Use the variation of parameters method to solve the DE y"+y'- 2y=1

Answer: (A) 30%

Step-by-step explanation:

Given : The probability that a bird will lay an egg =0.75

The probability that the egg will hatch  =0.50

Now, the probability that the egg will lay an egg and hatch = [tex]0.75\times0.50[/tex]

Also,The probability that the chick will be eaten by a snake before it fledges =0.20

Then, the probability that the chick will not  be eaten by a snake before it fledges : 1-0.20=0.80

Now, the probability that a parent will have progeny that survive to adulthood will be :-

[tex]0.75\times0.50\times0.80=0.30=30\%[/tex]

Hence, the probability that a parent will have progeny that survive to adulthood =  30%

Answer with Step-by-step explanation:

We are given that twos sets

[tex]S_1[/tex]={a+bx:[tex]a,b\in R[/tex]}=All polynomials which can expressed as  a linear combination of 1 and x.

[tex]S_2[/tex]={ax+b(2+x):[tex]a,b\in R[/tex]}=All polynomials which can be expressed as   a linear combination of x and 2+x.

We have to prove that given two sets are same.

[tex]S_2[/tex]={ax+2b+bx}={(a+b)x+2b}={cx+d}

[tex]S_2[/tex]={cx+d}=All polynomials which can be expressed as  a linear combination of 1 and x.

Because a+b=c=Constant

2b= Constant=d

Hence, the two sets are same .

slope-intercept formula y=mx+b

Answer:

The vertical height, h = 34.64 feets

Step-by-step explanation:

Given that,

Length of the ladder, l = 40 ft

The ladder makes an angle of  60 degrees with the ground, [tex]\theta=60^{\circ}[/tex]

We need to find the vertical height of of which the ladder will reach. Let it iss equal to h. Using trigonometric equation,      

[tex]sin\theta=\dfrac{perpendicular}{hypotenuse}[/tex]

Here, perpendicular is h and hypotenuse is l. So,

[tex]sin(60)=\dfrac{h}{40}[/tex]

[tex]h=sin(60)\times 40[/tex]

h = 34.64 feets

So, the vertical height of which the ladder will reach is 34.64 feets. Hence, this is the required solution.

Answer:

Taking into account the information presented in the image attached for this food.  

Naturally sugar are

2 g

Step-by-step explanation:

Information presented regarding to the composition of this food is attached.

One serving has a weight of 55 g.

Total carbohydrates in one serving are 37 g. This value includes fiber and sugars.

If we detail sugars, it has 12 g of total sugars (St), where 10 g are added sugars (Sa) and all the rest Natural sugars (Sn).

Expressing sugars as an equation

St = Sa + Sn

Isolating Sn value

Sn = St - Sa

Sn = 12 g - 10 g  

Sn = 2 g

Finally, Natural sugars (Sn) is 2 g.  

All 22 grams of total sugars per serving in the food are naturally occurring, as the label specifies there are 0g of added sugars.

To determine the grams of naturally occurring sugar in a serving of food, we need to examine the nutritional information provided.

According to the label, it includes 0g added sugars, which implies that all of the sugars listed are naturally occurring.

Therefore, if there are 22g total sugars in one serving of the food, and none of these are added sugars, then all 22 grams can be attributed to naturally occurring sugars.

The volume of the ark in cubic feet is  697500 feet³

Calculation:-

1 cubit= 6 palm

1 palm=3.10 inches (given in the question)

⇒the volume of a cuboid is =(lenght*breath*height)

lenght=300 cubit

wide=50 cubit

height=30 cubit

volume=300*50*30

            450000 cubit³

since 1 cubit = 6 palm

        450000 cubit = 6*450000 palm

                                  2700000 palm

Again 1 palm = 3.10 inches (given in question)

       ∴ 2700000 palm= 2700000*3.10 inches

                                    =    8370000 inches³

   12 inch = 1 feet

  8370000 inch = 1/12*8370000 feet³

                             = 697500 feet³ (answer)

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The estimated volume of Noah's Ark, assuming a shoe-box shape and given dimensions in cubits with one cubit equaling 1.55 feet, is approximately 1,583,737.5 cubic feet.

