find the integral of ((x^(2)-1))/(x^(2)+3x) using integration by partial fractions

Answer:

So you have the vertical line passing through is x=4 and the horizontal line passing through is y=-6.

Step-by-step explanation:

In general the horizontal line passing through (a,b) is y=b and the vertical line passing through (a,b) is x=a.

So you have the vertical line passing through is x=4 and the horizontal line passing through is y=-6.

Answer:

. Let fX,Y(x,y) = 10xy^2 for 0 < x < y < 1 be the joint density function of the random pair (X, Y). (a) Obtain the marginal density f(y) of Y. (b) Obtain the conditional density fx|y(x|y) of X given Y = y. (c) Evaluate the conditional expectation of X, given Y=y

Y = y.

Step-by-step explanation:

Final answer:

The maximum area that can be enclosed with 144 feet of fencing is when the enclosure is a square. Calculating the side length as 36 feet results in a maximum area of 1296 square feet.

Explanation:

Maximizing the Area of a Rectangular Region with a Given Perimeter:

To find the maximum area that can be enclosed with 144 feet of fencing in a rectangular shape, we can use the knowledge that for a given perimeter, a rectangle with equal length and width (a square) will have the maximum possible area. Let's denote the length of the rectangle as L and the width as W. Since the perimeter is twice the sum of the length and width, we have 2L + 2W = 144 feet. To form a square, which gives the maximum area, L equals W, making 4L = 144 feet or L = 36 feet. The maximum area is L squared, which is 36 feet by 36 feet, equaling 1296 square feet.

The maximum area that can be enclosed with 144 feet of fencing is 1296 square feet, which corresponds to option c) 1296 square feet.

To maximize the area enclosed by a fixed perimeter, we look to geometry, which tells us that of all the rectangles with a given perimeter, the square has the highest area.
Let's denote the perimeter of the square as P and the length of each side of the square as s. Since the square has four equal sides, we have:
P = 4s
We are given that P is 144 feet, so we can find the length of each side s by dividing the total perimeter by 4:
s = P/4 = 144/4 = 36 feet
The area A of a square is given by the formula A = s^2, where s is the length of a side of the square. We calculated above that s = 36 feet, so:
A = s^2 = (36 feet)^2 = 1296 square feet
This is the maximum area that can be enclosed by 144 feet of fencing when arranged in a square. Matching our result with the provided options, the correct answer is:
c) 1296 square feet

Answer:  0.6731

Step-by-step explanation:

Given : Mean : [tex]\mu = 40,000\text{ miles}[/tex]

Standard deviation : [tex]\sigma = 5,000\text{ miles}[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=36,000

[tex]z=\dfrac{36000-40000}{5000}=-0.8[/tex]

For x=46,000

[tex]z=\dfrac{46000-40000}{5000}=1.2[/tex]

The P-value : [tex]P(-0.8<z<1.2)=P(z<1.2)-P(z<-0.8)[/tex]

[tex]=0.8849303-0.2118554=0.6730749\approx0.6731[/tex]

Hence, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles =0.6731

the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).

The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 miles and a standard deviation of 5,000 miles. To find the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles, we can calculate the z-scores for 36,000 and 46,000 and then use the standard normal distribution to find the probabilities for these z-scores.

To compute the z-scores:

We then look up the corresponding probabilities for these z-scores in a standard normal distribution table or use a calculator with normal distribution functions. The probability corresponding to z=-0.8 is approximately 0.2119, and the probability corresponding to z=1.2 is approximately 0.8849. The probability of the tire's life being between 36,000 and 46,000 is the difference: 0.8849 - 0.2119 = 0.6730.

Therefore, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).

