A simple random sample of men over age 18 is taken to estimate the mean weight of all adult males.
Is this study
A. REPRESENTATIVE?
B. NON-REPRESENTATIVE?

Answer:

The reduced row-echelon form of the linear system is [tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right][/tex]

Step-by-step explanation:

We will solve the original system of linear equations by performing a sequence of the following elementary row operations on the augmented matrix:

To find the reduced row-echelon form of this augmented matrix

[tex]\left[\begin{array}{cccc}2&3&-1&14\\1&2&1&4\\5&9&2&7\end{array}\right][/tex]

You need to follow these steps:

[tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\1&2&1&4\\5&9&2&7\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\0&1/2&3/2&-3\\5&9&2&7\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&3/2&-1/2&7\\0&1/2&3/2&-3\\0&3/9&9/2&-28\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&16\\0&1/2&3/2&-3\\0&3/9&9/2&-28\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&16\\0&1/2&3/2&-3\\0&0&0&-19\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&16\\0&2&3&-6\\0&0&0&-19\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&16\\0&2&3&-6\\0&0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&-6\\0&0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right][/tex]

Answer:

Step-by-step explanation:

Let's remember the equation of a line:

y = mx + b

m: slope (we know it's 3)

b: the y-intercept

So far, we have y = 3x + b, now we need to find b.

Replacing y and x for the given points (-3,-5):

-5 = 3*(-3) + b

-5 = -9 + b

b = -5 + 9

b = 4

The equation of the line that passes through the point (-3,-5) with a slope of 3 is y = 3x + 4

Answer:

y = 3x + 4

Step-by-step explanation:

Answer:

A. The variable​ Yr_Built is​ quantitative, with units years.

Step-by-step explanation:

Since, Yr_Built means the year built, Also year has a numerical value like 2008, 2000, 1865, etc.

So, it is quantitative.

Further, Data is said to be quantitative if it has a numerical value.

Categorical data is that data which has categories. For example, Gender: It has two categories Female and Male.

The data which as ordering is called ordinal data. Example, Bad, Average, Good.

Identifier Variable is the variable that is used for identity. Example, Employee No., etc.

Answer:

The proof makes use of congruences as follows:

Step-by-step explanation:

We can prove this result using congruences module 3. First of all we shall show that

[tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n\in \mathbb{N}[/tex]. By induction we have

Then induction we proved that [tex]2^{2n-1}\equiv 2 \pmod{3}[/tex] for all [tex]n>1[/tex]. Then

[tex]2^{2n-1}+1\equiv 2+1\equiv 3\equiv 0 \pmod{3}[/tex]

From here we conclude that the expression [tex]2^{2n-1}+1[/tex] is divisible by 3.

Answer:

Probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]    

Step-by-step explanation:

If we roll one die then probability to get any one side is [tex]\frac{1}{6}[/tex]

Therefore, probability to get 2 by rolling the die will be P(A) = [tex]\frac{1}{6}[/tex]

Now we flip a coin then getting head or tale probability is [tex]\frac{1}{2}[/tex]

Or probability to get head by flipping the coin P(B) = [tex]\frac{1}{2}[/tex]

Probability of happening both the events (rolling a 2 and flipping a head) will be denoted by

P(A∩B) = P(A)×P(B)

           = [tex]\frac{1}{6}\times \frac{1}{2}[/tex]

           = [tex]\frac{1}{12}[/tex]

Therefore, probability of rolling a 2 and flipping a head will be [tex]\frac{1}{12}[/tex]        

Answer:

The probability of rolling a 2 and flipping a head is [tex]\frac{1}{12}[/tex]

Step-by-step explanation:

Notice that rolling a die and flipping a coin are two independent events this means that the probability that one event occurs in no way affects the probability of the other event occurring. When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.

[tex]P(X and \:Y) =P(X) \cdot P(Y)[/tex]

When you roll a die there six outcomes from 1 to 6 and when you flip a coin are two possible outcomes (heads or tails).

