Suppose a measurement on a population is assumed to be distributed N(u, 2) where u e Rl is unknown and that the size of the population is very large. A researcher wants to determine a 0.95-confidence interval for u that is no longer than 1. What is the minimum sample size that will guarantee this?

Therefore, if there were 40 dogs and the ratio stayed the same, there would be 120 cats at the pet daycare.

If the ratio of dogs to cats stays the same, then the ratio of dogs to cats in the two situations will be equal.

The initial ratio of dogs to cats is:

dogs : cats = 20 : 60

= 1 : 3

To maintain the same ratio, the new number of cats (C) can be found by setting up the proportion:

dogs : cats = 40 : C

Using the initial ratio of dogs to cats, we can substitute and simplify:

1 : 3 = 40 : C

Cross-multiplying gives:

C = (3 x 40) / 1

= 120

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Any k-cycle (a1.....ak) can be written as a product of (k-1) 2-cycles. Therefore, any permutation can be written as a product of some number of 2-cycles.

To prove that any k-cycle can be written as a product of (k-1) 2-cycles, we use induction on k.

Base case: For k=2, the 2-cycle (a1 a2) is already a product of (2-1) = 1 2-cycle.

Inductive step: Assume that any (k-1)-cycle can be written as a product of (k-2) 2-cycles. Consider a k-cycle (a1 a2 ... ak).

First, we can write this k-cycle as a product of two cycles: (a1 ak) and (a1 a2 ... ak-1).

Then, by the induction hypothesis, the cycle (a1 a2 ... ak-1) can be written as a product of (k-2) 2-cycles.

Finally, we can express the original k-cycle as a product of (k-1) 2-cycles:

(a1 a2)(a2 a3)...(ak-2 ak-1)(ak-1 ak)(a1 ak)

Therefore, any k-cycle can be written as a product of (k-1) 2-cycles.

Since any permutation can be written as a product of cycles, and each cycle can be written as a product of 2-cycles, it follows that any permutation can be written as a product of some number of 2-cycles.

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To find the 90% confidence interval for the standard deviation of lactation periods of grey seals, we can use the chi-squared distribution with n-1 degrees of freedom, where n is the sample size.

First, we need to calculate the chi-squared statistic. Using the formula:

chi-squared = (n-1)*s^2 / sigma^2

where s is the sample standard deviation (s = 2.501), sigma is the population standard deviation (which we don't know), and n is the sample size (n = 14), we can rearrange the formula to solve for sigma:

sigma^2 = (n-1)*s^2 / chi-squared

We want a 90% confidence interval, which means we need to find the chi-squared values that correspond to the 5% and 95% tails of the distribution with 13 degrees of freedom (n-1 = 13). Using a chi-squared distribution table or calculator, we find that these values are approximately 5.229 and 22.362, respectively.

Plugging these values into the formula above, we get:

sigma^2_lower = (n-1)*s^2 / 22.362
sigma^2_upper = (n-1)*s^2 / 5.229

Taking the square roots of these values, we get:

sigma_lower = 1.89
sigma_upper = 4.12

Therefore, the 90% confidence interval for the standard deviation of lactation periods of grey seals is (1.89, 4.12). We can interpret this interval as follows: if we were to take many samples of size 14 from the population of female grey seals, and calculate the standard deviation of lactation periods for each sample, then 90% of these sample standard deviations would fall within the range of 1.89 to 4.12.

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The surface area of the figure is 480mm2.

We are given that;

Dimensions of the figure=  5 mm 6 mm 5 mm 8 mm 4 mm

Now,

Area of base= 8 x 5

=40mm

Area of figure= 5 x 6 x 4 x 40

= 30 x 160

= 480

Therefore, by the area the answer will be 480mm2.

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a) The probability that the third car arrives within 3 minutes of the first car is 0.6331.

b) The probability that in a given ten-minutes interval, 15 cars arrive at the booth, and 10 of these are Japanese imports is 0.2023

a) The arrival of cars at the toll booth follows a Poisson process with a rate of 3 arrivals per minute. Let X be the time between the first and third car arrivals. Then X is exponentially distributed with a mean of 1/3 minutes. We want to find the probability that X is less than or equal to 3.