To estimate the volume of Noah's Ark using the measurements provided in cubits and converting them to a modern unit like feet, we first need to determine the length of a cubit in inches. As we are given that one cubit is equal to six palms and one palm is 3.10 inches, we can calculate the length of one cubit as follows:

Now, to convert inches to feet, we know that:

Therefore, one cubit in feet is:

Using this conversion, we can calculate the dimensions of the ark in feet:

To find the volume of the ark, we will multiply these dimensions together:

Therefore, the estimated volume of Noah's Ark is approximately 1,583,737.5 cubic feet.

Answer:

The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units

Step-by-step explanation:

In calculating the sample size to estimate a population proportion in which there is no information on an initial sample proportion, the principle of maximum uncertainty is assumed and a ratio [tex]P = 1/2[/tex] is assumed. The expression to calculate the size is:

[tex] n=\frac{z_{\alpha /2}^2}{4 \epsilon^2} [/tex]

With

Z value (for 0.005) [tex] Z _ {\alpha / 2} = 1.64485 [/tex]

Significance level [tex] \alpha = 0.01 [/tex]

Estimation error [tex] \epsilon = 0.03 [/tex]

[tex] n=\frac{(1.64485)^2}{(4)(0.03)^2} = 751.5398[/tex]

The minimum approximate size to reach a maximum estimation error of 0.03 and a 99% confidence is 752 units

To determine the percentage of male customers with a margin of error of 3% at a 99% confidence level, the manager would need to survey approximately 1847 customers.

The question is related to the concept of statistics, and more specifically to the idea of a confidence interval

for a population proportion. When you want to be very sure about your estimates, you use a high level of confidence. The standard formula for calculating the sample size needed in order to get a certain margin of error at a certain confidence level is n = [Z^2 * P * (1-P)] / E^2. In this formula, n is the sample size, Z is the z-score associated with your desired level of confidence, P is the preliminary estimate of the population proportion, and E is the desired margin of error. If the manager doesn't have a precursory idea of what the proportion of male customers is, it's standard to use P = 0.5. The Z score for a 99% confidence interval is approximately 2.576. Substituting these values, the manager would need a sample size of approximately 1846 customers. For more accuracy, it's better to round up to the next nearest whole number, so the minimum sample size required would be 1847.

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Answer:

Protein

Step-by-step explanation:

According to U.S. Food and Drug Administration (FDA) information, the current scientific evidence indicates that protein intake is not a matter of public concern, this is why the FDA does not request to add a % Daily Value on the Nutrition Facts panel, unless if the product is made for protein such as 'high protein' products or if this food is meant for use by childrend under 4 years old.

Answer:Protein

Step-by-step explanation: took quiz :)

Answer:

2400 laborers

Step-by-step explanation:

Let N be the amount of laborers employed at the plant. The amount of heaters produced by all workers in an hour is:

heaters = 0.15heaters/hour * N

In a month the total amunt of heaters will be:

heaters = 0.15 * N * 160

Since this has to be 57600 to meet the expected demand:

57600 = 0.15 * N * 160   Solving for N we get:

N = 57600 / (0.15 * 160) = 2400 laborers

Answer: 52$

Step-by-step explanation:

The money Dave spent on a t-shirt is obtained multiplying 120$ by the fraction 2/5

120(2/5) = 48$

And the money he spent on mountain bike magazines is obtained multiplying 120$ by the fraction 1/6

120(1/6)= 20$

The money he has left is:

120$ - 48$ -20$ = 52$

Answer:

x = 2100 - 20p

Step-by-step explanation:

Let the quantity demanded be 'x'

unit price be 'p'

thus, from the given relation in the question, we have

p (100) = $100

and,

p (1100) = $50

now, from the standard equation for the line

[tex]\frac{\textup{p - p(100)}}{\textup{x - 100}}[/tex]  = [tex]\frac{\textup{p(1100) - p(100)}}{\textup{50 - 100}}[/tex]

or

[tex]\frac{\textup{p - 100}}{\textup{x - 100}}[/tex]  = [tex]\frac{\textup{50 - 100}}{\textup{1100 - 100}}[/tex]

or

1000 × (p - 100) = - 50 × ( x - 100 )

or

20p - 2000 = - x + 100

or

x = 2100 - 20p

The demand equation for the MP3 players can be determined using the given data points. By setting up a system of linear equations and solving for the values of a and b, we can find the demand equation Qd = 1500 - 10P.