We know that the probability of an event is

[tex]P=\frac{the \:number \:of \:wanted\:outcomes}{the \:number \:of \:possible \:outcomes}[/tex]

So the probability of rolling a 2 and flipping a head is

[tex]P(2 \:and \:H) = P(2) \cdot P(H)\\P(2 \:and \:H) = \frac{1}{2} \cdot \frac{1}{6}\\P(2 \:and \:H) = \frac{1}{12}[/tex]

Answer:

15,384 such tablets may be prepared from 5 kg of aspirin

Step-by-step explanation:

The problem states that aspirin tablets generally contain 325 mg of aspirin. And asks how many such tablets may be prepared from 5 kg of aspirin.

Since the problem measures the weight of a tablet in kg, the first step is the conversion of 325mg to kg.

Each kg has 1,000,000mg. So

1kg - 1,000,000mg

xkg - 325mg.

1,000,000x = 325

[tex]x = \frac{325}{1,000,000}[/tex]

x = 0.000325kg

Each tablet generally contains 0.000325kg of aspirin. How many such tablets may be prepared from 5 kg of aspirin?

1 tablet - 0.000325kg

x tablets - 5kg

0.000325x = 5

[tex]x = \frac{5}{0.000325}[/tex]

x = 15,384 tablets

15,384 such tablets may be prepared from 5 kg of aspirin

Around 15,385 tablets of aspirin can be created from 5 kilograms of aspirin, using the assumption that one tablet typically contains 325 milligrams of aspirin.

This question requires a basic understanding of the conversion from kilograms to milligrams. 5 kilograms of aspirin is equal to 5,000,000 milligrams (as 1 kilogram = 1,000,000 milligrams). So, if one aspirin tablet contains 325 mg of aspirin, we can calculate the number of tablets from 5kg, or 5,000,000 mg, of aspirin by simply dividing the total milligrams of aspirin by the milligrams per tablet.

To calculate:
Number of tablets = Total mass / Mass per tablet
= 5,000,000 mg / 325 mg/tablet

So, roughly 15,385 tablets of aspirin can be produced from 5 kg of aspirin.

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Answer:  (45.79, 51.21)

Step-by-step explanation:

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

Sample size : n= 51 , which is a large sample (n>30), so we use z-test.

Critical value: [tex]z_{\alpha/2}=2.576[/tex]

Sample mean : [tex]\overline{x}= 48.5\text{ hours}[/tex]

Standard deviation : [tex]\sigma=7.5\text{ hours}[/tex]

The confidence interval for population means is given by :-

[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]

i.e. [tex]48.5\pm(2.576)\dfrac{7.5}{\sqrt{51}}[/tex]

i.e.[tex]48.5\pm2.70534112234\\\\\approx48.5\pm2.71\\\\=(48.5-2.71, 48.5+2.71)=(45.79, 51.21)[/tex]

Hence, 99% confidence interval for the standard deviation of the number of hours this firm’s employees spend on work-related activities in a typical week =  (45.79, 51.21)

Final answer:

To construct a 99% confidence interval for the standard deviation of firm's employees' work hours, follow the steps to calculate the interval between 6.26 and 9.08 hours.

Explanation:

Confidence Interval Calculation:

Calculate the degrees of freedom (df) using the formula df = n - 1. For a sample size of 51, df = 51 - 1 = 50.

Determine the critical values from the chi-squared distribution for a 99% confidence level with 50 degrees of freedom. These critical values are 30.984 and 73.361.

Next, calculate the confidence interval for the standard deviation using the formula CI = sqrt((n-1) ×s² / chi-squared upper) to sqrt((n-1) ×s² / chi-squared lower), which results in 6.26 to 9.08 hours.

Answer:

E. t - 2t - 3t =32, t=8

Step-by-step explanation:

A. 2x + 5 (x-1)= 9, X= 2 it's ok because

[tex]2x+5(x-1)=9\\ 2x+5x-5=9\\ 7x=14\\ x=2[/tex]

B. p - 3(p-5)= 10, p= 2.5 it's ok as well because

[tex]p-3(p-5)=10\\ p-3p+15=10\\ -2p=-5\\ p=\frac{-5}{-2} \\ p=2.5[/tex]

C. 4 y +3 y=28, y=4 also it's ok because

[tex]4y+3y=28\\ 7y=28\\ y=\frac{28}{7} \\ y=4[/tex]

E. t - 2t - 3t =32, t=8 it's not ok because

[tex]t-2t-3t=32\\-4t=32\\t=\frac{32}{-4} \\t=-8[/tex]

a (-) is missing!

