(1 point) Find y as a function of t if 8y" + 27y = 0, = y(0) = 8, y'(0) = 6. y(t) = Note: This particular webWork problem can't handle complex numbers, so write your answer in terms of sines and cosines, rather than using e to a complex power.

The line integral of this vector which lies between the points. f = x i +y j , from (2, 0) to (8, 0) is 30.

To calculate the line integral of the vector field F(x, y) = xi + yj along the line between the points (2, 0) and (8, 0), we can parameterize the line segment and then evaluate the integral.

1. Parameterize the line segment:
Let r(t) = (1-t)(2, 0) + t(8, 0) for 0 ≤ t ≤ 1.

Then r(t) = (2 + 6t, 0).

2. Find the derivative of the parameterization:
r'(t) = (6, 0)

3. Evaluate the vector field F along the line segment:
F(r(t)) = (2 + 6t)i + (0)j

4. Take the dot product of F(r(t)) and r'(t):
F(r(t)) • r'(t) = (2 + 6t)(6) + (0)(0) = 12 + 36t

5. Integrate the dot product over the interval [0, 1]:
∫(12 + 36t) dt from 0 to 1 = [12t + 18t^2] evaluated from 0 to 1 = 12(1) + 18(1)^2 - 0 = 12 + 18 = 30

The line integral of the vector field along the line between the given points is 30.

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The distance traveled by the car when the tire makes one complete turn is 56.52 inches. The distance traveled by the car is equivalent to the wheel's circumference.

Given that the diameter of a wheel is 18 inches and the value of Pi is 3.14. To find the distance traveled by the car when the tire makes one complete turn, we need to find the circumference of the wheel.

Circumference of a wheel = πd, where d is the diameter of the wheel. Substituting the given values in the above formula, we get:

Circumference of a wheel = πd

                                 = 3.14 × 18

                                 = 56.52 inches.

Therefore, the distance traveled by the car when the tire makes one complete turn is 56.52 inches. When a wheel rolls over a surface, it creates a circular path. The length of this circular path is known as the wheel's circumference. It is directly proportional to the diameter of the wheel.

A larger diameter wheel covers a larger distance in one complete turn. Similarly, a smaller diameter wheel covers a smaller distance in one complete turn. Therefore, to find the distance covered by a car when the tire makes one complete turn, we need to find the wheel's circumference. The formula to find the wheel's circumference is πd, where d is the diameter of the wheel. The value of Pi is generally considered as 3.14.

The wheel's circumference is 56.52 inches. Therefore, the distance traveled by the car when the tire makes one complete turn is 56.52 inches.

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Answer:

Inferences

Step-by-step explanation:

If you can assume that a variable is at least approximately normally distributed, then you can use certain statistical techniques to make a number of inferences about the values of that variable.

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c-9 would be an equation that means 9 less than c

If you were writing a newspaper article that ended with recommendations to fans about buying tickets and the research showed that the average local baseball fan plans to attend 67 games during the season,

You would recommend the average fan to purchase season tickets since they plan to attend 67 games during the season. Season tickets guarantee the fan a seat for every game they plan to attend. Single-game tickets may not be available, or if they are, may be for an unfavorable seat.

Season tickets often provide a discount compared to single-game tickets, and they save the fan time and effort to look for individual tickets. Additionally, season tickets holders are typically given priority seating options for post-season games and have access to exclusive team events and merchandise discounts.To sum up, you should recommend purchasing season tickets to the average local baseball fan since they plan to attend 67 games during the season.

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This is closest to option d) 343 cm³,  The volume of the cube is 343 cm³. which is the correct answer.

The volume of a cube is given by the formula [tex]V = s^3,[/tex] where s is the length of any side of the cube. In this case, the height, length, and width are all equal to 7 cm. Thus, the length of any side of the cube is also 7 cm.

Substituting s = 7 cm into the formula for the volume of a cube, we get:

V = s^3 = 7^3 = 343 cm³

Therefore, the volume of the cube is 343 cm³. This is closest to option d) 343 cm³, which is the correct answer.

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The second function (2n/4) becomes larger than the first (n2) when n is equal to or greater than 2.

To compare the two function n2 and 2n/4, we need to plug in different values of n and see which function gives a larger output.

Let's start with n = 1.
- n2 = 1
- 2n/4 = 1/2

So, n2 is larger than 2n/4 for n = 1.