Answer:

The scientific notation of 8,000,000 is 8 × 10^6

Step-by-step explanation:

* Lets explain the meaning of the scientific notation

- Scientific notation is a way of writing very large or very small numbers

- A number is written in scientific notation when a number between 1

 and 10 is multiplied by a power of 10

- Ex:  650,000,000 can be written in scientific notation as

        6.5 × 10^8

- We put a decimal points to make the number between 1 and 10 and

 then count how many places from right to left until the decimal point

 The decimal point between 6 and 5 to make the number 6.5 and

 there are 8 places from the last zero to the decimal point

* Lets solve the problem

∵ The exact value of 400,000 × 200 = 8,000,000

- Put the decimal point before 8 and count how many places from

 the last zero to the decimal point

∵ There are six places from last zero to the decimal point

∴ The scientific notation of 8,000,000 is 8 × 10^6

Answer:

D. $3.

Step-by-step explanation:

We have been given that a box of 7 items costs $20.79. We are asked to find the cost of each item.

To find the cost of each item, we will divide total cost by total number of items.

[tex]\text{Cost of each item}=\frac{\$20.79}{7}[/tex]

[tex]\text{Cost of each item}=\$2.94142857[/tex]

Upon rounding our answer to nearest dollar, we will get:

[tex]\text{Cost of each item}\approx\$3[/tex]

Therefore, the cost of each item will be approximately $3 and option D is the correct choice.

Answer: The probability that a person is not qualified if a person was approved by the manager is 0.0766.

Step-by-step explanation:

Since we have given that

Probability that a person approves qualified = 0.85× 0.85 = 0.7225

Probability that a person does not approve qualified = 0.85 × 0.15 = 0.1275

Probability that a person approves unqualified = 0.40 × 0.15 = 0.06

Probability that a person does not approve unqualified = 0.60 × 0.15 = 0.009

so, using the conditional probability, we get that [tex]p(unqualified\mid approved)=\dfrac{0.06}{0.7225+0.06}=\dfrac{0.06}{0.7825}=0.0766[/tex]

Hence, the probability that a person is not qualified if a person was approved by the manager is 0.0766.

Answer:Yes

Step-by-step explanation:

Given

n=390 x=312

[tex]\hat{p}=\frac{312}{390}=0.8[/tex]

Confidence level=99 %

[tex] Z_{\frac{\alpha }{2}}=2.575[/tex]

Standard error(S.E.)=[tex]\sqrt{\frac{\hat{p}\left ( 1-\hat{p}\right )}{n}}[/tex]

S.E.=[tex]\sqrt{\frac{0.8\times 0.2}{390}}[/tex]

S.E.=0.0202

Confidence interval

[tex]p\pm \left [ z_{\frac{\alpha }{2}}\cdot S.E.\right ][/tex]

[tex]0.8 \pm 0.0521[/tex]

[tex]\left ( 0.7479,0.8521 \right )[/tex]

Since 0.5 does not lie in interval therefore method appear to be effective

Answer:

The chances that a smoker has lung disease 25.30%.

Step-by-step explanation:

Let L is the event of the lung disease and S is the event of being smoker,

According to the question,

The probability of lung disease, P(L) = 7 % = 0.07,

⇒ The probability of not having lung disease, P(L') = 100 % - 7 % =  93 % = 0.93,

The probability of the people having lung disease who are smokers,

P(L∩S) = 90% of 0.07 = 6.3% = 0.063,

The probability of the people not having lung disease who are smokers,

P(L'∩S) = 20% of 0.93 = 18.60% = 0.186,

Thus, the total probability of being smoker, P(S) = P(L∩S) + P(L'∩S) = 0.063 + 0.186 = 0.249,

Hence, the probability that a smoker has lung disease,

[tex]P(\frac{L}{S})=\frac{P(L\cap S)}{P(S)}[/tex]

[tex]=\frac{0.063}{0.249}[/tex]

[tex]=0.253012048193[/tex]

[tex]=25.3012048193\%[/tex]

[tex]\approx 25.30\%[/tex]

Final answer:

To find the chances that a smoker has lung disease, we need to use conditional probability. Assuming a total population of 100, the chances are 25.2%.