Final answer:

After reviewing the solutions provided by Mary, we can conclude that the incorrect solution is Option E, where the solution should be t = -8 instead of the provided answer t = 8.

Explanation:

Mary is reviewing her algebra quiz and needs to determine which one of her solutions is incorrect. We must go through each option and verify the results given.

Option A

2x + 5(x - 1) = 9, solution X = 2.

First, expand the equation: 2x + 5x - 5 = 9.

Then combine like terms: 7x - 5 = 9.

Next, add 5 to both sides: 7x = 14.

Finally, divide by 7: x = 2.

This solution is correct.

Option B

p - 3(p - 5) = 10, solution p = 2.5.

First, expand the equation: p - 3p + 15 = 10.

Then combine like terms: -2p + 15 = 10.

Next, subtract 15 from both sides: -2p = -5.

Finally, divide by -2: p = 2.5.

This solution is correct.

Option C

4y + 3y = 28, solution y = 4.

First, combine like terms: 7y = 28.

Then, divide by 7: y = 4.

This solution is correct.

Option E

t - 2t - 3t = 32, solution t = 8.

First, combine like terms: -4t = 32.

Next, divide by -4: t = -8.

This solution is incorrect because the given answer is t = 8. The correct answer should be t = -8.

Answer:

A bundle of goods that Country A has are 1000 iPods and 500 Tablets.

Step-by-step explanation:

Country A can produce either  iPods and Tablets.

Total man power  Country A has 100 workers.

As per information in the question w

100 worker make either ipod or 100 worker make tablets

1 worker can produce 10 iPods.

Therefore

100 workers can produce (100 * 10) = 1000 iPods.

on the other hand

1 worker can produce 5 tablets.

Therefore

100 workers can produce (100 * 5)=  500 Tablets

Thus,

A bundle of goods that Country A has are 1000 iPods and 500 Tablets.

Answer:

  B.  1/10 the value of

Step-by-step explanation:

In our base-10 place-value number system, the place value of a number is multiplied by 10 when it moves 1 place to the left. It is multiplied by 1/10 when it moves 1 place to the right.

A digit has 1/10 the value of the same digit one place to its left.

Answer:    B

Step-by-step explanation:

Answer: [tex]G=f(50)=14[/tex]

Step-by-step explanation:

Given : The amount of garbage, G, in tons per week, produced by a city with population p, measured in thousands of people, is given by G = f(p).

Also, The town of Tola has a population of 50,000 and produces 14 tons of garbage each week.

Then, the expression in terms of the function G will be :-

[tex]G=f(50)=14[/tex]

Answer:

[tex]-\frac{6}{37} + \frac{371}{37}i[/tex]

Step-by-step explanation:

We need to evaluate [tex]\frac{(5+6i)(5+6i)}{6+i}[/tex]

(5+6i)(5+6i) = (25 + 36i² + 60i) = (25 - 36 + 60i) = -11 + 60i

= [tex]\frac{-11+60i}{6+i}[/tex]

Now we rationalize the denominator.

Now, multiplying both the numerator and denominator by (6-i)

[tex]\frac{-66 + 11i + 360i - 60i^2}{36 - i^2} = \frac{-66 + 60  + 371i}{37} = \frac{-6 + 371i}{37}[/tex]

= [tex]-\frac{6}{37} + \frac{371}{37}i[/tex]

Formula used:

(a+b)² = a² + b² + 2ab

i² = -1

Answer: There are 196.85 cm in 500 inches.

Step-by-step explanation:

Since we have given that

1 inch = 2.54 centimeters

We need to find the number of centimeters are in 500 inches.

Since 1 inch = 2.54 centimeters

1 inch = [tex]\dfrac{1}{2.54}\ cm[/tex]

500 inches = [tex]\dfrac{500}{2.54}=196.85\ cm[/tex]

Hence, There are 196.85 cm in 500 inches.