Now let's try n = 2.
- n2 = 4
- 2n/4 = 1

In this case, 2n/4 is larger than n2.

We can continue this process for larger values of n and see when the second function becomes larger than the first.

For n = 3,
- n2 = 9
- 2n/4 = 3

In this case, 2n/4 is larger than n2.

For n = 4,
- n2 = 16
- 2n/4 = 4

Again, 2n/4 is larger than n2.

Therefore, the second function (2n/4) becomes larger than the first (n2) when n is equal to or greater than 2.

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The rent of the apartment during the 9th year of living in the apartment is approximately1538.54.

In order to find the rent of the apartment during the 9th year of living in the apartment, we need to first find the rent of the apartment during the 2nd year, 3rd year, 4th year, 5th year, 6th year, 7th year and 8th year.

Rent of apartment during the second year

Rent during the second year = (1 + 0.095) x 800

Rent during the second year = 1.095 x 800

Rent during the second year = $876

Rent of apartment during the third year

Rent during the third year = (1 + 0.095) x 876

Rent during the third year = 1.095 x 876

Rent during the third year = $955.62

Rent of apartment during the fourth year

Rent during the fourth year = (1 + 0.095) x 955.62

Rent during the fourth year = 1.095 x 955.62

Rent during the fourth year = $1043.78

Rent of apartment during the fifth year

Rent during the fifth year = (1 + 0.095) x 1043.78

Rent during the fifth year = 1.095 x 1043.78

Rent during the fifth year = $1141.08

Rent of apartment during the sixth year

Rent during the sixth year = (1 + 0.095) x 1141.08

Rent during the sixth year = 1.095 x 1141.08

Rent during the sixth year = $1248.07

Rent of apartment during the seventh year

Rent during the seventh year = (1 + 0.095) x 1248.07

Rent during the seventh year = 1.095 x 1248.07

Rent during the seventh year = $1365.54

Rent of apartment during the eighth year

Rent during the eighth year = (1 + 0.095) x 1365.54

Rent during the eighth year = 1.095 x 1365.54

Rent during the eighth year = $1494.96

Rent of apartment during the ninth year

Rent during the ninth year = (1 + 0.095) x 1494.96

Rent during the ninth year = 1.095 x 1494.96

Rent during the ninth year = $1538.54

Therefore, the rent of the apartment during the 9th year of living in the apartment is approximately 1538.54.

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Assuming that the nominal size of the square hole is referring to the diameter of the smallest circle that can fully enclose the square, the maximum size of the square hole would be approximately 0.177 inches (or 4.5 millimeters).

This is calculated by taking the nominal size (0.25) and multiplying it by the square root of 2 (approximately 1.414), and then subtracting that result from the nominal size.

Therefore, the maximum size of the square hole would be 0.25 - (0.25 x 1.414) = 0.177 inches (or 4.5 millimeters).

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The value of "a" that would make the inequality statement true is 9.54.

The inequality statement is: 9.53 < √a < 9.54

To find the value of "a" that satisfies this inequality, we need to determine the range of values for which the square root of "a" falls between 9.53 and 9.54.

We know that the square root of "a" must be greater than 9.53 and less than 9.54.

So, we can write the inequality as:

9.53 < √a < 9.54

To solve this inequality, we need to square both sides of the inequality:

[tex](9.53)^2 < a < (9.54)^2[/tex]

Simplifying, we have:

90.5209 < a < 90.7216

Therefore, the value of "a" that makes the inequality statement true lies between 90.5209 and 90.7216.

Looking at the provided answer choices (85, 88, 91, 94), we see that none of these values fall within the range 90.5209 and 90.7216.

Hence, the correct value of "a" that makes the inequality statement true is not provided in the given answer choices. It is important to note that the value of "a" would be 9.54, as the square root of 9.54 falls within the specified range.

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The answer is C 0.0079, rounded to four decimal places.  The probability that among the students in the sample  is  0.0079.

To solve this problem, we can use the binomial distribution. Let X be the number of male students in the sample. Then X follows a binomial distribution with n=8 and p=0.6, since 60% of the students are male. We want to find the probability that X is at most 2, i.e., P(X <= 2).

Using the binomial probability formula, we can compute:

P(X = 0) = (0.4)^8 = 0.0016384

P(X = 1) = 8(0.4)^7(0.6) = 0.015552

P(X = 2) = 28(0.4)^6(0.6)^2 = 0.051816

P(X <= 2) = P(X=0) + P(X=1) + P(X=2) = 0.069006

Therefore, the answer is c. 0.0079, rounded to four decimal places.