Let Y be the number of car arrivals in the first 3 minutes. Y follows a Poisson distribution with a mean of lambda = 3*3 = 9, since there are 3 minutes and 3 arrivals per minute. Then, the probability that the third car arrives within 3 minutes of the first car is equal to the probability that there are at least 3 arrivals in the first 3 minutes, which is given by:

P(Y >= 3) = 1 - P(Y < 3) = 1 - P(Y = 0) - P(Y = 1) - P(Y = 2)

= 1 - e^(-9) - 9e^(-9) - (81/2)e^(-9)

= 0.6331 (rounded to four decimal places)

Therefore, the probability that the third car arrives within 3 minutes of the first car is 0.6331.

b) Let Z be the number of car arrivals in a 10-minute interval. Z follows a Poisson distribution with a mean of lambda = 10*3 = 30, since there are 10 minutes and 3 arrivals per minute. Let W be the number of Japanese imports in the same 10-minute interval. We are given that 60% of the cars are Japanese imports, so the probability that a given car is a Japanese import is 0.6. Therefore, W follows a binomial distribution with parameters n = Z and p = 0.6.

We want to find the probability that 15 cars arrive at the booth and 10 of them are Japanese imports. This can be calculated using the binomial distribution as follows:

[tex]P(W = 10 | Z = 15) = (15 choose 10)(0.6)^10(0.4)^5[/tex]

= 0.2023 (rounded to four decimal places)

Here, we assumed that the arrivals are independent and identically distributed, which is a reasonable assumption for a Poisson process. We also assumed that the probability of a car being a Japanese import is the same for each car arrival, which may not be entirely accurate in practice.

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A common rule of thumb is to use the p-value of a statistical test to determine whether a difference in study outcomes is statistically significant.

If the p-value is less than the pre-determined level of significance (often set at 0.05), then the difference is considered statistically significant. This means that there is strong evidence to suggest that the observed difference is not due to chance alone, but rather a result of the variables being studied. However, it's important to keep in mind that statistical significance does not necessarily imply practical significance, and other factors such as effect size and clinical relevance should also be considered when interpreting study outcomes.

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Answer:

6

Step-by-step explanation:

Let x = amount of her pocket money.

Let b = price of 1 blouse.

Let j = price of 1 pair of jeans.

j = 3b

3/8 x was used for blouses

5/8 x was left after the blouses

3/5 of 5/8 x was used for 2 pairs of jeans

3/8 x was used for 2 pairs of jeans

1 pair of jeans costs 3/16 x

3 blouses cost 3/16 x

1 blouse costs 1/16 x

3/8 x was used for blouses

1 blouse costs 1/16 x

(3/8) / (1/16) = 3/8 × 16/1 = 6

Answer: 6

The equation which represents the relationship between rides and  total cost is c = 0.50r + 30.00

Let c represent the total cost, and

let's use the variable "r" to represent the number of rides Naomi goes on.

Naomi pays a fixed amount of $30.00 to enter the park, and then an additional $0.50 for every ride that she goes on.

So, the equation that shows the relationship between the number of rides and the total cost is:

c = 0.50r + 30.00

This equation represents a linear relationship between the number of rides and the total cost, where the slope of the line is $0.50 and the y-intercept is $30.00

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Answer:36

Step-by-step explanation:

36

The resultant velocity and acceleration of the skier at t=4.00 sec on the ski jump path are 12.8 m/s and 45.9 m/s², respectively.

To find the resultant velocity, first find the velocity vector components using the parametric equations:

vx = 7.00t, vy = 0.48t³ - 6.00t²/√(t⁴+1)

At t=4.00 s, vx = 28.0 m/s and vy = 10.50 m/s. The resultant velocity is the magnitude of the velocity vector, given by:

|v| = √(vx² + vy²) = 12.8 m/s

To find the acceleration vector components, differentiate the velocity vector components with respect to time:

ax = 7.00 m/s², ay = 1.44t² - 12.00t/√(t⁴+1) - 6.00t³(t⁴+1)^(-3/2)

At t=4.00 s, ax = 7.00 m/s² and ay = 45.9 m/s². The acceleration vector magnitude is:

|a| = √(ax² + ay²) = 46.1 m/s².

