Solve for X. Assume all lines that appear tangent are tangent.

a. All members of the Avery Fitness Club in Carrollton, GA. Subset of members chosen for the survey.

b. List of all members from which the sample will be drawn. A probability sample of Stratified sampling will be used to ensure the representation of different member categories.

c. 100 members to ensure a representative sample while keeping costs and time constraints in mind.

a. The population for the survey is all members of the Avery Fitness Club in Carrollton, GA. The sample is a subset of the population selected for the survey. The sampling frame is a list of all the members of the Avery Fitness Club who are eligible to be selected for the sample. A population is the entire group of people or objects that the researcher wants to study, while a sample is a smaller group selected from the population. A sampling frame is a list of all the individuals or objects that the sample can be drawn.

b. For this survey, a Simple Random Sample (SRS) would be the best choice. This is because each member of the population has an equal chance of being selected for the sample, and this helps to minimize bias. Other options such as Systematic, Stratified, or Cluster Samples may introduce bias or complexity in the sampling process that might not be necessary for this survey.

c. The sample size for the survey should be determined based on the desired level of precision, the margin of error, and confidence level. For example, if we want a 95% confidence level and a margin of error of 5%, we would need a sample size of 246 members of the Avery Fitness Club. This ensures that the sample is large enough to accurately represent the population, while also minimizing the potential for sampling error.

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Answer:

The y-intercept is (0, 2).

The approximation for f(1.4) using Euler's method with two steps of equal size is -0.632.

Euler's method is a numerical method for approximating the solutions to differential equations. It works by approximating the derivative at each step and using it to estimate the next value of the function.

In this case, we are given the differential equation dy/dx = x-y-1 and the initial condition f(1)=-2. We want to find an approximation for f(1.4) using Euler's method with two steps of equal size, starting at x=1.

To use Euler's method, we first need to determine the step size, which is the distance between x-values at each step. Since we have two steps of equal size, the step size is (1.4-1)/2 = 0.2.

Next, we use the initial condition to find the first approximation:

f(1.2) ≈ f(1) + f'(1)*0.2

= -2 + (1 - (-2) - 1)*0.2

= -1.2

Now, we can use this approximation to find the second approximation:

f(1.4) ≈ f(1.2) + f'(1.2)*0.2

= -1.2 + (1.2 - (-1.2) - 1)*0.2

= -0.632

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The statistical question is solved and

a) The null hypothesis is (H0) and alternative hypothesis is (Ha)

b) The z-test statistic is approximately 1.747.

c) The P-value for the z-test is 0.1614.

Given data,

(a)

The null hypothesis (H0): The mean femininity score for male hotel managers is equal to the mean femininity score for men in general (μ = 5.19).

The alternative hypothesis (Ha): The mean femininity score for male hotel managers is different from the mean femininity score for men in general (μ ≠ 5.19).

(b)

To calculate the z-test statistic, we'll use the formula:

z = (y - μ) / (σ / √n)

where:

y = sample mean femininity score (y = 5.29)

μ = population mean femininity score (μ = 5.19)

σ = standard deviation of the population (σ = 0.78)

n = sample size (n = 148)

Substituting the given values:

z = (5.29 - 5.19) / (0.78 / √148)

Calculating the expression:

z ≈ 1.747

Therefore, the z-test statistic is approximately 1.747.

(c)

To find the P-value for the z-test, we need to determine the probability of observing a z-value as extreme as 1.747 or more extreme in a two-tailed distribution.

Using a standard normal distribution table or a statistical calculator, we find that the P-value for a z-value of 1.747 is approximately 0.0807.

Since this is a two-tailed test, we multiply the P-value by 2:

P-value = 2 * 0.0807 ≈ 0.1614

Hence , the P-value for the z-test is approximately 0.1614.

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Design, the main effects of A, B, C, and D are not confounded with blocks because each block contains exactly one run for each level of each factor.

The 24 runs into four blocks, we can use a balanced incomplete block design (BIBD) with parameters (v, b, r, k) = (24, 4, 6, 2).

This means that there are 24 runs, divided into 4 blocks, each block contains 6 runs, and each pair of runs appears together in 2 blocks.

The runs can be divided into blocks:

Block 1:

ABCD, ABDC, ACBD, ADBC, ADBC, ACDB

Block 2:

BACD, BADC, BCAD, BDAC, BDCA, BCDA

Block 3:

CABD, CADB, CBAD, CDAB, CDBA, CBDA

Block 4:

DABC, DACB, DBAC, DCAB, DCBA, DBCA

The two-factor interactions are confounded with blocks because each pair of runs appears together in exactly two blocks.

Specifically, the AB, AC, AD, BC, BD, and CD interactions are confounded with blocks.

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If the three-vertices of a rectangular-piece of material are ​(-2.2​,-2.3​), ​(-2.2​,1.5​) and ​(1.5​,1.5​), then the fourth-vertex is (1.5, -2.3).

A "Rectangle" is defined as a quadrilateral shape which has "four-sides" and "four-angles", where the opposite sides are parallel and of equal length, and the four angles are all right angles.

Let the coordinates of fourth-vertex be = (x,y).

Since it's a rectangular-piece of material, the "opposite-sides" of  rectangle must be parallel and have same-length.

The three vertices of th rectangular piece are :

⇒ Vertex 1: (-2.2, -2.3),

⇒ Vertex 2: (-2.2, 1.5),

⇒ Vertex 3: (1.5, 1.5)

We see that first two vertices have the same "x-coordinate" of -2.2, and the last two vertices have same "y-coordinate" of 1.5.

So, the "fourth-vertex" should have the same x-coordinate as Vertex 3, and the same y-coordinate as Vertex 1.

Therefore, coordinates of fourth-vertex is (1.5, -2.3).

