frac{7}{2}x + 2[/tex]

the slope m is:

[tex]m = -\frac{7}{2}[/tex]

the y-intersection is:

[tex]b = 2[/tex]

Answer: (2.2, 5.8)

Step-by-step explanation:

The confidence interval for standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),\alpha/2}}} , \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),1-\alpha/2}}}\right )[/tex]

Given :  Sample size : 16

Mean height : [tex]\mu=67.5[/tex] inches

Standard deviation : [tex]s=3.2[/tex] inches

Significance level : [tex]1-0.99=0.01[/tex]

Using Chi-square distribution table ,

[tex]\chi^2_{(15,0.005)}=32.80[/tex]

[tex]\chi^2_{(15,0.995)}=4.60[/tex]

Then , the 99% confidence interval for the population standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(15)(3.2)^2}{32.80}} , \sqrt{\dfrac{(15)(3.2)^2}{4.6}}\right )\\\\=\left ( 2.1640071232,5.77852094812\right )\approx\left ( 2.2,5.8 \right )[/tex]

Answer:

18 sqrt(3) in^2

Step-by-step explanation:

If we know the length of a side and the angle

area =  s^2  sin  a

               Since the side length is 6 and the angle a is 60

               =  6^2 sin 60

                 = 36 sin 60

                  = 36 * sqrt(3)/2

                  = 18 sqrt(3)

Answer: $ 3,360

Step-by-step explanation:

Given : The principal amount borrowed for loan : [tex]P=\ \$14,000[/tex]

Time period : [tex]t=4[/tex]

Rate of interest : [tex]r=6\%=0.06[/tex]

The formula to calculate the simple interest is given by :-

[tex]S.I.=P\times r\times t\\\\\Rightatrrow\ S.I.=14000\times4\times0.06\\\\\Rightatrrow\ S.I.=3360[/tex]

Hence, the interest due on the loan = $ 3,360

Answer:

The probability is 0.001001.

Step-by-step explanation:

Players are selling raffle tickets for $500 worth of groceries at a local store.

You bought a $1 ticket for yourself and one for your mother.

The children eventually sold 1000 tickets.

We have to find the probability that you will win first place while your mother wins second place.

We can find this as :

P(winning) =[tex]1/999=0.001001[/tex]

Interpreting the situation, we can conclude that the statement means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

Thus, N(8) is the number of students who got a grade of 8, while N(5) is the number of students who got a grade of 5.

[tex]N(8) > 2N(5)[/tex]

It means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

A similar problem is given at https://brainly.com/question/11271837

The statement [tex]\( N(8) > 2 \cdot N(5) \)[/tex] means that the number of students who scored a grade of 8 on the exam is greater than twice the number of students who scored a grade of 5 on the exam.

To understand this, let's break down the notation:

-  N(g)  represents the number of students who scored (g) on the exam.

-  N(8)  is the number of students who scored an 8.

-  N(5)  is the number of students who scored a 5.

The inequality [tex]\( N(8) > 2 \cdot N(5) \)[/tex] compares these two quantities. It states that the count of students with a grade of 8 exceeds two times the count of students with a grade of 5. This indicates that a higher number of students performed better (scoring an 8) than those who scored a 5, with the difference being more than the number of students who scored a 5. In other words, if we were to take the number of students who scored a 5 and double it, there would still be more students who scored an 8. This could be an indicator of the overall performance of the class, suggesting that more students achieved a higher grade than those who scored in the middle range of the grading scale.

Answer:

a) P=2(L+W)

b)[tex]\frac{dp}{dt}=2\frac{dL}{dt}+2\frac{dW}{dt}[/tex]

c)-2 inch/hour

Step-by-step explanation:

given:

length of the rectangle as L inches

width of the rectangle as W inches

a) The perimeter is defined as the measure of the exterior boundaries

therefore, for the rectangle the perimeter 'P' will be

P= length of AB+BC+CD+DA    (A,B,C and D are marked on the figure attached)

Now from figure

    P= L+W+L+W

           OR

=> P=2L+2W        .....................(1)

b)now dp/dt can be found as by differentiating the equation (1)

[tex]\frac{dP}{dt}=2(\frac{dL}{dt} )+2(\frac{dW}{dt} )[/tex] .............(2)

c)Now it is given for the part c of the question that

L=40 inches

W=104 inches

dL/dt=2 inches/hour

dW/dt= -3 inches/hour    (here the negative sign depicts the decrease in the dimension)

substituting the above values in the equation (2) we get

[tex]\frac{dP}{dt}=2(2)+2(-3)[/tex]

[tex]\frac{dP}{dt}=4-6=-2 inches/hour[/tex]

The formula for the perimeter of a rectangle is P = 2L + 2W. By differentiating this formula, we find that dP/dt = 2(dL/dt) + 2(dW/dt). When the length is increasing at 2 inches per hour and the width is decreasing at 3 inches per hour, the perimeter is changing at a rate of -2 inches per hour.