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The solution for x is x ∈ (-∞, 11] ∪ (9, ∞)

We are given that;

The inequality − 36 < − 3− 9 or −36<−3x−9or − 42 ≥ − 3 − 9 −42≥−3x−9

Now,

You can solve this inequality by first adding 9 to both sides of each inequality to get:

-27 < -3x or -33 >= -3x

Then, divide both sides of each inequality by -3, remembering to reverse the inequality symbol when dividing by a negative number:

9 > x or 11 <= x

Therefore, by inequality the answer will be x ∈ (-∞, 11] ∪ (9, ∞).

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The value of c2-4mk is 76 and the position of mass after t seconds is x(t) = (1/√21)[(√21-5)e^(αt) + (5+√21)e^(βt)].

The value of c2-4mk can be calculated as follows:
c2-4mk = (damping constant)^2 - 4*(mass)*(spring constant)
c2-4mk = 10^2 - 4*(2 kg)*(2 N/m)
c2-4mk = 76

To find the position of the mass after t seconds, we first need to find the values of α and β. We can do this using the following equation:
mα^2 + cα + k = 0
mβ^2 + cβ + k = 0

Substituting the given values, we get:
2α^2 + 10α + 2 = 0
2β^2 + 10β + 2 = 0

Solving these equations, we get:
α = -5 + √21
β = -5 - √21

Therefore, the position of the mass after t seconds is given by:
x(t) = c1e^(αt) + c2e^(βt)

To find the values of c1 and c2, we use the initial conditions:
x(0) = 1 m (the spring is stretched 1 meter beyond its natural length)
x'(0) = 0 m/s (the mass is released with zero velocity)

Using these initial conditions, we get:
c1 + c2 = 1
αc1 + βc2 = 0

Solving these equations, we get:
c1 = (β-1)/2√21
c2 = (1-α)/2√21

Therefore, the position of the mass after t seconds is:
x(t) = [(β-1)/2√21]e^(αt) + [(1-α)/2√21]e^(βt)

Simplifying this expression, we get:
x(t) = (1/√21)[(√21-5)e^(αt) + (5+√21)e^(βt)]

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The area of the shaded sector with a central angle of 120 degrees and radius 12 units is 150.72 sq units

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

central angle = 120 degrees

Radius = 12 units

Using the above as a guide, we have the following:

Sector area = central angle/360 * 3.14 * Radius^2

Substitute the known values in the above equation, so, we have the following representation

Sector area = 120/360 * 3.14 * 12^2

Evaluate

Sector area = 150.72

Hence, the area of the sector is 150.72 sq units

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"The eigenvalues of the square of two matrices are not necessarily equal to the square of the eigenvalues of each of the matrices". The statement is incorrect.

Eigenvalues of the square of two matrices (A*B) are not necessarily equal to the square of the eigenvalues of each matrix (A and B).

In general, eigenvalues of the product of two matrices do not follow the same relationship as their individual eigenvalues.

In fact, there is no simple relationship between the eigenvalues of a matrix and the eigenvalues of its square. The eigenvalues of a matrix and its square can be different, and even if they are the same, their relationship is not necessarily as simple as taking the square root.

However, if the two matrices commute, meaning A*B = B*A, their eigenvalues may exhibit some specific relationships, but this is not guaranteed in all cases.

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f(x) is always increasing on R and there are no intervals on which it is decreasing.

To determine where the function f(x) is increasing or decreasing, we need to analyze the sign of its derivative f'(x).

[tex]f'(x) = e^{x^2-4x+3} - 1[/tex]

The derivative is always positive since [tex]e^{x^2-4x+3}[/tex] is always greater than 1 for all real values of x.

A derivative is a fundamental concept in calculus that measures the rate at which a function changes. It represents the slope of a function at a given point and provides information about how the function is changing with respect to its input variable.

The derivative of a function f(x) is denoted as f'(x) or dy/dx and is defined as the limit of the ratio of the change in the function's output to the corresponding change in its input, as the change in the input approaches zero. Geometrically, the derivative represents the slope of the tangent line to the graph of the function at a particular point.

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The normal component of the acceleration vector (a_n) is a_n = √(|a(t)|^2 - a_t^2).