To convert 500 inches to centimeters, use the conversion factor 2.54 cm = 1 inch. Cross-multiply and solve for x to find that there are approximately 1270 centimeters in 500 inches.

To convert inches to centimeters, we use the conversion factor 2.54 cm = 1 inch. In this case, we have 500 inches, so we can set up the following proportion:

We can cross-multiply and solve for x:

Calculating x, we find that there are approximately 1270 centimeters in 500 inches.

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Answer: The value of t become 10000.

Step-by-step explanation:

Since we have given that

[tex]\dfrac{1}{2}\times 10t=50000[/tex]

We need to find the value of t from the above expression:

first we divide 10t by 2 to get 50000.

[tex]\dfrac{10t}{2}=50000\\\\5t=50000\\\\t=\dfrac{50000}{5}\\\\t=10000[/tex]

Hence, the value of t become 10000.

Answer:

[tex]\frac{Wm^2C^{\circ}}{m^2-C^{\circ}}[/tex]

Step-by-step explanation:

We are given that A=[tex]\frac{W}{m^2-C^{\circ}}[/tex]

B=[tex]m^2[/tex]

C=[tex]C^{\circ}[/tex]

We are given that [tex]D=A\times B\times C[/tex]

We have to find the unit if D

Substitute the values then we get

[tex]D=\frac{W}{m^2-C^{\circ}}\times m^2\times C^{\circ}=\frac{Wm^2C^{\circ}}{m^2-C^{\circ}}[/tex]

[tex]D=\frac{Wm^2C^{\circ}}{m^2-C^{\circ}}[/tex]

Hence, units of D=[tex]\frac{Wm^2C^{\circ}}{m^2-C^{\circ}}[/tex]

To figure out how much E5 should be mixed with 6000 gal of E10 to make an E9 mixture, we need to use a weighted average equation: 0.10*6000 + 0.05*X = 0.09 * (6000 + X). Solving this equation will provide the required amount of E5 gasoline.

To solve this problem, we use a technique known as a weighted average. The weight is the number of gallons and the 'value' is the percentage of ethanol. We can formulate an equation using the principle that the sum of the ethanol in the initial gasolines will equal to the ethanol in the final mixture.

Let's denote by X the needed amount of E5 gasoline. Therefore, the total ethanol before mixing would be 0.10*6000 (from the E10 gasoline) + 0.05*X (from the E5 gasoline). After mixing, the total ethanol will be 0.09 (6000 + X).

Equating these two gives us 0.10*6000 + 0.05*X = 0.09 * (6000 + X). Solving this equation will give the needed amount of E5 gasoline.

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To create an E9 mixture, you need to mix 1500 gallons of E5 with 6000 gallons of E10. This is calculated using the volume percentages of ethanol in each mixture and solving a linear equation. The final mixture will contain 9% ethanol.

To solve the problem of finding out how much E5 should be mixed with 6000 gallons of E10 to create an E9 mixture, we need to set up an equation based on the percentage volumes of ethanol.

Let's denote the amount of E5 to be added as x (in gallons).

E10 contains 10% ethanol and 90% gasoline. Therefore, in 6000 gallons of E10, the amount of ethanol is:

E5 includes 5% ethanol and 95% gasoline. Therefore, in x gallons of E5, the amount of ethanol is:

We want to create an E9 mixture, which means the final mixture should contain 9% ethanol. The total volume of the final mixture will be:

The amount of ethanol in the final mixture can be represented as:

This amount should be 9% of the total final volume:

Solving for x:

Therefore, you need to mix 1500 gallons of E5 with 6000 gallons of E10 to create an E9 mixture.

The explanatory variable is the web page design. The response variables are the amount spent by the visitor and the amount of time visiting the site. Confounding could be an issue if other factors are not accounted for.

(a) The explanatory variable in this study is the web page design. It is a qualitative variable because it represents different design options.

(b) The two response variables are the amount spent by the visitor and the amount of time visiting the site. The amount spent is a quantitative variable, while the amount of time is also quantitative.

(c) Confounding could be an issue in this study if there are other factors that could affect the response variables and are not accounted for. For example, if there are differences in the age groups of the visitors using mobile and desktop platforms, their preferences may affect the response variables.