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Thevenin's theorem states that the Thevenin voltage is equal to the open circuit voltage between two terminals of a linear, passive circuit.

In other words, it is the voltage difference measured between the two terminals when no current is flowing between them. The Thevenin voltage is often used as a simplified representation of a complex circuit when the circuit is being analyzed or modeled. By finding the Thevenin voltage and resistance, a complex circuit can be reduced to a single voltage source and a single resistor, making it much easier to analyze.

The theorem is named after French electrical engineer Léon Charles Thévenin, who first published the concept in 1883.

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The limit by substituting the Maclaurin series for the trig and inverse trig functions is -81/2.

To begin, we use the Maclaurin series for tan⁻¹(x) and cos(x):

tan⁻¹(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...

cos(x) = 1 - x²/2 + x⁴/24 - x⁶/720 + ...

Substituting x = 9 in the first equation, we get:

tan⁻¹(9) = 9 - 9³/3 + 9⁵/5 - 9⁷/7 + ...

= 9 - 243/3 + 6561/5 - 3,874,161/7 + ...

Simplifying the terms, we get:

tan⁻¹(9) = 9 - 81 + 1312.2 - 553091.6 + ...

Next, substituting x = 9 in the second equation, we get:

cos(9) = 1 - 9²/2 + 9⁴/24 - 9⁶/720 + ...

= 1 - 81/2 + 6561/24 - 3,874,161/720 + ...

Simplifying the terms, we get:

cos(9) = 1 - 40.5 + 273.375 - 5375.223 + ...

Finally, substituting the above expressions into the original limit and simplifying, we get:

lim_(x→0) [tan⁻¹(9) - 9cos(9)]/243235

= [(-71.5) - (-5374.448)]/243235

= -81/2.

Therefore, the limit by substituting the Maclaurin series for the trig and inverse trig functions is -81/2.

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The determinant of matrix C is 0.

(a) To compute the determinant of the matrix PAPT, we can use the property that the determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore:

det(PAPT) = det(P) * det(A) * det(P)

Substituting the given determinant values:

det(PAPT) = det(P) * det(A) * det(P) = 2 * 5 * 2 = 20

So, the determinant of the matrix PAPT is 20.

(b) To find the determinant of matrix C, we can expand along the first row or the first column. Let's expand along the first row :

C = | a 006 |

| 0 0 1 |

| 0 1 0 |

Using the expansion along the first row:

det(C) = a * det(0 1) - 0 * det(0 1) + 0 * det(0 0)

| 1 0 |

We can simplify this:

det(C) = a * (1 * 0 - 0 * 1) = a * 0 = 0

Therefore, the determinant of matrix C is 0.

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The probability of daily demand being less than 495 sodas is approximately 0.9893 or 98.93%.

To find the probability of daily demand being less than 495 sodas, given that the mean of demand is 410 sodas per day and the standard deviation of demand is 37 sodas per day, follow these steps:

1. Convert the demand value (495 sodas) to a z-score:
  z = (X - μ) / σ
  z = (495 - 410) / 37
  z ≈ 2.30

2. Use a z-table or a calculator with a normal distribution function to find the probability corresponding to the z-score:
  P(Z < 2.30) ≈ 0.9893

Thus, the probability of daily demand being less than 495 sodas is approximately 0.9893 or 98.93%.

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You must select 1,096 teenagers to ensure that 4 of them were born on the exact same date.

To ensure that 4 teenagers were born on the exact same date (mm/dd/yyyy), you must consider the total possible birthdates in a non-leap year, which is 365 days.

By using the Pigeonhole Principle, you would need to select 3+1=4 teenagers for each day, plus 1 additional teenager to guarantee that at least one group of 4 shares the same birthdate.

Therefore, you must select 3×365 + 1 = 1,096 teenagers to ensure that 4 of them were born on the exact same date.

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The total number of posts in the fence is 300.

A community garden is surrounded by a fence. The total length of the fence is 3000 feet. For every 40 8 PM defense, there are four posts.

To find the total number of posts in the fence, first, we need to find out the number of fence segments. Each segment has 1 post at the start and 1 post at the end. The number of posts between any two segments is given by 40/4 = 10 posts per segment.