The demand equation can be determined using the given information. We know that when the price is set at $100, the quantity demanded is 100 units, and when the price is $50, the quantity demanded is 1100 units.

We can set up a linear demand equation in the form Qd = a + bP, where Qd is the quantity demanded and P is the unit price. Using the two data points, we can solve for the values of a and b.

Therefore, the demand equation is Qd = 1500 - 10P.

Answer:

In kJ/kg.K - 1.005  kJ/kg degrees Kalvin.

In  J/g.°C  -  1.005 J/g °C

In kcal/ kg °C  0.240 kcal/kg °C

In Btu/lbm-°F   0.240 Btu/lbm degree F

Step-by-step explanation:

given data:

specific heat of air = 1.005 kJ/kg °C

In kJ/kg.K

1.005 kJ./kg °C = 1.005 kJ/kg degrees Kalvin.

In  J/g.°C

[tex]1.005 kJ/kg C \times (1000 J/1 kJ) \times (1kg / 1000 g) = 1.005 J/g °C[/tex]

In kcal/ kg °C

[tex]1.005 kJ/kg C \times (\frac{1 kcal}{4.190 kJ}) = 0.240 kcal/kg C[/tex]  

For   kJ/kg. °C to Btu/lbm-°F  

Need to convert by taking following conversion ,From kJ to Btu, from kg to lbm and from degrees C to F.

[tex]1.005 kJ/kg C \frac{1 Btu}{1.055 kJ} \times \frac{0.453 kg}{1 lbm} \times \frac{(5/9)\ degree C}{ 1\ degree F}  = 0.240 Btu/lbm degree F[/tex]

1.005 kJ/kg C =  0.240 Btu/lbm degree F

Answer:

[tex]CV=\frac{0.1252}{1.57}=0.07975[/tex]

[tex]CV=\frac{0.1317}{1.72}=0.07657[/tex]

The relative variability is almost equal in both samples a slight greater variability can be noticed in the first sample.

Step-by-step explanation:

The coefficient of variation of a sample is defined as the ratio between the mean standard deviation and the sample mean. And it represents the percentage relation of the variation of the data with respect to the average.

[tex]CV=\frac{S}{\bar X}[/tex]

In the case of the first sample you have:

[tex]CV=\frac{0.1252}{1.57}=0.07975[/tex]

In the case of the second sample you have:

[tex]CV=\frac{0.1317}{1.72}=0.07657[/tex]

The relative variability is almost equal in both samples a slight greater variability can be noticed in the first sample.

Step-by-step explanation:

For each case we have the next step by step solution.

Answer:

0.0559

Step-by-step explanation:

Cant prove it but its right

To test the claim that more than 65% homeowners in Omaha possess a lawnmower, a one-tailed test of proportion was performed. After calculating a test statistic (z =1.52), it was determined that the P-value is 0.064, suggesting there is a 6.4% probability that a sample proportion this high could happen by chance given the null hypothesis is true.

In this question, you're asked to calculate the P-value for a test of the claim that the proportion of homeowners with lawn mowers in Omaha is higher than 65%. The proportion from the sample size of Omaha is given as 340/497 = 0.68 or 68%. To test this claim, you would employ a one-tailed test of proportion. The null hypothesis (H0) is that the proportion is equal to 65%, while the alternative hypothesis (Ha) is that the proportion is greater than 65%.