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To find the value of 'p' that satisfies the equation, we use Bayes' theorem and substitute the given values into the equation.

In this question, we need to find the value of p which satisfies the equation P(A|B) = P(A' | B'), where A is the event 'breath analyser indicates that legal limit is exceeded' and B is the event 'driver's blood alcohol percentage exceeds legal limit.'

Therefore, the value of p that should be used, rounded to 4 decimal places, can be calculated using the above steps.

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Answer:

[tex]\dfrac{2}{s^3}e^{-3s}\ +\ \dfrac{6}{s^2}e^{-3s}\ +\ \dfrac{9}{s}e^{-3s}\ -\ \dfrac{3}{s}[/tex]

Step-by-step explanation:

Given polynomial,

[tex]f(t)\ =\ t^2.u(t-3)\ -\ 3[/tex]

we can write above polynomial as

[tex]f(t)\ =\ (t-3+3)^2.u(t-3)\ -\ 3[/tex]

     [tex]=\ ((t-3)^2\ +\ 2\times 3\times (t-3)\ +\ 3^2).u(t-3)-3[/tex]

     [tex]=\ (t-3)^2.u(t-3)\ +\ 6(t-3).u(t-3)\ +\ 9.u(t-3)\ -\ 3[/tex]

Now, we have to calculate the Laplace of above polynomial

according to shifting property of Laplace transform, we can write

[tex]f(t-t_0)\ =\ F(s).e^{-st_0}[/tex]

So, we can write the Laplace transform of above polynomial as

[tex]L[f(t)]\ =\ L[(t-3)^2.u(t-3)\ +\ 6(t-3).u(t-3)\ +\ 9.u(t-3)\ -\ 3][/tex]

          [tex]=\ \dfrac{2}{s^3}e^{-3s}\ +\ \dfrac{6}{s^2}e^{-3s}\ +\ \dfrac{9}{s}e^{-3s}\ -\ \dfrac{3}{s}[/tex]

So, the Laplace transform of the given polynomial will be[tex]\ \dfrac{2}{s^3}e^{-3s}\ +\ \dfrac{6}{s^2}e^{-3s}\ +\ \dfrac{9}{s}e^{-3s}\ -\ \dfrac{3}{s}[/tex]

Answer:

750 units

Step-by-step explanation:

As given in question,

speed of solution = 30 mL/hr

1 L of solution contains heparin = 25000 units

=> 1000 mL of solution contains heparin = 25000 units

[tex]=>\ \textrm{1 mL of solution contains heparin}\ =\ \dfrac{25000}{1000}\ units[/tex]

                                                                             = 25 units

Hence, 30 mL of solutions contains heparin = 30 x 1 mL of solution contains heparin

                                                                         = 30 x 25

                                                                         = 750 units

Hence, heparin administered per hour = 750 units

Final answer:

To find the amount of heparin administered per hour at a rate of 30mL/hr in a 1L solution with 25,000 units, calculate the concentration in units/mL and multiply by the hourly rate. This results in 750 units of heparin administered per hour.

Explanation:

To calculate the amount of heparin administered per hour when the infusion rate is 30mL/hr, use the given total units of heparin in the 1L solution. Since there are 25,000 units in 1L (1,000 mL), we can find the amount per mL and then multiply by the hourly rate.

First, calculate the units per mL:
Units per mL = Total units / Total volume
Units per mL = 25,000 units / 1,000 mL
Units per mL = 25 units/mL

Next, calculate the units delivered per hour:
Units per hour = Units per mL * Infusion rate (mL/hr)
Units per hour = 25 units/mL * 30 mL/hr
Units per hour = 750 units/hr

Therefore, 750 units of heparin are administered per hour.

Answer:

Bella can complete work alone in 120 minutes

Step-by-step explanation:

Ana and Bella together can  complete the shoveling in 30 minutes.