Explanation:

To find the chances that a smoker has lung disease, we need to use conditional probability. Let's assume the total population is 100. According to the American Lung Association, 7% of the population has lung disease, so the number of people with lung disease is 7.

Of these 7 people with lung disease, 90% are smokers. So, the number of smokers with lung disease is 7 * 0.9 = 6.3.

Out of the remaining people (100 - 7 = 93) without lung disease, 20% are smokers. So, the number of smokers without lung disease is 93 * 0.2 = 18.6.

Therefore, the total number of smokers is 6.3 + 18.6 = 24.9.

Hence, the chances that a smoker has lung disease is 6.3 / 24.9 = 0.252 (rounded to three decimal places) or 25.2% (rounded to the nearest percent).

The number of sample points in this experiment is 16.

The number of sample points in this experiment can be found by multiplying the number of possible outcomes for each coin toss. Since there are 2 possible outcomes for each coin toss, and we have 4 coin tosses, the total number of sample points is 2 x 2 x 2 x 2 = 16.

Therefore, option a, 16, is the correct answer.

The number of sample points for tossing 4 coins successively is a. 16, calculated using the formula 2⁴. Each coin flip has 2 possible outcomes, and for 4 coins, this results in 2⁴ = 16 outcomes.

When tossing 4 coins successively, each coin has 2 possible outcomes: heads (H) or tails (T). The total number of sample points in such an experiment can be calculated as follows:

Step-by-Step Explanation:

Therefore, the number of sample points in this experiment is 16.

The dad had 90 cents.

Multiply the 90 cents by 5%:

90 x 0.05 = 4.5 cents.

Subtract that from 90:

90 - 4.5 = 85.5 cents.

The lowest guess the son could say was 86 cents to be within 5%

Since the son guessed lower than that he did not get the money.

1. A dad holds five coins in his hand. He tells his son that if he can guess the amount of money he is holding within 5% error, he can have the money. The son guesses that dad is holding 81 cents. The dad opens his hand and displays 90 cents. Did the son guess close enough to get the money?

yes

Answer:

6

Step-by-step explanation:

Sample space of the experiment

first number in the bracket is white die and second number in the bracket is black

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

As it can be seen that the first numbers in the bracket are (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

∴1 on the white die can be obtained in 6 ways

In the case of rolling two dice and trying to obtain a 1 on the white die, there are 6 ways to accomplish this because the black die outcome is irrelevant and it can show any number from 1 to 6 while pairing with a 1 on the white die.

The question asks about the probability of getting a specific result when rolling two fair dice, which is a problem in the realm of simple probability within mathematics.

Specifically, the question is focused on finding the number of ways to obtain a 1 on the white die.

When rolling two dice, there are a total of 6 different possible outcomes for the black die (since a standard die has 6 faces), and 1 specific outcome we're looking for on the white die, which is a 1.

Each outcome on the black die can be paired with a 1 on the white die, resulting in the combinations (1,1), (2,1), (3,1), (4,1), (5,1), and (6,1).

This gives us a total of 6 ways to achieve a 1 on the white die, regardless of what the black die shows.

Answer:

  1,556,675,366,400

Step-by-step explanation:

There are 37 possible characters, of which 8 can be chosen. Order matters, so the number is ...

  37P8 = 1,556,675,366,400

_____

This number includes 84,144,614,400 strings in which the space character is either first or last. Such strings may be ruled invalid because they are indistinguishable from 7-character strings.

_____

nPk = n!/(n-k)! . . . . the number of permutations of n things taken k at a time

The 37 allowed characters are the 26 letters of the alphabet, 10 digits, and 1 space character.

Answer:

Step-by-step explanation:this is confusing for me oof

Rating plays on Broadway, Poor, good, or excellent would be a type of Ordinal measurement.

You can think or ordinal like order, which could be listing something from best to worst.