To determine the P-value, you need to first calculate the test statistic (z) using the formula: z = (p - P) / sqrt [(P(1 - P)) / n], where p is the sample proportion, P is the population proportion, and n is the sample size. Substituting into the formula, z = (0.683 - 0.65) / sqrt [(0.65 * 0.35) / 497] = 1.52. The P-value is the probability of getting a z-score that is greater than or equal to 1.52.

Using a standard normal distribution table, or an online z-score calculator, you can find that P(Z > 1.52) = 0.064 (approximately). This is the P-value for the test, which indicates that if the null hypothesis is true, there is a 6.4% probability that a sample proportion this high could occur by chance.

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Answer: Our required probability is [tex]\dfrac{11}{20}[/tex]

Step-by-step explanation:

Since we have given that

Number of red marbles = 8

Number of blue marbles = 2

Number of yellow marbles = 7

Number of green marbles = 3

So, Total number of marbles = 8 + 2 + 7 + 3 = 20

So, Probability for selecting a red marble and then selecting a green marbles is given by

P(red) + P(green) is equal to

[tex]\dfrac{8}{20}+\dfrac{3}{20}\\\\=\dfrac{8+3}{20}\\\\=\dfrac{11}{20}[/tex]

Hence, our required probability is [tex]\dfrac{11}{20}[/tex]

Answer: 0.144

Step-by-step explanation:

Formula to find standard deviation: [tex]\sigma=\sqrt{\dfrac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}}[/tex]

Given : The growth rate of a random sample of knockout lines:-

0.8, 0.98, 0.72, 1, 0.82, 0.63, 0.63, 0.75, 1.02, 0.97, 0.86

Here ,

[tex]\overline{x}=\dfrac{\sum_{i=1}^{10}x_i}{n}\\\\=\dfrac{0.8+0.98+0.72+1+0.82+0.63+0.63+ 0.75+1.02+ 0.97+ 0.86}{10}\\\\=\dfrac{9.18}{10}\approx0.83[/tex]

[tex]\sum_{i=1}^n(x_i-\overline{x})^2=(-0.03)^2+(0.15)^2+(-0.11)^2+(0.17)^2+(-0.01)^2+(-0.2)^2+(-0.2)^2+(-0.08)^2+(0.19)^2+(0.14)^2+(0.03)^2\\\\=0.2075[/tex]

Now, the standard deviation:

[tex]\sigma=\sqrt{\dfrac{0.2075}{10}}=0.144048602909\approx0.144[/tex]

Hence, the standard deviation of growth rate this sample of yeast lines =0.144

The standard deviation of the growth rates for the given sample of yeast knockout lines is 0.148 when rounded to three decimal places.

To calculate the standard deviation of the growth rates of the yeast knockout lines, we first need to compute the mean (average) of the given data. Then, following the steps, we find the variance by calculating the difference between each value and the mean, squaring those differences, and finding their average. Finally, we take the square root of the variance to find the standard deviation.

Here are the calculations using the given growth rates:

Mean (average) = (0.8 + 0.98 + 0.72 + 1 + 0.82 + 0.63 + 0.63 + 0.75 + 1.02 + 0.97 + 0.86) / 11 = 0.836

Variance = 0.021918

Standard Deviation = sqrt(variance) = sqrt(0.021918) ≈ 0.148

Therefore, the standard deviation of the sample growth rates, rounded to three decimal places, is 0.148.

Answer:

Ans. the present value of $1,300/month, at 6.4% compounded monthly for 360 months (30 years) is $207,831.77

Step-by-step explanation:

Hi, first, we have to turn that 6.4% compound monthly rate into an effective rate, one that meets the units of the payment, in our case, effective monthly, that is:

[tex]r(EffectiveMonthly)=\frac{r(CompMonthly)}{12} =\frac{0.063}{12} =0.005333[/tex]

Therefore, our effective monthly rate is 0.5333%, and clearly the time of the investment is 30 years*12months=360 months.

Now, we need to use the following formula.

[tex]Present Value=\frac{A((1+r)^{n}-1) }{r(1+r)^{n} }[/tex]

Everything should look like this.