To find the tangential and normal components of the acceleration vector for the given position vector r(t) = 7e^t*i + 7√2^t*j + 7e^(-t)*k, follow these steps:

1. Differentiate the position vector r(t) to find the velocity vector v(t):

v(t) = dr(t)/dt = (7e^t)*i + (7√2^t * ln(√2))*j - (7e^(-t))*k

2. Differentiate the velocity vector v(t) to find the acceleration vector a(t):

a(t) = dv(t)/dt = (7e^t)*i + (7√2^t * ln^2(√2))*j + (7e^(-t))*k

3. Calculate the magnitude of the velocity vector |v(t)|:

|v(t)| = √((7e^t)^2 + (7√2^t * ln(√2))^2 + (7e^(-t))^2)

4. Find the tangential component of the acceleration vector (a_t):

a_t = (a(t) • v(t)) / |v(t)|

Here, '•' denotes the dot product.

5. Find the normal component of the acceleration vector (a_n):

a_n = √(|a(t)|^2 - a_t^2)

By following these steps, you can find the tangential and normal components of the acceleration vector for the given position vector r(t).

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For f(x) = sin(Tr), there exists at least one point 0 < c < 2 such that f (2) - f(0) = f'(c) (2 - 0)^2. However, for f(x) = |x|, there is no such c that satisfies f(1) - f(-1) = f'(c) (2). The Mean Value Theorem does not apply over the interval [-1, 1] for f(x) = |x|.

For f(x) = sin(Tr), we can apply the Mean Value Theorem which states that for a function f(x) that is continuous on the interval [a, b] and differentiable on (a, b), there exists at least one point c in (a, b) such that:

f(b) - f(a) = f'(c) (b - a)

Here, a = 0, b = 2, and f(x) = sin(Tr). Thus,

f(2) - f(0) = f'(c) (2 - 0)

sin(2T) - sin(0) = cos(cT) (2)

2 = cos(cT) (2)

cos(cT) = 1

cT = 2nπ, where n is an integer

0 < c < 2, so 0 < cT < 2π

Thus, cT = π/2, and c = π/4

Therefore, f'(π/4) satisfies the Mean Value Theorem condition.

For f(x) = |x|, we can find f'(x) for x ≠ 0:

f'(x) = d/dx|x| = x/|x| = ±1

However, at x = 0, the function f(x) is not differentiable because the left and right derivatives do not match:

f'(x=0-) = lim(h->0-) (f(0) - f(0-h))/h = -1

f'(x=0+) = lim(h->0+) (f(0+h) - f(0))/h = 1

Thus, the Mean Value Theorem does not apply over the interval [-1, 1] for f(x) = |x|.

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The area of a regular polygon inscribed in a circle is given by A = (1/2)nr^2sin(2π/n), where n is the number of sides and r is the radius of the circle.

For a square, n = 4, so A(square) = 2r^2.

For a regular octagon, n = 8, so A(octagon) = 2(2+√2)r^2.

The ratio of the areas is:

A(octagon)/A(square) = [2(2+√2)r^2]/(2r^2) = 2+√2 ≈ 3.83

Therefore, the octagon covers approximately 283% more of the circle's area than the square.

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The frequency of each range in the table is as follows:-

Range             Frequency

0–4                         2        

5–9                         3

10–14                       6

15–19                       1

25–29                     2

The frequency (f) of a particular value is the number of times the value occurs in the data. The distribution of a variable is the pattern of frequencies, meaning the set of all possible values and the frequencies associated with these values.

Raphael surveyed his co-workers to find out their spent hours on the internet each week.

The results are:-

14, 22, 10, 6, 9, 3, 13, 7, 12, 2, 26, 11, 13, 25

We have to find the number of times the particular value occurs in the data.

Thus, the number of occurrence of a particular range can be written as follows:-

Range        Hours in given data       Frequency

0–4                         3, 2                             2

5–9                     6 , 9 , 7                           3

10–14                 14, 10 ,13, 12, 11, 13           6

15–19                        0                               0

20–24                      22                             1

25–29                      25, 26                      2

The frequency of each range in the table is as follows:-

Range             Frequency

0–4                         2        

5–9                         3

10–14                       6

15–19                       1

25–29                     2

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The given question is incomplete, complete question is:

Raphael surveyed his coworkers to find out how many hours they spend on the Internet each week.

The results are shown below.

14, 22, 10, 6, 9, 3, 13, 7, 12, 2, 26, 11, 13, 25

Drag numbers to record the frequency for each range in the table.

Numbers may be used once, more than once, or not at all.