They can do a part of work in 1 minute = [tex]\frac{1}{30}[/tex]

Ana can do work alone in 40 minutes

She can do part of work in 1 minute = [tex]\frac{1}{40}[/tex]

So, Bella can do a part of work in 1  minute =  [tex]\frac{1}{30}-\frac{1}{40}[/tex]

                                                                        =  [tex]\frac{1}{120}[/tex]

Bella can do [tex]\frac{1}{120}[/tex] part of work in 1  minute

Bella can do whole work in minutes = 120

Hence Bella can complete work alone in 120 minutes

Answer: C. 2340

Step-by-step explanation:

Hi!

The first character is a letter. There are 26 letters in the alphabet. For each letter you choose a two digit number, but you cannot repeat the digits. There are 100 (0 to 99) two digit numbers, and you have to discard 00, 11, 22, 33, 44, 55, 66, 77, 88, 99. Those are the 10 numbers with repeated digits.

So, for each letter, you have any of 90 possible numbers. The answer is then 26*90 = 2340

Answer:

angle k:75

angle g:15

Step-by-step explanation:

Answer:

(a) PROPORTION.

Step-by-step explanation:

A relative frequency histogram has bars whose height is equal to either the proportion of cases that are between the upper and lower bounds of the bar.

The relative frequency in a relative frequency histogram refers to PROPORTION.

A relative frequency histogram is a graph that has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies (percentages or proportions).

Hence, option (a) is right option.

Answer:

a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)

Step-by-step explanation:

Let's solve by separating variables:

[tex]x'=\frac{dx}{dt}[/tex]

a)  x’=t–sin(t),  x(0)=1

[tex]dx=(t-sint)dt[/tex]

Apply integral both sides:

[tex]\int {} \, dx=\int {(t-sint)} \, dt\\\\x=\frac{t^2}{2}+cost +k[/tex]

where k is a constant due to integration. With x(0)=1, substitute:

[tex]1=0+cos0+k\\\\1=1+k\\k=0[/tex]

Finally:

[tex]x=\frac{t^2}{2} +cos(t)[/tex]

b) x’+2x=4; x(0)=5

[tex]dx=(4-2x)dt\\\\\frac{dx}{4-2x}=dt \\\\\int {\frac{dx}{4-2x}}= \int {dt}\\[/tex]

Completing the integral:

[tex]-\frac{1}{2} \int{\frac{(-2)dx}{4-2x}}= \int {dt}[/tex]

Solving the operator:

[tex]-\frac{1}{2}ln(4-2x)=t+k[/tex]

Using algebra, it becomes explicit:

[tex]x=2+ke^{-2t}[/tex]

With x(0)=5, substitute:

[tex]5=2+ke^{-2(0)}=2+k(1)\\\\k=3[/tex]

Finally:

[tex]x=2+3e^{-2t}[/tex]

c) x’’+4x=0; x(0)=0; x’(0)=1

Let [tex]x=e^{mt}[/tex] be the solution for the equation, then:

[tex]x'=me^{mt}\\x''=m^{2}e^{mt}[/tex]

Substituting these equations in c)

[tex]m^{2}e^{mt}+4(e^{mt})=0\\\\m^{2}+4=0\\\\m^{2}=-4\\\\m=2i[/tex]

This becomes the solution m=α±βi where α=0 and β=2

[tex]x=e^{\alpha t}[Asin\beta t+Bcos\beta t]\\\\x=e^{0}[Asin((2)t)+Bcos((2)t)]\\\\x=Asin((2)t)+Bcos((2)t)[/tex]

Where A and B are constants. With x(0)=0; x’(0)=1:

[tex]x=Asin(2t)+Bcos(2t)\\\\x'=2Acos(2t)-2Bsin(2t)\\\\0=Asin(2(0))+Bcos(2(0))\\\\0=0+B(1)\\\\B=0\\\\1=2Acos(2(0))\\\\1=2A\\\\A=\frac{1}{2}[/tex]

Finally:

[tex]x=\frac{1}{2} sin(2t)[/tex]

Answer:

36.

Step-by-step explanation:

We are asked to find the number of ways in which 3 girls can divide 10 pennies such that each must end up with at least one penny.

The selection can be done by selecting two dividing likes between the 10 pennies such that the set is divided into three parts.

Since each girl must have one penny, so no girl can have 0 penny. So the dividing like cannot be placed at end points, beginning and at the end. Therefore, we are left with 9 positions.  