[tex]Present Value=\frac{1,300((1+0.005333)^{360}-1) }{0.005333(1+0.005333)^{360} }[/tex]

Therefore

[tex]PresentValue=207,831.77[/tex]

Best of luck.

Answer:

4 parameters are necessary to specify all solutions and correspond to the number of free variables of the system.

Step-by-step explanation:

Remember that the number of free variables of a system is equal to m-rank(A) where m is the number of unknowns variables and A is the matrix of the system.

Since the system is consistent and the rank of the matrix is 3 then echelon form of the augmented matrix has two rows of zeros.

Then m-rank(A)=7-3=4.

Answer:

See explanation below.

Step-by-step explanation:

The prime numbers are bold:

1  2  3 4  5 6  7  8  9  10  11 12  13  14  15  16  17

18  19  20  21  22  23  24  25  26  27  28  29  30  31

a) We can see that as we go higher, twin primes seem less frequent but even considering that, there is an infinite number of twin primes. If you go high enough you will still eventually find a prime that is separated from the next prime number by just one composite number.

b) I think it's interesting the amount of time that has been devoted to prove this conjecture and the amount of mathematicians who have been involved in this. One of the most interesting facts was that in 2004 a purported proof (by R. F. Arenstorf) of the conjecture was published but a serious error was found on it so the conjecture remains open.

Answer:

[tex]volume = 1536 m^3[/tex]

Step-by-step explanation:

given data;

B = 16m

b =8 m

height   H = 4 m

length  L = 32 m

volume of any right cylinder = (Area of bottom) \times (length)

Volume = A* L

The area of a trapezoid is

[tex]A=\frac{1}{2} H*(b+B)[/tex]

[tex]A =\frac{1}{2} 4*(8+16)[/tex]

[tex]A = 48 m^2[/tex]

therefore volume is given as  

volume = 48*32

[tex]volume = 1536 m^3[/tex]

Final answer:

To find the volume of the right trapezoidal cylinder, calculate the area of the trapezoid base and multiply by the cylinder's length. With B = 16m, b = 8m, and height = 4m, the trapezoid's area is 48m². The volume of the cylinder is 1536m³.

Explanation:

The volume of a right trapezoidal cylinder can be calculated using the area of the trapezoid as the base area and then multiplying by the height of the cylinder. Firstly, to calculate the area of the trapezoid (the base of the cylinder), we use the formula for the area of a trapezoid, which is ½ × (sum of the parallel sides) × height of the trapezoid. In this case, the parallel sides are B = 16m and b = 8m, and the height is 4m.

The area A of the trapezoid is then ½ × (16m + 8m) × 4m = ½ × 24m × 4m = 48m2. To find the volume of the trapezoidal cylinder, we multiply this area by the length of the cylinder, which is 32m. So, the volume V = 48m2 × 32m = 1536m3.

Answer:

59 coins

Step-by-step explanation:

The number of coins in Anthony's collection is 294

Using the parameters given for our Calculation;

Total number of coins now :

Six times as many coins can be calculated thus :

Therefore, the number of coins in Anthony's collection is 294

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Answer:  A ∪ B = {a, b, c, d}

               (A ∪ B) ∪ C =  = {a, b, c, d, e}

               B ∪ C =  = {a, b, d, e}

               (B ∪ C) ∪ A  =  = {a, b, c, d, e}

Step-by-step explanation:

A = {a, b, c}       B = {a, b, d}         C = {b, d, e}

Union means "to join" so combine the sets to form a union.

A ∪ B = {A & B}

         = {a, b, c & a, b, d}

         = {a, b, c, d}          because we do not need to list a & b twice

(A ∪ B) ∪ C = {(A ∪ B) & C)

                  = {a, b, c, d & b, d, e}

                  = {a, b, c, d, e}     because we do not need to list b & d twice

B ∪ C = {A & B}

         = {a, b, d & b, d, e}

         = {a, b, d, e}           because we do not need to list b & d twice

(B ∪ C) ∪ A = {(B ∪ C) & A)

                  = {a, b, d, e & a, b, c}

                  = {a, b, c, d, e}     because we do not need to list a & b twice

Final answer:

We utilized set theory to find the union of sets A, B, and C in different orders and observed that the associative property of union holds true, meaning the order of union operations does not change the outcome which is {a, b, c, d, e}.