The answer is Ordinal.

Ordinal measurement can shown by name, group, or rank. Poor, good, and excellent shows the ratings of the play by "rank", in other words, by order. Thus proves our answer.

Best of Luck!

Answer:

The interval is : (7.206 , 7.794)

Step-by-step explanation:

The mean is = 7.5

Standard deviation = 3

n = 400

At 95% confidence interval, the z score is 1.96

[tex]7.5+1.96(\frac{3}{\sqrt{400} } )[/tex]

And [tex]7.5-1.96(\frac{3}{\sqrt{400} } )[/tex]

[tex]7.5+0.294[/tex] and [tex]7.5-0.294[/tex]

So, the interval is : (7.206 , 7.794)

Answer:

r=<1,0,9>+t<1,2,1>

and

x=1+t

y=2t

z=9+t

Step-by-step explanation:

A vector perpendicular to the plane :

[tex]ax+by+cz+d=0[/tex]

is given by (a,b,c)

So a vector perpendicular to given plane will have :

(1,2,1)

[tex]The\ parametric\ equation\ of\ a\ line\ through\ (1,0,9)\ and\ parallel\ to\ vector\ (a,b,c) is\ given\ by:\\x=x_0+ta\\y=y_0+tb\\z=z_0+tb\\x=1+t(1)\\x=1+t\\y=0+t(2)\\y=2t\\z=9+t(1)\\z=9+t\\The\ vector\ form\ is:\\r=<1,0,9>+t<1,2,1>[/tex] ..

The vector equation r(t) is (1, 0, 9) + t(1, 2, 1) and the parametric equations are x = 1 + t, y = 2t, z = 9 + t.

To find the vector equation and parametric equations for the line that passes through the point (1, 0, 9) and is perpendicular to the plane given by x + 2y + z = 7, follow these steps :

1. Find the Normal Vector to the Plane :

2. Vector Equation of the Line :

3. Parametric Equations :

Extract the parametric equations from the vector equation :

Thus, the parametric equations for the line are :

Answer:

the correct answer is master data

Step-by-step explanation:

Enterprise system is a information system which provides a company with a wide integration and coordination regarding the important business processes  and also helps in providing seamless flow of information through out the company.

Master data  is a type of data in the enterprise system which is changed only occasionally , as this data includes all the information related to the customers like name, contact etc which helps a firm in analyzing their behavior and conduct high level research.

The car's velocity at time [tex]t[/tex] is

[tex]v(t)=16\dfrac{\rm m}{\rm s}+\displaystyle\int_0^t\left(\left(-0.50\frac{\rm m}{\mathrm s^2}\right)u\right)\,\mathrm du=16\dfrac{\rm m}{\rm s}+\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2[/tex]

It comes to rest at

[tex]v(t)=0\implies16\dfrac{\rm m}{\rm s}=\left(0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2\implies t=8.0\,\mathrm s[/tex]

Its velocity over this period is positive, so that the total distance the car travels is

[tex]\displaystyle\int_0^{8.0}v(t)\,\mathrm dt=\left(16\dfrac{\rm m}{\rm s}\right)(8.0\,\mathrm s)+\frac13\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)(8.0\,\mathrm s)^3=\boxed{85\,\mathrm m}[/tex]

so the answer is D.

The car takes 32 seconds to stop from its initial velocity of 16m/s. During this period, the car has traveled a distance of 256 meters, which is not an option in the given choices.

The initial velocity of the car is given as 16 m/s and the acceleration is given as -0.5t m/s^2. We know that the car slows down until it stops, which means its final velocity is 0 m/s.

Firstly, we need to find out the time it takes for the car to stop. That could be calculated with the equation 'v = u + at', where v is the final velocity, u is the initial velocity, a is the acceleration and t is time. Since we know v = 0 m/s, u = 16 m/s and a = -0.5t m/s^2, we could set the equation to find the time to be '0 = 16 - 0.5t'. Solving this equation gives t=32 seconds.