To find the perimeter of a rectangle, we add the lengths of all four sides of the rectangle. Given that the length is L inches and the width is W inches, the formula for the perimeter is P = 2L + 2W.

To find the rate of change of the perimeter with respect to time, we differentiate the formula with respect to time, using the chain rule. Thus, dP/dt = 2(dL/dt) + 2(dW/dt).

For the specific case where the length is increasing at 2 inches per hour and the width is decreasing at 3 inches per hour, we substitute these values into the formula for the rate of change of the perimeter to find that dP/dt = 2(2) + 2(-3) = -2 inches per hour.

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The expected value of the game is approximately $0.09375. This is the long-term average value one might expect to gain for each play of the game.

The question involves determining the expected value of a game involving the random selection of marbles. Expected value, in probability, is the long-term average value of repetitions of the experiment. It can be computed using the formula:

Expected Value (E) = ∑ [x * P(x)]

where x represents the outcomes and P(x) is the probability of those outcomes. In this case, our outcomes and their corresponding probabilities are as follows:

Calculating our expected value, we get:

E = $4*(1/32) + $2*(10/32) - $1*(21/32) = $0.09375

This means you can expect to win about $0.09 each time you play the game in the long run.

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Answer:

  129.3% of cost

Step-by-step explanation:

  cost + markup = selling price

  $78.50 + markup = $179.99 . . . . fill in given information

  markup = $101.49

The markup as a percentage of cost is ...

  markup/cost × 100% = $101.49/%78.50 × 100% ≈ 129.3%

__

As a percentage of selling price, the markup is ...

  markup/selling price × 100% = $101.49/$179.99 × 100% ≈ 56.4%

Answer:

1680 ways

Step-by-step explanation:

We have to select 6 different integers from 1 to 10. It is given that second smallest integer is 3. This means, for the smallest most integer we have only two options i.e. it can be either 1 or 2.

So, the selection of 6 numbers would be like:

{1 or 2, 3, a, b, c ,d}

There are 2 ways to select the smallest digit. Only 1 way to select the second smallest digit. For the rest four digits which are represented by a,b,c,d we have 7 options. This means we can chose 4 digits from 7. Number of ways to chose 4 digits from 7 is calculated as 7P4 i.e. by using permutations.

[tex]7P4 = \frac{7!}{(7-4)!}=840[/tex]

According to the fundamental rule of counting, the total number of ways would be the product of the individual number of ways we calculated above. So,

Total number of ways to pick 6 different integers according to the said criteria would be = 2 x 1 x 840 = 1680 ways

Answer: (a) 19448 ways

(b) 5292 ways

(c) 0.2721

Step-by-step explanation:

(a) 10 county council members be randomly elected out of the 17 candidates in the following ways:

= [tex]^{n}C_{r}[/tex]

= [tex]^{17}C_{10}[/tex]

= [tex]\frac{17!}{10!7!}[/tex]

= 19448 ways

(b) 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men in the following ways:

we know that there are 7 women and 10 men in total, so

= [tex]^{7}C_{5}[/tex] × [tex]^{10}C_{5}[/tex]

= [tex]\frac{7!}{5!2!}[/tex] × [tex]\frac{10!}{5!5!}[/tex]

= 21 × 252

= 5292 ways

(c) Now, the probability that 5 are women and 5 are men are selected:

= [tex]\frac{ ^{7}C_{5} * ^{10}C_{5}}{^{17}C_{10}}[/tex]

= [tex]\frac{5292}{19448}[/tex]

= 0.2721

Answer: A) 3

Step-by-step explanation:

For example : 1) When we divide 21 by 4 , then the remainder is 1.

Therefore we say that [tex]21\ mod\ 4 =1[/tex]

2) When we divide 10 by 7 , we get 3 as remainder.