01234567

Hours on the Internet

Hours Frequency

0–4

5–9

10–14

15–19

20–24

25–29

Vector equation of the line tangent to the graph of r(t) at the point p0 on the curve r(t) = (3t - 1) i + 13t j + 4 k.

To find a vector equation of the line tangent to the graph of r(t) at the point P0 on the curve r(t) = (3t - 1) i + 13t j + 16 k, where P0 is given as (-1,4), we can use the following steps:

Step 1: Find the derivative of r(t) with respect to t:

r'(t) = 3 i + 13 j

Step 2: Evaluate the derivative at the point P0:

r'(-1) = 3 i + 13 j

Step 3: Use the point P0 and the vector r'(-1) to form the vector equation of the tangent line:

r(t) = P0 + r'(-1) t

where t is a scalar parameter.

Plugging in the values, we get:

r(t) = (-1)i + 4j + (3i + 13j)t

Simplifying, we get:

r(t) = (3t - 1) i + 13t j + 4 k

Therefore, the vector equation of the line tangent to the graph of r(t) at the point P0 on the curve

r(t) = (3t - 1) i + 13t j + 16 k  is

r(t) = (3t - 1) i + 13t j + 4 k.

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The tangent line to the point (0,2): where (x1, y1) is the point (0, 3), and m is the slope of the tangent line, which is f'(0).

To find the tangent line to the curve y = f(x) with the initial condition f(0) = 3 at the point (0, 2), we need to first determine the derivative of the function f(x), which represents the slope of the tangent line. However, you provided an initial condition of f(0) = 3, but the point given is (0, 2). These two pieces of information seem contradictory.

Assuming you meant to find the tangent line at the point (0, 3) instead, we would need the derivative f'(x). Without knowing the function f(x), we cannot compute its derivative. However, if we were given the derivative, we would use the point-slope form of the linear equation to find the tangent line:

y - y1 = m(x - x1),

where (x1, y1) is the point (0, 3), and m is the slope of the tangent line, which is f'(0).

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Maximizing the likelihood function is the same as minimizing the least sqaure objective function can be proved by the linear regression model.

Let's consider a linear regression model with the following equation:

Y = β0 + β1X + ε

where Y is the response variable, X is the predictor variable, β0 and β1 are the intercept and slope coefficients, respectively, and ε is the error term. We assume that ε follows a normal distribution with mean 0 and variance σ^2.

The maximum likelihood estimation of β0 and β1 is based on the likelihood function:

L(β0, β1) = f(Y | β0, β1, X)

where f(Y | β0, β1, X) is the probability density function of Y given β0, β1, and X. Assuming ε follows a normal distribution, we have:

f(Y | β0, β1, X) = (2πσ^2)^(-1/2)exp(-(Y-β0-β1X)^2/(2σ^2))

The likelihood function can be written as:

L(β0, β1) = (2πσ^2)^(-n/2)exp(-SSR/(2σ^2))

where SSR is the sum of squared residuals, given by:

SSR = Σ(Yi-β0-β1Xi)^2

Minimizing SSR is equivalent to maximizing the likelihood function, as the value of σ^2 that maximizes L(β0, β1) is the same value that minimizes SSR. This can be seen by taking the derivative of SSR with respect to β0 and β1 and setting them to 0, which yields the following normal equations:

ΣYi = nβ0 + β1ΣXi

ΣXiYi = β0ΣXi + β1Σ(Xi^2)

Solving these equations for β0 and β1 gives the least squares estimators:

β1 = Σ(Xi - Xbar)(Yi - Ybar) / Σ(Xi - Xbar)^2

β0 = Ybar - β1Xbar

which minimize SSR.

Therefore, maximizing the likelihood function is equivalent to minimizing the least squares objective function.

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The required number is 936.

We have,

Since the number has 3 different digits and the hundreds and ones place digits are both divisible by 3, this means that the number must be in the form of ABC, where A and C are divisible by 3 and A ≠ C.

We also know that the hundreds place digit is the third multiple of 3, so it must be 9.

This leaves us with two options for the ones and tens place digits: 3 and 6.

However, since the digits must be different, the one's place digit must be 3, and the tens place digit must be 6.

Therefore,

The number is 936.

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The total amount of money earned by Kadeem over 22 years is approximately $983,332.11.