Explanation:

To answer this student's question, we can apply set theory concepts, specifically the concepts of the union and associativity.

Given sets A = {a, b, c}, B = {a, b, d}, and C = {b, d, e}, the union of these sets can be found as follows:

From these operations, we observe that regardless of the order in which we take the union of the three sets, the result is the same. This demonstrates the associative property of union in set theory, where the order in which unions are performed does not affect the final outcome. We can generalize this as A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C.

Answer:

b.

917 sheet sets.

Step-by-step explanation:

We are asked to find the total number of sheet sets received by linens department.

To find the total number of sheet sets received by linens department, we will add the number of dozens of each sheet.

[tex]\text{Total number of sheet sets received in dozens}=19\frac{1}{2}+33\frac{2}{3}+23\frac{1}{4}[/tex]

[tex]\text{Total number of sheet sets received in dozens}=\frac{39}{2}+\frac{101}{3}+\frac{93}{4}[/tex]

Make common denominator:

[tex]\text{Total number of sheet sets received in dozens}=\frac{39*6}{2*6}+\frac{101*4}{3*4}+\frac{93*3}{4*3}[/tex]

[tex]\text{Total number of sheet sets received in dozens}=\frac{234}{12}+\frac{404}{12}+\frac{279}{12}[/tex]

[tex]\text{Total number of sheet sets received in dozens}=\frac{234+404+279}{12}[/tex]

[tex]\text{Total number of sheet sets received in dozens}=\frac{917}{12}[/tex]

To find the number of sheets, we will multiply number of dozens of sheets by 12 as 1 dozen equals 12.

[tex]\text{Total number of sheet sets received}=\frac{917}{12}\times 12[/tex]

[tex]\text{Total number of sheet sets received}=917[/tex]

Therefore, the total number of sheet sets received is 917.

Answer:

According to the rule of 72, the doubling time for this interest rate is 8 years.

The exact doubling time of this amount is 8.04 years.

Step-by-step explanation:

Sometimes, the compound interest formula is quite complex to be solved, so the result can be estimated by the rule of 72.

By the rule of 72, we have that the doubling time D is given by:

[tex]D = \frac{72}{Interest Rate}[/tex]

The interest rate is in %.

In our exercise, the interest rate is 9%. So, by the rule of 72:

[tex]D = \frac{72}{9} = 8[/tex].

According to the rule of 72, the doubling time for this interest rate is 8 years.

Exact answer:

The exact answer is going to be found using the compound interest formula.

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this exercise, we have:

We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.

is double the initial amount, double the principal.

[tex]A = 2P[/tex]

[tex]r = 0.09[/tex]

The interest is compounded anually, so [tex]n = 1[/tex]

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]2P = P(1 + \frac{0.09}{1})^{t}[/tex]

[tex]2 = (1.09)^{t}[/tex]

Now, we apply the following log propriety:

[tex]\log_{a} a^{n} = n[/tex]

So:

[tex]\log_{1.09}(1.09)^{t} = \log_{1.09} 2[/tex]

[tex]t = 8.04[/tex]

The exact doubling time of this amount is 8.04 years.

Answer:

The T-score is 2.49216

Step-by-step explanation:

A 98% confidence interval should be estimated for the end times of cyclists. Since the sample is small, a T-student distribution should be used, in such an estimate. The confidence interval is given by the expression:

[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]

[tex]n = 25\\\alpha = 0.02\\T_{(n-1;\frac{\alpha}{2})}= T_{(24;0.01)} = 2.49216[/tex]

Then the T-score is 2.49216

Answer:

2.485

Step-by-step explanation:

Answer:

Step-by-step explanation:

The given information can be tabulated as follows:

                          Graphics G     Printing P            Total

X                                      3                 2                    5

Y                                      1                  1                      2

Available                       21                19

We have the constraints as

[tex]3x+y\leq 21\\2x+y\leq 19\\x\leq 2\\y\leq 15[/tex]