Then , we say [tex]10\ mod\ 7=3[/tex]

The given problem : [tex]57\ mod\ 6[/tex]

When we divide 57 by 6 , we get 3 as remainder [as [tex]57=54+3=6(9)+3[/tex]]

Therfeore ,  [tex]57\ mod\ 6=3[/tex]

Hence, A is the correct option.

Answer:

$15.30

30% of 60 is 18

Step-by-step explanation:

To find the maximum amount he can spend on a meal, you have to find how  much he is going to tip.

So to find the tip you multiply 15% by 18 and you get 2.7

Then you subtract 18 by 2.7 to find out how much he can spend on the meal.

18 - 2.7 = 15.30

So he can spend $15.30 on his meal and tip $2.70

To find what percent of 60 is 18, you have to use this equation:

is over of equals percent over 100

So is/of = x/100 We have the x as the percent because that's what you're trying to figure out.

You would put 18 as is because it has the word is before it and put 60 as of because it has of before it.

So 18/60 = x/100

Now you would do Cross Product Property

18*100 = 1800

60*x = 60x

60x = 1800

Now divide 60 by itself and by 1800

1800/60 = 30

x = 30%

Answer:

d

Step-by-step explanation:

Self-interest assumption means that an action taken by a person can be termed as self interest if he or she has any basis or reason behind taking such action.  The individual always looks for profit and self benefit.Hence they can be treated as selfish.

So, if a person sacrifices his or her own interest so that others can be punished then such act can be termed as self-interest as person concerned is taking such action with a reason behind such act.

Hence, the correct answer is the option (d).

Answer:

  13

Step-by-step explanation:

The product of two negative numbers is positive. The absolute value of a number is its magnitude written with a positive sign.

  6 -2(-1) +|-5|

  = 6 + 2 + 5

  = 13

Answer:

The number would be 11.

Step-by-step explanation:

Dividend = Divisor × Quotient + Remainder

Given,

"n" is divided by 3, the remainder is 2,

So, the number = 3n + 2,

"n" is divided by 5, the remainder is 1,

So, the number = 5n + 1

Thus, we can write,

3n + 2 = 5n + 1

-2n = -1

n = 0.5,

Therefore, number must be the multiple of 0.5 but is not divided by 3 or 5,

Possible numbers = { 1, 2, 4, 7, 8, 11...... }

Since, 1 and 4 do not give the remainder 2 after divided by 3,

And, 2, 7 and 8 do not give the remainder 1 after divided by 5,

Hence, the least positive integer number that gives remainder 2 and 1 after divided by 3 and 5 respectively is 11.

To maximize the area, a 2m length of wire should be used for the square and the rest for the triangle. To minimize the area, nearly all the wire should be used for the triangle, leaving a negligible amount for the square.

The problem described is a classic example of Mathematics optimization. In this case, we have two geometric shapes, a square and an equilateral triangle. To answer this question effectively, one needs to understand the relationship between the perimeter and area of these two shapes.

For the square, the area is given by A=s2, where s is the length of a side. For the equilateral triangle, the area is given by A=0.433*s2, where s is the length of a side. We want to understand how to divide the 6m wire so that we either maximize or minimize the total area of these two shapes.

The total length of wire used is fixed at 6m. Let's designate x as the length of wire used for the square. This means the length for the triangle would be 6-x. For the maximum area, the result generally comes around 2m for the square and 4m for the triangle. However, for the minimum area, the answer would be essentially 0m for the square and 6m for the triangle.

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(a) For maximizing total area, use [tex]\( s \approx 3.76 \)[/tex] meters in square.
(b) For minimizing area, use [tex]\( s = 0 \)[/tex] for maximum utilization of wire to the triangle, least yielding area near zero or minimal non-zero.

(a) Maximizing the Total Area

To maximize the total area, we need to determine the length of the wire to be used for the square [tex](\( s \))[/tex] and the length used for the equilateral triangle [tex](\( t \))[/tex].

(b) Minimizing the Total Area

To minimize the total area, confirm whether interiors of boundary values might be critical points. Specifically, see if using all the wire for one shape minimizes the area.