Use the formula for the sum of a geometric series to find the total amount of money earned by Kadeem over 22 years.

The salary in the first year is $31,500 and it increases by 6% every year, so the salary in the second year will be:

$31,500 + 0.06 × $31,500 = $33,390

The salary in the third year will be:

$33,390 + 0.06 × $33,390 = $35,316.40

And so on. The salary in the 22nd year will be:

$31,500 × 1.06^21 ≈ $87,547.31

So the total amount of money earned over the 22 years is the sum of the salaries for each year:

$31,500 + $33,390 + $35,316.40 + ... + $87,547.31

This is a geometric series with a first term of $31,500, a common ratio of 1.06, and 22 terms. The formula for the sum of a geometric series is:

[tex]S = \dfrac{a(1 - r^n)} { (1 - r)}[/tex]

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

Plugging in the values, we get:

[tex]S ={$31,500\dfrac{(1 - 1.06^{22})} { (1 - 1.06) }[/tex]

S = $983,332.11

So the total amount of money earned by Kadeem over 22 years is approximately $983,332.11.

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Substitute these values into the equation: cos(7π/12) = (√2/2)(1/2) - (√2/2)(√3/2) = √2/4 - √6/4 = (√2 - √6)/4. Therefore, cos(7π/12) = (√2 - √6)/4.

To compute cos[7W/12), we can use the sum formula for cosine:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

In this case, let a = pi/4 and b = pi/3, so that a + b = 7pi/12:

cos(7pi/12) = cos(pi/4)cos(pi/3) - sin(pi/4)sin(pi/3)

cos(7pi/12) = (sqrt(2)/2)(1/2) - (sqrt(2)/2)(sqrt(3)/2)

cos(7pi/12) = (sqrt(2) - sqrt(6))/4

For the second question, we can use the Pythagorean identity to find cos(a):

cos^2(a) + sin^2(a) = 1

cos^2(a) = 1 - sin^2(a)

cos(a) = -sqrt(1 - (3/5)^2) = -4/5

Then, we can use the fact that sin(pi - a) = sin(a) to find sin(B - a):

sin(B - a) = sin(pi/2 - a - B) = cos(a + B)

sin(B - a) = cos(a)cos(B) - sin(a)sin(B)

sin(B - a) = (-4/5)(12/13) - (3/5)(5/13)

sin(B - a) = -63/65

To compute cos(7π/12) without using a calculator, we can use the sum formula for cosine and the given fact that π/4 + π/3 = 7π/12. Let angle A be π/4 (second quadrant) with sin(A)=3/5, and angle B be π/3 (first quadrant) with cos(B)=12/13. We want to compute sin(A-B).

The sum formula for cosine is cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B). Since we want to compute cos(7π/12), we have:

cos(7π/12) = cos(π/4 + π/3) = cos(π/4)cos(π/3) - sin(π/4)sin(π/3).

Now we need to find the cosine and sine values for the given angles:
cos(π/4) = √2/2,
sin(π/4) = √2/2,
cos(π/3) = 1/2,
sin(π/3) = √3/2.

Substitute these values into the equation:

cos(7π/12) = (√2/2)(1/2) - (√2/2)(√3/2) = √2/4 - √6/4 = (√2 - √6)/4.

Therefore, cos(7π/12) = (√2 - √6)/4.

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The local extrema of xy^2 subject to x+y=4 is f(x,y) = (16λ^3)/(27λ^2-8λ^2)

This is the function we would call g(x,y) in the Lagrange multiplier method. To find the local extrema of f(x,y), we would take the partial derivatives of g(x,y) with respect to x, y, and lambda, set them equal to zero, and solve for x, y, and lambda. The critical points would then be evaluated to determine if they are local maxima, minima, or saddle points.