Thus we have solutions as

[tex]0\leq x\leq 2\\0\leq y\leq 15[/tex]

Answer:

One proof can be as follows:

Step-by-step explanation:

We have that [tex]g.c.d(a,b)=1[/tex] and [tex]a=cp, b=dq[/tex] for some integers [tex]p, q[/tex], since [tex]c[/tex] divides [tex]a[/tex] and [tex]d[/tex] divides [tex]b[/tex].  By the Bezout identity two numbers [tex]a,b[/tex] are relatively primes if and only if there exists integers [tex]x,y[/tex] such that

[tex]ax+by=1[/tex]

Then, we can write

[tex]1=ax+by=(cp)x+(dq)y=c(px)+d(qy)=cx'+dy'[/tex]

Then [tex]c[/tex] and [tex]d[/tex] are relatively primes, that is to say,

[tex]g.c.d(c,d)=1[/tex]

Answer:

Since 2 and -1 are eigenvalues of the differential equation,

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

is a solution to the differential equation

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

The solution to the initial value problem is:

[tex]x(t) = \frac{20}{3}e^{-t} +  \frac{10}{3}e^{2t}[/tex]

Step-by-step explanation:

We have the following differential equation:

[tex]x'' - x' - 2x = 0[/tex]

The first step is finding the eigenvalues for this differential equation, that is, finding the roots of the following second order equation:

[tex]r^{2} - r - 2 = 0[/tex]

[tex]\bigtriangleup = (-1)^{2} -4*1*(-2) = 1 + 8 = 9[/tex]

[tex]r_{1} = \frac{-(-1) + \sqrt{\bigtriangleup}}{2*1} = \frac{1 + 3}{2} = 2[/tex]

[tex]r_{2} = \frac{-(-1) - \sqrt{\bigtriangleup}}{2*1} = \frac{1 - 3}{2} = -1[/tex]

Since 2 and -1 are eigenvalues of the differential equation,

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

is a solution to the differential equation.

Solution of the initial value problem:

[tex]x(t) = c_{1}e^{-t} + c_{2}e^{2t}[/tex]

[tex]x(0) = 10[/tex]

[tex]10 = c_{1}e^{-0} + c_{2}e^{2*0}[/tex]

[tex]c_{1} + c_{2} = 10[/tex]

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

[tex]x'(t) = -c_{1}e^{-t} + 2c_{2}e^{2t}[/tex]

[tex]x'(0) = 0[/tex]

[tex]0 = -c_{1}e^{-0} + 2c_{2}e^{2*0}[/tex]

[tex]-c_{1} + 2c_{2} = 0[/tex]

[tex]c_{1} = 2c_{2}[/tex]

So, we have to solve the following system:

[tex]c_{1} + c_{2} = 10[/tex]

[tex]c_{1} = 2c_{2}[/tex]

[tex]2c_{2} + c_{2} = 10[/tex]

[tex]3c_{2} = 10[/tex]

[tex]c_{2} = \frac{10}{3}[/tex]

[tex]c_{1} = 2c_{2} = \frac{20}{3}[/tex]

The solution to the initial value problem is:

[tex]x(t) = \frac{20}{3}e^{-t} +  \frac{10}{3}e^{2t}[/tex]

Answer:

$ 293.40

Step-by-step explanation:

Given,

The allowance given by the store = $ 10.50,

The original cost of the dress = $ 285.99,

So, the cost of the dress after allowance = 285.99 - 10.50

= $ 275.49,

Now, the sales tax = 6.5 %,

Hence, the final amount of the dress after tax = (100+6.5)% of 275.49

= 106.5% of 275.49

[tex]=\frac{106.5\times 275.49}{100}[/tex]

[tex]=\frac{29339.685}{100}[/tex]

= $ 293.39685

≈ $ 293.40

Final answer:

The customer paid $293.40 for the dress after subtracting the allowance and adding the sales tax, which was calculated and rounded to the nearest penny.