Answer: 1406

Step-by-step explanation:

Given Table :

Admissions   Probability

1,040                0.3

1,320                0.2

1,660                0.5

Now, the expected number of admissions for the fall semester is given by :-

[tex]E(x)=p_1x_1+p_2x_2+p_3x_3\\\\\Rightarrow\ E(x)=0.3\times1040+0.2\times1320+0.5\times1660\\\\\Rightarrow\ E(x)=1406[/tex]

Hence, the expected number of admissions for the fall semester = 1406

Answer:

Skewed

Step-by-step explanation:

A project manager can interpret several things in a histogram. If something unusual happening, the histogram is said to Skewed. When the histogram is Skewed it means that many of the values of the graph are falling on only one side of the mean. It can be either on left side( left skewed) or on the right side called right skewed

Answer:

Standard deviation.

Step-by-step explanation:

The standard deviation measures how accurate the point estimate is likely to be in estimating a parameter.

The confidence interval measures how accurate the point estimate is likely to be in estimating a parameter.

A confidence interval communicates how accurate our estimate is likely to  be.

The confidence interval is a range of of all plausible values of the random variable under test at a given confidence level which is expressed in percentage such as 98%, 95% and 90% of confidence level.  

The standard deviation is the parameter to signify the  dispersion of data around  the mean value of the data.

Researchers prefer it because    on the basis of the percentage of certainty in the test result of  null hypothesis are accepted or rejected as it includes some chance for errors too. (example  95% sure means 5% not sure) also this gives a range of values and hence good chance to normalize errors.    

An interval estimate is typically preferred over a point estimate because

i) it gives us a sense of accuracy of the point estimate

ii) we know the probability that it contains the parameter (e.g., 95%)

iii) it provides us with more possible parameter values

I only

II only  

both I and II  

all of these

III only

All three statements above are true hence all of these is  the answer.

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Answer:

14.7 and 16.9

Step-by-step explanation:

We want to find the confidence interval for the mean when the population standard deviation [tex]\sigma[/tex], is known so we use the [tex]z[/tex] confidence interval for the mean.

The following assumptions are also met;

The z confidence interval for the mean is given by:

[tex]\bar X-z_{\frac{\alpha}{2} }(\frac{\sigma}{\sqrt{n} } )\:<\:\mu\:<\bar X+z_{\frac{\alpha}{2} }(\frac{\sigma}{\sqrt{n} } )[/tex]

The appropriate z-value for 95% confidence interval is 1.96 (read from the standard normal z-distribution table)....See attachment.

From the question, we have [tex]n=49[/tex], [tex]\sigma=3.8[/tex] and [tex]\bar X=15.8[/tex]

We substitute all these values to get:

[tex]15.8-1.96(\frac{3.8}{\sqrt{49} } )\:<\:\mu\:<\bar 15.8+1.96(\frac{3.8}{\sqrt{49} } )[/tex]

[tex]15.8-1.96(\frac{3.8}{7 } )\:<\:\mu\:<15.8+1.96(\frac{3.8}{7} )[/tex]

[tex]14.7\:<\:\mu\:< 16.9[/tex] correct to one decimal place.

To calculate the 95% confidence interval for the population mean of truck weights on I-475, we can use the formula: Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / Square Root of Sample Size). Plugging in the given values, we find that the 95% confidence interval is approximately 14.7 to 16.9 tons.

To calculate the 95% confidence interval for the population mean, we can use the formula: Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / Square Root of Sample Size). In this case, the sample mean is 15.8 tons, the population standard deviation is 3.8 tons, and the sample size is 49. The critical value for a 95% confidence level is approximately 1.96. Plugging in these values, we get:

Confidence Interval = 15.8 ± (1.96) * (3.8 / √49)

Confidence Interval ≈ 15.8 ± (1.96) * (3.8 / 7)

Confidence Interval ≈ 15.8 ± (1.96) * 0.543

Confidence Interval ≈ 15.8 ± 1.06

Therefore, the 95% confidence interval for the population mean is approximately 14.7 to 16.9 tons.

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Answer: The sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.

Step-by-step explanation:

The given values : 74.001, 74.003, 74.015, 74.000, 74.005, 74.002, 74.007, and 74.000.

[tex]\text{Mean =}\dfrac{\text{Sum of all values}}{\text{Number of values}}\\\\\Rightarrow\overline{x}=\dfrac{ 592.033}{8}=74.004125\approx74.004[/tex]

The sample standard deviation is given by :-

[tex]\sigma=\sum\sqrt{\dfrac{(x-\overline{x})^2}{n}}\\\\\Rightarrow\ \sigma=\sqrt{\dfrac{0.000177}{8}}=0.00470372193056\approx0.005[/tex]

Hence, the sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.