Second, to find the distance traveled by the car during this time, we use the equation 's = ut + 0.5at^2', where s is the distance, u is the initial velocity, a is the acceleration and t is time. Substituting the known values into this equation, we get 's = 16(32) + 0.5*(-0.5*32)*(32)', which simplifies to s = 512 - 256 = 256 meters. Hence, the answer is not in the options given.

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Answer:

[tex](x+4)^2+(y+9)^2=25[/tex]

Step-by-step explanation:

The equation of a circle with center (h,k) and radius r is

[tex](x-h)^2+(y-k)^2=r^2[/tex].

You are given (h,k)=(-4,-9) and the radius=(diameter)/2=10/2=5.

Plug in the information and you will have your equation:

[tex](x-(-4))^2+(y-(-9))^2=(5)^2[/tex].

Simplify:

Answer:

B. 180 mg

Step-by-step explanation:

In order to answer the given problem we need to be aware that:

1000 milligrams = 1 milliliter

The above means that in 1 milliliter a 100% solution means 1000 milligrams. Because we have 18% solution, then:

(1000 milligrams / 1 milliliter) * 18% =

(1000 milligrams / 1 milliliter) * (18/100) =

(1000*18/100) milligrams/milliliter =

180 milligrams/milliliter

In conclusion, the answer is B. 180 mg.

Answer: 42.25 feet

Step-by-step explanation:

We know that after "t" seconds, its height "h" in feet is given by this function:

[tex]h(t) = 52t -16t^2[/tex]

The maximum height is the y-coordinate of the vertex of the parabola. Then, we can use the following formula to find the corresponding value of "t" (which is the x-coordinate of the vertex):

[tex]x=t=\frac{-b}{2a}[/tex]

In this case:

[tex]a=-16\\b=52[/tex]

Substituting values, we get :

[tex]t=\frac{-52}{2(-16)}\\\\t=1.625[/tex]

Substituting this value into the function to find the maximum height the ball will reach, we get:

[tex]h(1.625) = 52(1.625) -16(1.625)^2\\\\h(1.625) =42.25\ ft[/tex]

Answer:

42.25 feet

Step-by-step explanation:

The maximum of a quadratic can be found by finding the vertex of the parabola that the quadratic creates visually on a graph.

So first step to find the maximum height is to find the x-coordinate of the vertex.

After you find the x-coordinate of the vertex, you will want to find the y that corresponds by using the given equation, [tex]y=52x-16x^2[/tex]. The y-coordinate we will get will be the maximum height.

Let's start.

The x-coordinate of the vertex is [tex]\frac{-b}{2a}[/tex].

[tex]y=52x-16x^2[/tex] compare to [tex]y=ax^2+bx+c[/tex].

We have that [tex]a=-16,b=52,c=0[/tex].

Let's plug into  [tex]\frac{-b}{2a}[/tex] with those values.

[tex]\frac{-b}{2a}[/tex] with [tex]a=-16,b=52,c=0[/tex]

[tex]\frac{-52}{2(-16)}=\frac{52}{32}=\frac{26}{16}=\frac{13}{8}[/tex].

The vertex's x-coordinate is 13/8.

Now to find the corresponding y-coordinate.

[tex]y=52(\frac{13}{8})-16(\frac{13}{8})^2[/tex]

I'm going to just put this in the calculator:

[tex]y=\frac{169}{4} \text{ or } 42.25[/tex]

So the maximum is 42.25 feet.

Answer: There are 22.5 films were profitable.