We can then use this information to solve the problem as follows:Let the number of fence segments be n.Each segment is 8 pm = 1/3 day long.The total length of the fence is 3000 feet.So, the length of one segment of the fence = (3000/n) feet.There are 10 posts per segment.

So, the number of posts in one segment of the fence = 10 x (1/3) = (10/3) posts.Since there is one post at the start and end of each segment, the total number of posts in one segment of the fence = (10/3) + 2 = (16/3) posts.

So, the total number of posts in the fence, n = Total length of the fence / Length of one segmentNumber of segments = n = 3000 / (3000/n)Number of segments = n = (3000 * n) / 3000Number of segments = n = n

Number of segments = n²

Number of segments = 900/16 = 56.25 ~ 56

The total number of posts in the fence = Number of segments x Number of posts per segmentTotal number of posts = 56 x (16/3)Total number of posts = 299.67 ~ 300 posts.

Therefore, the total number of posts in the fence is 300.

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The MAD is approximately 15.4. The MAD tells us that on average, the forecast errors are about 15.4 units away from the mean forecast error.

The Mean Absolute Deviation (MAD) is a measure of the variability of a set of data. It represents the average distance of the data points from the mean of the data set.

To calculate the MAD, we need to first find the mean of the forecast errors. The mean is the sum of the forecast errors divided by the number of errors:

Mean = (-22 - 10 + 15)/3 = -4/3

Next, we find the absolute deviation of each error by subtracting the mean from each error and taking the absolute value:

|-22 - (-4/3)| = 64/3

|-10 - (-4/3)| = 26/3

|15 - (-4/3)| = 49/3

Then, we find the average of these absolute deviations to get the MAD:

MAD = (64/3 + 26/3 + 49/3)/3 = 139/9

Therefore, the MAD is approximately 15.4. The MAD tells us that on average, the forecast errors are about 15.4 units away from the mean forecast error.

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The area under the standard normal curve between z1 = -2.02 and z2 = 2.02 is approximately 0.9566.

To find the area under the standard normal curve between the given z-values, z1 = -2.02 and z2 = 2.02, follow these steps:

1. Look up the corresponding probabilities in a standard normal distribution table (or use a calculator or software with a built-in z-table) for each z-value.
2. Subtract the probability of z1 from the probability of z2 to find the area between the two z-values.

Step 1: Look up probabilities for z1 and z2
- For z1 = -2.02, the probability is 0.0217
- For z2 = 2.02, the probability is 0.9783

Step 2: Subtract probabilities
- Area between z1 and z2 = P(z2) - P(z1) = 0.9783 - 0.0217 = 0.9566

So, the area under the standard normal curve between z1 = -2.02 and z2 = 2.02 is approximately 0.9566.

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Mrs. Falkener has written a company report every 3 months for the last 6 years, resulting in a total of 24 reports. Among these reports, 2/3 of them show the company earning more money than it spends. Therefore, 1/3 of the reports, or 8 reports, show the company spending more money than it earns.

In 6 years, there are 12 quarters since there are 4 quarters in a year. Mrs. Falkener has written a company report every 3 months, which means there are 12 * 3 = 36 periods in total. However, since each report covers a 3-month period, the total number of reports is 36 / 3 = 12.

Given that 2/3 of the reports show the company earning more money than it spends, we can calculate the number of reports showing the company spending more money than it earns. Since 2/3 of the reports represent the earnings being greater, the remaining 1/3 represents the expenses being greater. Therefore, 1/3 of 12 reports is 12 * (1/3) = 4 reports.

In conclusion, among the 24 company reports written by Mrs. Falkener in the last 6 years, 2/3 of them, or 16 reports, show the company earning more money than it spends. The remaining 1/3, or 8 reports, show the company spending more money than it earns.

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The magnitude of the vector product is at most 2sin(θ/2), with equality if and only if u and v are antiparallel.

Let u and v be unit vectors with an angle of θ between them. We want to compute the vector product uv.

The vector product of two vectors u and v is defined as:

u × v = |u| |v| sin(θ) n

where |u| and |v| are the magnitudes of u and v, respectively, θ is the angle between them, and n is a unit vector perpendicular to both u and v (the direction of n is determined by the right-hand rule).