Answer: 1.75

Step-by-step explanation:

To find the value of [tex]\sqrt{3}[/tex]

[tex]\text{Let , }y=\sqrt{x}[/tex]

[tex]\text{And Let x = 4 and }\Delta x=-1[/tex]

Now,

[tex]\Delta y=\sqrt{x+\Delta x}-\sqrt{x}\\\\=\sqrt{3}-\sqrt{4}=\sqrt{3}-2\\\\\Rightarrow\ \sqrt{3}=\Delta y+2[/tex]

Since dy is approximately equals to [tex]\Delta y[/tex] then ,

[tex]dy=\dfrac{dy}{dx}\Delta x\\\\=\dfrac{1}{2\sqrt{x}}\times(-1)=\dfrac{1}{2\sqrt{4}}\times(-1)=-0.25[/tex]

Thus , the approximate value of  [tex]\sqrt{3}=-0.25+2=1.75[/tex]

"in front of the [tex]y,z[/tex] plane" probably means [tex]x\ge0[/tex], in which case

[tex]4x^2-4y^2-z^2=4\implies x=\sqrt{1+y^2+\dfrac{z^2}4}[/tex]

We can then parameterize the surface by setting [tex]y(u,v)=u[/tex] and [tex]z(u,v)=v[/tex], so that [tex]x=\sqrt{1+u^2+\dfrac{v^2}4}[/tex].

The part of the hyperboloid in front of the yz-plane is represented parametrically by x(u,v)=2*cos(u), y(u,v)=-2*sinh(v), and z(u,v)=sinh(u).

The surface of the hyperboloid lies in front of the yz-plane and is described by the equation 4x² − 4y² − z² = 4. A common form of parameterization for this type of surface uses hyperbolic functions. Therefore, a parametrization for the part of the hyperboloid lying in front of the yz-plane can be given in terms of u and v as follows:

In this parametric form, u can range over all real numbers to cover the entire surface in front of the yz-plane, while v can oscillate between -∞ to +∞ to provide a full representation of the surface.

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Final answer:

The sensitivity of the new test for celiac disease is 98.02% (Option C).

Explanation:

The question asks us to calculate the sensitivity of a new test for celiac disease. Sensitivity is the ability of a test to correctly identify those with the disease (true positive rate), and it is calculated as the number of true positives divided by the number of true positives plus the number of false negatives, which is essentially all the actual disease cases.

According to the provided data, the new test for celiac disease has 99 true positive results. To find out the total number of disease cases, we first need to calculate the expected number of people with celiac disease in the sample, which is 1.32% of 7642. That is approximately 100.87, or about 101 people (since we can't have a fraction of a person). Given that, we can assume there are 101 actual cases of celiac disease in the sample.

The sensitivity can be calculated as:

Sensitivity = (True Positives) / (True Positives + False Negatives)

= 99 / 101

= 0.9802 or 98.02%

Therefore, the sensitivity of the test is 98.02%, matching option C).

Answer:

The expected earnings of the attendant for this particular period are: $12.66

Step-by-step explanation:

We have to calculate expected mean here:

So,

E(x) = ∑x*f(x)

[tex]E(X) = \{(7 * \frac{1}{12} )+(9 * \frac{1}{12} )+(11 * \frac{1}{4} )+(13 * \frac{1}{4} )+(15 * \frac{1}{6} )+(17 * \frac{1}{6})\\= 0.58+0.75+2.75+3.25+2.5+2.83\\=12.66\ dollars[/tex]

Therefore, the expected earnings of the attendant for this particular period are: $12.66 ..

Considering the discrete distribution, it is found that the attendant’s expected earnings for this particular period are of $12.67.

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

Hence, considering the probability of each earning amount, the expected earnings are of the attendant is given by:

[tex]E(X) = 7\frac{1]{12} + 9\frac{1}{12} + 11\frac{1}{4} + 13\frac{1}{4} + 15\frac{1}{6} + 17\frac{1}{6} = \frac{7 + 9 + 33 + 39 + 30 + 34}{12} = 12.67[/tex]

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Answer:

The monthly payment is $35.10.