To find the local extrema of xy^2 subject to x+y=4, we can use the Lagrange multiplier method. This involves introducing a new variable, lambda, and setting up the equations:

f(x,y) = xy^2
g(x,y) = x+y-4
∇f(x,y) = λ∇g(x,y)

Taking the partial derivatives of f and g, we get:

∂f/∂x = y^2
∂f/∂y = 2xy
∂g/∂x = 1
∂g/∂y = 1

So the equation for ∇f(x,y) is:

(∂f/∂x, ∂f/∂y) = (y^2, 2xy)

And the equation for ∇g(x,y) is:

(∂g/∂x, ∂g/∂y) = (1, 1)

Multiplying the equations for ∇g(x,y) by lambda, we get:

(λ, λ)

Setting these equations equal to each other, we get the system of equations:

y^2 = λ
2xy = λ
x + y = 4

Solving for x and y in terms of lambda, we get:

x = (4λ)/(3λ+2)
y = (4λ)/(3λ-2)

Substituting these expressions for x and y into the equation for f(x,y), we get:

f(x,y) = (16λ^3)/(27λ^2-8λ^2)

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The length of the path described by the parametric equations

 is 3/2units.

We can find the length of the path described by the parametric equations x=cos³t and y=sin³t by using the arc length formula.

The arc length formula for a parametric curve given by:

x=f(t) and y=g(t) is given by:

L = ∫[a,b] √[f'(t)² + g'(t)²] dt

where f'(t) and g'(t) are the derivatives of f(t) and g(t), respectively.

In this case, we have:

x = cos³t, so x' = -3cos²t sin t

y = sin³t, so y' = 3sin²t cos t

Therefore,

f'(t)² + g'(t)² = (-3cos²t sin t)² + (3sin²t cos t)²

= 9(cos⁴t sin²t + sin⁴t cos²t)

= 9(cos²t sin²t)(cos²t + sin²t)

= 9(cos²t sin²t)

Thus, we have:

L = ∫[0,2π] √[f'(t)² + g'(t)²] dt

= ∫[0,2π] √[9(cos²t sin²t)] dt

= 3∫[0,2π] sin t cos t dt

Using the identity sin 2t = 2sin t cos t, we can rewrite the integral as:

L = 3/2 ∫[0,2π] sin 2t dt

Integrating, we get:

L = 3/2 [-1/2 cos 2t] from 0 to 2π

= 3/4 (cos 0 - cos 4π)

= 3/2

Therefore, the length of the path described by the parametric equations x=cos³t and y=sin³t is 3/2 units.

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The probability that there are more than 5 left-handed students in a class of 30 students is 0.1049

The given parameters are:

Sample size, n = 30

Probability of success, p = 0.11

x > 5

To determine the required probability, we make use of the following complement rule:

P(x > 5) = 1 - P(x ≤ 5)

Using a binomial calculator, we have:

P(x ≤ 5) = 0.89508640002

Substitute P(x ≤ 5) = 0.89508640002 in P(x > 5) = 1 - P(x ≤ 5)

P(x > 5) = 1 - 0.89508640002

Evaluate the difference

P(x > 5) = 0.10491359998

Approximate

P(x > 5) = 0.1049

Hence, the probability that there are more than 5 left-handed students in a class of 30 students is 0.1049

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Answer:

[tex]q = \dfrac{4v}{5}[/tex]

Step-by-step explanation:

We can solve for q by cross-multiplying.

[tex]\dfrac{q}{4} = \dfrac{v}{5}[/tex]

↓ cross-multiplying

[tex]5q = 4v[/tex]

↓ dividing both sides by 5

[tex]\boxed{q = \dfrac{4v}{5}}[/tex]

[tex]\boxed{\sf q=\dfrac{4}{5}v}.[/tex]

[tex]\sf \dfrac{q}{4} =\dfrac{v}{5}[/tex]

[tex]\sf (4)\dfrac{q}{4} =\dfrac{v}{5}(4)\\ \\\\ \boxed{\sf q=\dfrac{4}{5}v}.[/tex]

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We can say with 95% confidence that the true mean fluid ounces of the soda bottles being produced lies within a range of 67.6 ± 0.0276 fluid ounces, where margin of error is 0.0276.

To calculate the margin of error, we need to use the formula:

Margin of Error = Critical value x Standard error

The critical value can be found using a Z-table at a 95% confidence level, which gives a value of 1.96.

The standard error can be calculated using the formula:

Standard error = Standard deviation / Square root of sample size

Plugging in the given values, we get:

Standard error = 0.2 / √(200)

Standard error = 0.0141

Now we can find the margin of error:

Margin of Error = 1.96 x 0.0141

Margin of Error = 0.0276

Therefore, we can say with 95% confidence that the true mean fluid ounces of the soda bottles being produced lies within a range of 67.6 ± 0.0276 fluid ounces.

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