Explanation:

To calculate the final price the customer paid for the dress after receiving an allowance and including sales tax, we follow these steps:

Therefore, the customer paid $293.40 for the dress after the allowance and including the 6.5% sales tax.

[tex]y=\displaystyle\sum_{n\ge0}a_nx^n=a_0+\sum_{n\ge1}a_nx^n[/tex]

[tex]y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n=a_1+\sum_{n\ge0}(n+1)a_{n+1}x^n[/tex]

[tex]y''=\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n[/tex]

Notice that [tex]y(0)=-3=a_0[/tex], and [tex]y'(0)=2=a_1[/tex].

Substitute these series into the ODE:

[tex](x-1)y''-xy'+y=0[/tex]

[tex]\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^{n+1}-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge0}(n+1)a_{n+1}x^{n+1}+\sum_{n\ge0}a_nx^n=0[/tex]

Shift the indices to get each series to include a [tex]x^n[/tex] term.

[tex]\displaystyle\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge0}a_nx^n=0[/tex]

Remove the first term from both series starting at [tex]n=0[/tex] to get all the series starting on the same index [tex]n=1[/tex]:

[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge1}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge1}a_nx^n=0[/tex]

[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}\bigg[n(n+1)a_{n+1}-(n+1)(n+2)a_{n+2}-(n-1)a_n\bigg]x^n=0[/tex]

The coefficients are given recursively by

[tex]\begin{cases}a_0=-3\\a_1=2\\\\a_n=\dfrac{(n-2)(n-1)a_{n-1}-(n-3)a_{n-2}}{n(n-1)}&\text{for }n>1\end{cases}[/tex]

Let's see if we can find a pattern to these coefficients.

[tex]a_2=\dfrac{a_0}2=-\dfrac32=-\dfrac3{2!}[/tex]

[tex]a_3=\dfrac{2a_2}{3\cdot2}=-\dfrac12=-\dfrac3{3!}[/tex]

[tex]a_4=\dfrac{2\cdot3a_3-a_2}{4\cdot3}=-\dfrac18=-\dfrac3{4!}[/tex]

[tex]a_5=\dfrac{3\cdot4a_4-2a_3}{5\cdot4}=-\dfrac1{40}=-\dfrac3{5!}[/tex]

[tex]a_6=\dfrac{4\cdot5a_5-3a_4}{6\cdot5}=-\dfrac1{240}=-\dfrac3{6!}[/tex]

and so on, suggesting that

[tex]a_n=-\dfrac3{n!}[/tex]

which is also consistent with [tex]a_0=3[/tex]. However,

[tex]a_1=2\neq-\dfrac3{1!}=-3[/tex]

but we can adjust for this easily:

[tex]y(x)=-3+2x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]

[tex]y(x)=5x-3-3x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]

Now all the terms following [tex]5x[/tex] resemble an exponential series:

[tex]y(x)=5x-3\displaystyle\sum_{n\ge0}\frac{x^n}{n!}[/tex]

[tex]\implies\boxed{y(x)=5x-3e^x}[/tex]

The given differential equation is a second-order homogeneous differential equation. The power series method may not straightforwardly work for this equation due to the x dependence in the coefficients. Even using advanced techniques like the Frobenius method, the solution cannot be expressed as an elementary function.

The given differential equation (x − 1)y'' - xy' + y = 0 is an example of a second order homogeneous differential equation. To solve the equation using the power series method, let's assume a solution of the form y = ∑(from n=0 to ∞) c_n*x^n. Substituting this into the equation and comparing coefficients, we can find a relationship for the c_n's and thus the power series representation of the solution.

However, for this particular differential equation, the power series method is not straightforward because of the x dependence in the coefficients of y'' and y'. Therefore, the standard power series approach would not work, and we would need more advanced techniques like the Frobenius method which allows for non-constant coefficients.

Unfortunately, even with the Frobenius method, the solution isn't an elementary function, meaning the solution cannot be expressed in terms of a finite combination of basic arithmetic operations, exponentials, logarithms, constants, and solutions to algebraic equations.

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