Since u and v are unit vectors, we have |u| = |v| = 1. Therefore, the vector product simplifies to:

u × v = sin(θ) n

Multiplying both sides by |u| = |v| = 1, we get:

|u| u × v = sin(θ) u n

|v| u × v = sin(θ) v n

Since u and v are unit vectors, we have |u| = |v| = 1. Therefore, we can add these two equations to get:

(u × v)(|u| + |v|) = sin(θ) (u + v) n

Since |u| = |v| = 1, we have |u| + |v| = 2. Therefore, we can simplify further to get:

u × v = sin(θ/2) (u + v) n

Finally, multiplying both sides by 2/sin(θ/2), we get:

2u × v/sin(θ/2) = 2(u + v)n

Since u and v are unit vectors, we have |u + v| ≤ 2, with equality if and only if u and v are parallel. Therefore, the magnitude of the vector product is at most 2sin(θ/2), with equality if and only if u and v are antiparallel.

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The pair of square roots that e fall is √1 and √9

From the question, we have the following parameters that can be used in our computation:

e = 2.718

Represent as an interval

So, we have

a < e < b

This means that

a < 2.718 < b

The number 2.718 is between 1 and 3

So, we have

1 < 2.718 < 3

Express 1 and 3 as square roots

√1 < 2.718 < √9

Hence, the pair of square roots that e fall is √1 and √9

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The pair of square roots that e falls between is √7 and √8.

The range that the figure falls between is √7 and √8. To get the range, we will find the roots of all the numbers and see the one that the figure falls between.

√2 = 1.414

√3 = 1.732

√4 = 2

√5 = 2.236

√7 = 2.645

√8 = 2.828

Now we will look at the ranges and see the one that figures 2.718 falls between. This is √7 to √8.

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One adult ticket costs $122.

Given that Gavin paid $334 for 2 adult tickets and 1 child ticket last year and will spend $392 for 1 adult ticket and 3 child tickets this year, we have to determine how much one adult ticket costs.

To calculate the cost of an adult ticket, we need to use the concept of proportionality. We know that the total cost of the tickets is proportional to the number of tickets bought.

The cost of 2 adult tickets and 1 child ticket is $334, so we can write:

334 = 2x + y,

Where x is the cost of an adult ticket and y is the cost of a child ticket.

Next, we can use the information given about the cost of tickets this year:

392 = x + 3y

We can now solve the system of equations using substitution:

334 = 2x + y

y = 334 - 2x

392 = x + 3y

392 = x + 3(334 - 2x)

392 = x + 1002 - 6x

392 - 1002 = -5x

-610 = -5x

122 = x

Therefore, one adult ticket costs $122.

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The function f(x) = {2 - x if x ≤ 1, x if x > 1} is one-to-one but not onto.

To prove that a function f(x) is one-to-one but not onto, we need to show that it satisfies the following conditions:

One-to-one: For any two different values x1 and x2 in the domain, if f(x1) ≠ f(x2), then x1 ≠ x2.

Not onto: There exists at least one value y in the codomain that is not the image of any value x in the domain.

Let's analyze the function f(x) = {2 - x if x ≤ 1, x if x > 1}.

One-to-one:

To show that f(x) is one-to-one, we need to demonstrate that if f(x1) ≠ f(x2), then x1 ≠ x2.

Consider two different values x1 and x2 in the domain such that f(x1) ≠ f(x2).

If both x1 and x2 are less than or equal to 1, then f(x1) = 2 - x1 and f(x2) = 2 - x2. Since x1 and x2 are different, f(x1) and f(x2) will also be different. Therefore, x1 ≠ x2.

If both x1 and x2 are greater than 1, then f(x1) = x1 and f(x2) = x2. Since x1 and x2 are different, f(x1) and f(x2) will also be different. Therefore, x1 ≠ x2.

If one value is less than or equal to 1 and the other is greater than 1, then f(x1) = 2 - x1 and f(x2) = x2. In this case, f(x1) and f(x2) will always be different because 2 - x1 will never be equal to x2. Therefore, x1 ≠ x2.

In all cases, we have shown that if f(x1) ≠ f(x2), then x1 ≠ x2. Hence, f(x) is one-to-one.

Not onto:

To show that f(x) is not onto, we need to find at least one value y in the codomain that is not the image of any value x in the domain.

The codomain of f(x) is the set of all real numbers. Let's consider the value y = 3. No matter what value of x we choose from the domain, the function f(x) will never be equal to 3. Therefore, there is no x in the domain such that f(x) = 3.

Since we have found a value y (3) in the codomain that is not the image of any value x in the domain, we can conclude that f(x) is not onto.

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The sum of the arithmetic sequence 4, 11, 18, 25, ..., 249 is 4554 and there are 36 terms in the sequence.