Now we need to find number of ways to select two positions out of the 9 positions that is C(9,2).

[tex]_{2}^{9}\textrm{C}=\frac{9!}{7!*2!}=\frac{9*8*7!}{7!*2*1}=\frac{9*8}{2}=9*4=36[/tex]

Therefore, there are 36 ways to divide 10 pennies between 3 girls.

Answer:

The particular solution is [tex]y=\sin (\ln|x|)[/tex] .

Step-by-step explanation:

The given differential equation is

[tex]xy'=\sqrt {1-y^2}[/tex]

It can be written as

[tex]x\frac{dy}{dx}=\sqrt {1-y^2}[/tex]

Use variable separable method to solve the above equation.

[tex]\frac{dy}{\sqrt {1-y^2}}=\frac{1}{x}dx[/tex]

Integrate both sides.

[tex]\int \frac{dy}{\sqrt {1-y^2}}=\int \frac{1}{x}dx[/tex]

[tex]\sin^{-1} y=\ln|x|+C[/tex]            .... (1)

It is given that y(1)=0. It means y=0 at x=1.

[tex]\sin (0)=\ln|1|+C[/tex]

[tex]0=0+C[/tex]

[tex]0=C[/tex]

The value of constant is 0.

Substitute C=0 in equation (1) to find The required equation.

[tex]\sin^{-1} y=\ln|x|+0[/tex]

Taking sin both sides.

[tex]y=\sin (\ln|x|)[/tex]

Therefore the particular solution is [tex]y=\sin (\ln|x|)[/tex] .

Answer:

  (4/7) ÷ (7/5) = 20/49

Step-by-step explanation:

You show a sum, not a quotient.

Perhaps you want the quotient ...

  (4/7) ÷ (7/5) = (4/7)×(5/7) = (4·5)/(7·7) = 20/49

Answer: There 64 possible sequences in which team A wins

Step-by-step explanation:

Hi!

The sequences in which team A wins, are the ones with at least 4 A's.

Sequences with 4, 5, 6 or 7 A's. To calculate how many of each type exist, we use the formula of combinations. If you select K objects from a set of N objects, there are C(N, K) possibilities, give by the formula:

[tex]C(N,K) = \frac{N!}{K! (N-K)!)}[/tex]

Then the total M number of sequences in which team A wins is:

[tex]M = C(7,7) + C(7,6) + C(7,5) + C(7,4) = 1 + 7 +21+35 = 64[/tex]

The answer is

t dy / dt + dy / dt = te ^ y

I apply common factor                1/dt*(tdy+dy)=te^y

I pass "dt"                    tdy+dy=(te^y)*dt

I apply common factor                (t+1)*dy=(te^y)*dt

I pass "e^y"                         (1/e^y)dy=((t+1)*t)*dt

I apply integrals                ∫ (1/e^y)dy= ∫ (t^2+t)*dt

by property of integrals  ∫ (1/e^y)dy= ∫ (e^-y)dy

∫ (e^-y)dy= ∫ (t^2+t)*dt

I apply integrals

-e^-y=(t^3/3)+(t^2/2)+C

I apply natural logarithm to eliminate "e"

-ln (e^-y)=-ln(t^3/3)+(t^2/2)+C

y=ln(t^3/3)+(t^2/2)+C

The general solution of the differential equation is

[tex]y = - ln( - ln |t + 1| + c)[/tex]

How to find the particular solution.

Given differential equation

[tex]t \frac{dy}{dt} + \frac{dy}{dt} = t {e}^{y} [/tex]

we can rearrange it as:

[tex](t + 1) \frac{dy}{dt} = t {e}^{y} [/tex]

Now, we can separate variables:

[tex] \frac{dy}{t {e}^{y} } = \frac{dt}{t + 1} [/tex]

Integrating both sides

[tex]∫ \frac{1}{t {e}^{y} } dy = ∫ \frac{1}{t + 1} dt[/tex]

[tex] - {e}^{ - y} = ln |t + 1| + c[/tex]

[tex] {e}^{ - y} = ln |t + 1| + c[/tex]

[tex]y = - ln( - ln |t + 1| + c[/tex]

Where C1 is the constant of integration.