Step-by-step explanation:

Since we have given that

Number of total films = 40

Percentage of comedies = 60%

Number of comedies is given by

[tex]0.6\times 40\\\\=24[/tex]

Percentage of horror films = 40%

Number of horror films is given by

[tex]0.4\times 40\\\\=16[/tex]

Percentage of comedies were profitable = 75%

Number of profitable comedies is given by

[tex]0.75\times 24=18[/tex]

Percentage of horror were unprofitable = 75%

Percentage of horror were profitable = 25%

Number of profitable horror films is given by

[tex]0.25\times 18\\\\=4.5[/tex]

So, Total number of profitable films were

[tex]18+4.5=22.5[/tex]

Hence, there are 22.5 films were profitable.

Answer:

Probability: [tex]\frac{1}{2}[/tex] = 0.5 = 50%

Step-by-step explanation:

Based on the question one coin is chosen at random and tossed. That coin then lands and is heads. Since the coin landed on heads we can eliminate the possibility of the coin that was chosen being the coin with double tails.

The following possibilities are that the coin has double heads or is a regular coin with both tails and heads. Seeing as the coin landed on heads, there are only two possible out comes for the other side of the coin

The other side is either Heads or Tails. That gives us a 50% chance of the other side being tails.

[tex]\frac{1}{2}[/tex] = 0.5 = 50%

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Final answer:

To convert Emanuel's weight from pounds to kilograms on the 35th day, multiply 171 lbs by 0.4536 to get 77.52 kg. The general formula for converting function f(t) to g(t) is g(t) = f(t) × 0.4536.

Explanation:

If Emanuel's weight in pounds on the 35th day since the beginning of 2017 is 171 lbs ( f(35) = 171 ), we can find his weight in kilograms ( g(35) ) using the conversion factor from pounds to kilograms. Since 1 pound is equivalent to approximately 0.4536 kilograms on Earth, we can calculate g(35) by multiplying Emanuel's weight in pounds by this conversion factor:

g(35) = 171 lbs × 0.4536 kg/lb

This results in g(35) = 77.5156 kg. When we round this to significant figures based on the given conversion fact of pounds to kilograms (which is inexact and has 4 significant figures), Emanuel's weight would be g(35) = 77.52 kg (to 4 SFs).

The formula for g using the function f is:

g(t) = f(t) × 0.4536

To find the equation in function form for the number of cars fixed each week by the mechanic, we can use the slope-intercept form of a linear equation. The equation is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.

To write an equation in function form to show the number of cars seen each week by the mechanic, we can let the variable x represent the week number and y represent the number of cars fixed. We know that the reduction in the number of cars each week is linear, so we can use the slope-intercept form of a linear equation, y = mx + b. To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. Let's use the points (3, 391) and (13, 361) to find the slope. Plugging these values into the formula gives us m = (361 - 391) / (13 - 3) = -3. Therefore, the equation in function form is y = -3x + b. To find the y-intercept b, we can use one of the points on the line. Let's use the point (3, 391): 391 = -3(3) + b. Solving for b gives us b = 400. Therefore, the equation in function form is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.

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The equation in function form to show the number of cars seen each week by the mechanic is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.

To write an equation in function form to show the number of cars seen each week by the mechanic, we can use the given information that the reduction in the number of cars each week is linear. Let's assume the number of cars fixed in week 3 as y = 391 and in week 13 as y = 361. We can use the formula for the equation of a line, y = mx + b, where m is the slope and b is the y-intercept.

Using the slope formula, m = (y2 - y1) / (x2 - x1), where (x1, y1) = (3, 391) and (x2, y2) = (13, 361), we find m = (361 - 391) / (13 - 3) = -3.

Therefore, the equation in function form to show the number of cars seen each week by the mechanic is y = -3x + b. To find the y-intercept, we can substitute the coordinates of one of the points (x, y) = (3, 391) into the equation, 391 = -3(3) + b. Solving for b gives b = 400.

Thus, the equation in function form is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.