(a) To determine the number of terms in the sum, we can find the pattern in the terms.  we observe that each term is obtained by adding 7 to the previous term. Starting from 4 and incrementing by 7, we can write the sequence of terms as 4, 11, 18, 25, ..., and so on.

To find the number of terms, we need to determine the value of n in the equation 4 + 7(n-1) = 249. Solving this equation, we find n = 36. There are 36 terms in the sum.

(b) To compute the sum using a technique discussed in this section, we can use the formula for the sum of an arithmetic series. The formula is given by Sn = (n/2)(2a + (n-1)d), where Sn represents the sum of the series, n is the number of terms, a is the first term, and d is the common difference.

In this case, the first term a is 4, the number of terms n is 36, and the common difference d is 7.

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The initial population of lynx on the island is 1000 lynx.

To find the initial population of lynx on the island, we need to look at the equation for P(t) when t = 0.

This is because t represents the time since observations of the island began, so when t = 0, this is the starting point of the observations.

Therefore, we can substitute t = 0 into the equation for P(t):
P(0) = 121(0) - 0.4(0)⁴ + 1000
P(0) = 0 - 0 + 1000
P(0) = 1000

So the initial population of lynx on the island is 1000 lynx.

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In the recipe for a fruit smoothie drink, 20% of all pieces of fruit used are strawberries.

A recipe for a fruit smoothie drink calls for strawberries and raspberries. The ratio of strawberries to raspberries in the drink is 5:20.

The ratio of strawberries to raspberries in the drink is 5:20, i.e., the total parts are 5 + 20 = 25.

The fraction representing strawberries is: 5/25 = 1/5.

Now we have to convert this fraction to percent form.

This can be done using the following formula:

Percent = (Fraction × 100)%

Therefore, the percent of all pieces of fruit used that are strawberries is:

1/5 × 100% = 20%

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The coordinates of the vertices of the image after rotating shape A 180° with centre of rotation (3,-1) are as follows :Vertex A' : (4,-3)Vertex B' : (-1,-1)Vertex C' : (-2,-4)

To rotate a shape in the Cartesian plane, you need to know the centre of rotation and the angle of rotation. Here, the centre of rotation is given as (3,-1) and the angle of rotation is 180°.To rotate a shape 180° about the centre of rotation, we need to find the mirror image of the shape about the line passing through the centre of rotation. This mirror image will be the required image. We can find the mirror image by simply negating the x and y coordinates of each point with respect to the centre of rotation.

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The rectangular equation given is x + 7√(x² + y²) = x² + y², which can be converted to the polar equation r = 7 - cos(θ).

Using the trigonometric identity cos(θ) = x/r, we can write:

r = 7 - x/r

Multiplying both sides by r, we get:

r² = 7r - x

Using the polar to rectangular conversion formulae x = r cos(θ) and y = r sin(θ), we can express r in terms of x and y:

r² = x² + y²

Substituting r² = x² + y² into the previous equation, we get:

x² + y² = 7r - x

Substituting cos(θ) = x/r, we can write:

x = r cos(θ)

Substituting this into the previous equation, we get:

x² + y² = 7r - r cos(θ)

Simplifying, we get:

x² + y² = 7√(x² + y²) - x

Rearranging, we get:

x + 7√(x² + y²) = x² + y²

This is the rectangular form of the polar equation r = 7 - cos(θ).

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the probability that the teacher will pick exactly five girls out of seven students is approximately 0.307, or 30.7%.

We can use the binomial probability formula to calculate the probability of picking exactly five girls out of seven students:

P(exactly 5 girls) = (number of ways to pick 5 girls out of 8) * (number of ways to pick 2 boys out of 6) / (total number of ways to pick 7 students out of 14)

The number of ways to pick 5 girls out of 8 is given by the binomial coefficient:

C(8, 5) = 8(factorial)/ (5(factorial) * 3(factorial)) = 56

The number of ways to pick 2 boys out of 6 is also given by the binomial coefficient:

C(6, 2) = 6(factorial) / (2(factorial)* 4(factorial)) = 15

The total number of ways to pick 7 students out of 14 is:

C(14, 7) = 14(factorial) / (7(factorial) * 7(factorial)) = 3432

Therefore, the probability of picking exactly 5 girls out of 7 students is:

P(exactly 5 girls) = (56 * 15) / 3432 ≈ 0.307

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