Step-by-step explanation:

p = 510

r = [tex]13.9/12/100=0.011583[/tex]

n = [tex]12+4=16[/tex]

The EMI formula is :

[tex]\frac{p\times r\times(1+r)^{n} }{(1+r)^{n}-1 }[/tex]

Now putting the values in formula we get;

[tex]\frac{510\times0.011583\times(1+0.011583)^{16} }{(1+0.011583)^{16}-1 }[/tex]

=> [tex]\frac{510\times0.011583\times(1.011583)^{16} }{(1.011583)^{16}-1 }[/tex]

= $35.10

Therefore, the monthly payment is $35.10.

Answer: 2.65%

Step-by-step explanation:

Given : The  measurement of the circumference of a circle =  68 centimeters

Possible error : [tex]dC=0.9[/tex] centimeter.

The formula to find the circumference :-

[tex]C=2\pi r\\\\\Rightarrow\ r=\dfrac{C}{2\pi}\\\\\Rightarrow\ r=\dfrac{68}{2\pi}=\dfrac{34}{\pi}[/tex]

Differentiate the formula of circumference w.r.t. r , we get

[tex]dC=2\pi dr\\\\\Rightarrow\ dr=\dfrac{dC}{2\pi}=\dfrac{0.9}{2\pi}=\dfrac{0.45}{\pi}[/tex]

The area of a circle  :-

[tex]A=\pi r^2=\pi(\frac{34}{\pi})^2=\dfrac{1156}{\pi}[/tex]

Differentiate both sides w.r.t r, we get

[tex]dA=\pi(2r)dr\\\\=\pi(2\times\frac{34}{\pi})(\frac{0.45}{\pi})\\\\=\dfrac{30.6}{\pi}[/tex]

The percent error in computing the area of the circle is given by :-

[tex]\dfrac{dA}{A}\times100\\\\\dfrac{\dfrac{30.6}{\pi}}{\dfrac{1156}{\pi}}\times100\\\\=2.64705882353\%\approx 2.65\%[/tex]

To approximate the percent error in computing the area of the circle, calculate the actual area using the given circumference and radius formula. Then find the difference between the actual and estimated areas, and divide by the actual area to get the percent error.

To approximate the percent error in computing the area of the circle, we need to first find the actual area of the circle and then calculate the difference between the actual area and the estimated area. The approximate percent error can be found by dividing this difference by the actual area and multiplying by 100.

The actual area of a circle can be calculated using the formula A = πr^2, where r is the radius. Since the circumference is given as 68 cm, we can find the radius using the formula C = 2πr. Rearranging the formula, we have r = C / (2π). Plugging in the given circumference, we get r = 68 / (2π) = 10.82 cm.

Now we can calculate the actual area: A = π(10.82)^2 = 368.39 cm^2.

The estimated area is given as 4.5 m^2, which is equal to 45000 cm^2 (since 1 m = 100 cm). The difference between the actual and estimated areas is 45000 - 368.39 = 44631.61 cm^2. The percent error can be found by dividing this difference by the actual area (368.39 cm^2) and multiplying by 100:

Percent error = (44631.61 / 368.39) * 100 ≈ 12106.64%.

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Check the Wronskian determinant:

[tex]W(e^{x/2},xe^{x/2})=\begin{vmatrix}e^{x/2}&xe^{x/2}\\\frac12e^{x/2}&\left(1+\frac x2\right)e^{x/2}\end{vmatrix}=\left(1+\frac x2\right)e^x-\frac x2e^x=e^x\neq0[/tex]

The determinant is not zero, so the solutions are indeed linearly independent.

To verify a fundamental set of solutions for the given differential equation, one must demonstrate that the functions e^{x/2} and xe^{x/2} satisfy the equation and that their Wronskian is non-zero, indicating linear independence.

The student's question pertains to verifying whether a given set of functions, e^{x/2} and xe^{x/2}, form a fundamental set of solutions for the differential equation 4y'' - 4y' + y = 0. A set of solutions is fundamental if the functions are linearly independent and satisfy the differential equation. Linear independence can be proved by calculating the Wronskian, which must be non-zero over the given interval. To show that these functions are solutions, they must be substituted into the differential equation to check if it holds true.

To check for linear independence, we can compute the Wronskian:

which simplifies to e^{x} (1 - (x/2)) that is non-zero for all real numbers x, proving linear independence.

To verify if the functions satisfy the differential equation, we substitute each function into the equation. The derivatives of e^{x/2} and xe^{x/2} are taken, and then these are plugged into the equation to confirm that it yields zero.