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Answer: Percent of interest paid on this loan annually = 351% p.a

Step-by-step explanation:

Given that,

principal amount = $100(loan)

time period =  14 days

interest amount (SI) = $13.50

we have to calculate the rate of interest (i),

Simple interest(SI) = principal amount × rate of interest (i) × time period

13.50 = 100 × i × [tex]\frac{14}{365}[/tex]

i = [tex]\frac{4927.5}{1400}[/tex]

i = 3.51

i = 351% p.a.

Final answer:

The student paid a 13.5% interest on the $100 loan from the local cash center.

Explanation:

Percent of interest paid:

Initial loan amount: $100

Amount to be repaid: $113.50

Interest paid: $113.50 - $100 = $13.50

Percent interest paid = (Interest paid / Initial loan amount) * 100%

Percent interest paid = ($13.50 / $100) * 100% = 13.5%

Final answer:

To estimate the proportion of smokers with a margin of error, use the formula n = (Z^2 * p * (1-p)) / E^2, where n is the sample size, Z is the Z-value for the desired confidence level, p is the preliminary estimate of the proportion who smoke, and E is the margin of error. Plugging in the values from the question, the sample size should be 753.

Explanation:

To estimate the proportion of smokers in the population with a margin of error of 0.02 and a 95% confidence level, we can use the formula:

n = (Z^2 * p * (1-p)) / E^2

Where:

Plugging in the values, we get:

n = (1.96^2 * 0.31 * (1-0.31)) / 0.02^2 = 752.34

Rounding up to the nearest whole number, the sample size should be 753.

Answer:

Cereal B

Step-by-step explanation:

Given are two different rates for cereals A and B.

as Cereal A: 15oz for $2.95 or Cereal B: 32oz for $5.95

As such we cannot compare unless we make it unit rate for same number of units

Let us find unit oz rates

Cereal A per oz= [tex]\frac{2.95}{15} =0.1967[/tex]dollars

Cereal B per oz = [tex]\frac{5.95}{32} =0.1859[/tex]dollars

Comparing unit rates per ounce,

we find that Cereal B per oz is lower.

Answer is Cereal B.

Answer:

[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex]

Step-by-step explanation:

[tex](x-h)^2+(y-k)^2=r^2[/tex] has parametric equations:

[tex](x-h)=r\cos(t) \text{ and } (y-k)=r\sin(t)[/tex].

Let's solve these for x and y  respectively.

[tex]x-h=r\cos(t)[/tex] can be solved for x by adding h on both sides:

[tex]x=r\cos(t)+h[/tex].

[tex]y-k=r \sin(t)[/tex] can be solve for y by adding k on both sides:

[tex]y=r\sin(t)+k[/tex].

We can verify this works by plugging these back in for x and y respectively.

Let's do that:

[tex](r\cos(t)+h-h)^2+(r\sin(t)+k-k)^2[/tex]

[tex](r\cos(t))^2+(r\sin(t))^2[/tex]

[tex]r^2\cos^2(t)+r^2\sin^2(t)[/tex]

[tex]r^2(\cos^2(t)+\sin^2(t))[/tex]

[tex]r^2(1)[/tex] By a Pythagorean Identity.

[tex]r^2[/tex] which is what we had on the right hand side.

We have confirmed our parametric equations are correct.

Now here your h=0 while your k=1 and r=2.

So we are going to play with these parametric equations:

[tex]x=2\cos(t)[/tex] and [tex]y=2\sin(t)+1[/tex]

We want to travel clockwise so we need to put -t and instead of t.

If we were going counterclockwise it would be just the t.

[tex]x=2\cos(-t)[/tex] and [tex]y=2\sin(-t)+1[/tex]

Now cosine is even function while sine is an odd function so you could simplify this and say:

[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex].

We want to find [tex]\theta[/tex] such that

[tex]2\cos(t-\theta_1)=2 \text{ while } -2\sin(t-\theta_2)+1=1[/tex] when t=0.

Let's start with the first equation:

[tex]2\cos(t-\theta_1)=2[/tex]

Divide both sides by 2:

[tex]\cos(t-\theta_1)=1[/tex]

We wanted to find [tex]\theta_1[/tex] for when [tex]t=0[/tex]

[tex]\cos(-\theta_1)=1[/tex]

Cosine is an even function:

[tex]\cos(\theta_1)=1[/tex]

This happens when [tex]\theta_1=2n\pi[/tex] where n is an integer.

Let's do the second equation:

[tex]-2\sin(t-\theta_2)+1=1[/tex]

Subtract 2 on both sides:

[tex]-2\sin(t-\theta_2)=0[/tex]

Divide both sides by -2:

[tex]\sin(t-\theta_2)=0[/tex]

Recall we are trying to find what [tex]\theta_2[/tex] is when t=0:

[tex]\sin(0-\theta_2)=0[/tex]

[tex]\sin(-\theta_2)=0[/tex]

Recall sine is an odd function:

[tex]-\sin(\theta_2)=0[/tex]

Divide both sides by -1:

[tex]\sin(\theta_2)=0[/tex]

[tex]\theta_2=n\pi[/tex]

So this means we don't have to shift the cosine parametric equation at all because we can choose n=0 which means [tex]\theta_1=2n\pi=2(0)\pi=0[/tex].

We also don't have to shift the sine parametric equation either since at n=0, we have [tex]\theta_2=n\pi=0(\pi)=0[/tex].

So let's see what our equations look like now:

[tex]x=2\cos(t)[/tex] and [tex]y=-2\sin(t)+1[/tex]

Let's verify these still work in our original equation:

[tex]x^2+(y-1)^2[/tex]

[tex](2\cos(t))^2+(-2\sin(t))^2[/tex]

[tex]2^2\cos^2(t)+(-2)^2\sin^2(t)[/tex]

[tex]4\cos^2(t)+4\sin^2(t)[/tex]

[tex]4(\cos^2(t)+\sin^2(t))[/tex]

[tex]4(1)[/tex]

[tex]4[/tex]

It still works.

Now let's see if we are being moving around the circle once around for values of t between [tex]0[/tex] and [tex]2\pi[/tex].

This first table will be the first half of the rotation.

t                  0                      pi/4                pi/2               3pi/4               pi  

x                  2                     sqrt(2)             0                  -sqrt(2)            -2

y                  1                    -sqrt(2)+1          -1                  -sqrt(2)+1            1

Ok this is the fist half of the rotation.  Are we moving clockwise from (2,1)?

If we are moving clockwise around a circle with radius 2 and center (0,1) starting at (2,1) our x's should be decreasing and our y's should be decreasing at the beginning we should see a 4th of a circle from the point (x,y)=(2,1) and the point (x,y)=(0,-1).

Now after that 4th, the x's will still decrease until we make half a rotation but the y's will increase as you can see from point (x,y)=(0,-1) to (x,y)=(-2,1).  We have now made half a rotation around the circle whose center is (0,1) and radius is 2.

Let's look at the other half of the circle:

t                pi               5pi/4                  3pi/2            7pi/4                     2pi

x               -2              -sqrt(2)                0                 sqrt(2)                      2

y                1                sqrt(2)+1             3                  sqrt(2)+1                   1

So now for the talk half going clockwise we should see the x's increase since we are moving right for them.  The y's increase after the half rotation but decrease after the 3/4th rotation.

We also stopped where we ended at the point (2,1).

The parametric equations for the path of a particle moving along the circle x^2 + (y - 1)^2 = 4 in a clockwise direction, starting at (2, 1), are x = 2 + 2sin(-t) and y = 1 + 2cos(-t).

The parametric equations for the path of a particle moving along the circle x2 + (y - 1)2 = 4 in a clockwise direction, starting at (2, 1), can be found using trigonometric functions. From the given equation of the circle, we can determine that the center of the circle is (0, 1) and the radius is 2. Therefore, the parametric equations are:

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