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The probability that a randomly selected worker at the fast food restaurant earns more than $6.95/hr is approximately 9.18%.

To calculate the probability that a randomly selected worker earns more than $6.95/hr, we will use the z-score formula to standardize the value and then find the corresponding probability from a standard normal distribution table.

Step 1: Calculate the z-score
z = (X - μ) / σ
where X is the value we're interested in ($6.95/hr), μ is the average hourly wage ($6.35/hr), and σ is the standard deviation ($0.45/hr).

z = (6.95 - 6.35) / 0.45
z = 0.6 / 0.45
z ≈ 1.33

Step 2: Find the probability
Using a standard normal distribution table, we find that the probability of a z-score being less than 1.33 is approximately 0.9082. Since we want the probability that a worker earns more than $6.95/hr, we need to find the complement of this probability.

P(X > 6.95) = 1 - P(X ≤ 6.95)
P(X > 6.95) = 1 - 0.9082
P(X > 6.95) ≈ 0.0918

Therefore, the probability that a randomly selected worker at the fast food restaurant earns more than $6.95/hr is approximately 9.18%.

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Answer:

The simplified expression is [tex]\frac{(\sqrt{(a+2)}-2)^2}{(a-3)}[/tex]

Step-by-step explanation:

Pls give me brainliest and let me know if this is incorrect. Thx

The container with dimensions 6 meters by 6 meters by 6  meters has

the minimum cost among all containers with a volume of 324  cubic

meters.

Let's first determine the dimensions of the square wooden base.

Let the side length of the square base be x meters. Then the height of

the container would also be x meters, since the container is made of a

square base and the sides are made of cardboard.

Therefore, the volume of the container can be expressed as[tex]V = x^2 \times x = x^3[/tex] cubic meters.

We want to find the dimensions of the cheapest container with a volume

of 324 cubic meters. Therefore, we need to minimize the cost of the

container, which is a function of the surface area of the cardboard sides.

The surface area of the cardboard sides is given by A = 4xh = 4x^2

square meters, where h is the height of the container.

Let's use the fact that the cost of wood is three times the cost of

cardboard to express the cost of the container in terms of x:

[tex]C(x) = 3x^2 + 4x^2 = 7x^2[/tex]

where the first term represents the cost of the wooden base and the

second term represents the cost of the cardboard sides.

Now we can express the cost of the container in terms of its volume:

[tex]C(V) = 7(V^{(2/3)})[/tex]

We want to find the value of x that minimizes C(V) subject to the

constraint that V = 324.

To do this, we can use the method of Lagrange multipliers:

[tex]L(x, \lambda) = 7(x^{(2/3)}) + \lambda(324 - x^3)[/tex]

Taking the partial derivative of L with respect to x and setting it equal to zero, we get:

[tex](14/3)x^{(-1/3)} - 3\lambda x^2 = 0[/tex]

Taking the partial derivative of L with respect to λ and setting it equal to zero, we get:

[tex]324 - x^3 = 0[/tex]

Solving for x, we get:

[tex]x = (324/3)^{(1/3)}[/tex] = 6 meters

Therefore, the dimensions of the cheapest container with a volume of

324 cubic meters are 6 meters by 6 meters by 6 meters. To verify that

this gives a minimum cost, we can take the second derivative of C(V)

with respect to V and evaluate it at V = 324:

C''(324) = -98/81 < 0

Since the second derivative is negative, this confirms that C(V) has a

local maximum at V = 324, and hence a local minimum at x = 6.

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Euler's method is a numerical method used to approximate solutions to first-order ordinary differential equations (ODEs). It is based on the idea of walking along tangent lines of nearby solutions for short periods of time. The given statement is true.

The basic idea of Euler's method is to approximate the solution to an ODE at discrete time steps using a simple iterative formula that involves the slope of the solution at each time step. At each time step, the slope of the solution is approximated by the slope of the tangent line to the solution at that point. The method then takes a small step along this tangent line to approximate the solution at the next time step.

This process is repeated over and over again, with each step approximating the solution at the next time point. While the method is not exact, it can provide a useful approximation of the true solution if the time steps are small enough.

In summary, Euler's method is based on the idea of approximating the solution to an ODE by walking along tangent lines of nearby solutions for short periods of time.

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you would end up paying a total of $1,422.72 over two years, with $222.72 of that being interest.

What is simple interest?

A quick and simple way to figure out interest on money is to use the simple interest technique, which adds interest at the same rate for each time cycle and always to the initial principal amount. Any bank where we deposit our funds will pay us interest on our investment. One of the different types of interest charged by banks is simple interest. Now, before exploring the idea of basic curiosity in further detail,

Plugging in these values, we get:

[tex]PMT = 1200 x*0.0158 / (1 - (1 + 0.0158)^{(-24)) = $59.28[/tex]

So you would need to pay about $59.28 each month to pay off your credit card in two years.

To find the total interest paid, we can subtract the original balance from the total amount paid:

Total interest = Total amount paid - Original balance

We can find the total amount paid by multiplying the monthly payment by the total number of months:

Total amount paid = PMT x n = $59.28 x 24 = $1,422.72

So the total interest paid is:

Total interest = $1,422.72 - $1200 = $222.72

Therefore, you would end up paying a total of $1,422.72 over two years, with $222.72 of that being interest.

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This means that the probability that x assumes a value between 94.0 and 148.2 is 0.9032 or 90.32% (rounded to four decimal places).

To find the probability that x assumes a value between 94.0 and 148.2, we need to find the area under the normal curve between these two values. We can do this by standardizing the values using the z-score formula and then using a table or calculator to find the area under the standard normal curve.
First, we calculate the z-scores for 94.0 and 148.2:
z1 = (94.0 - 119.8) / 12.5 = -2.05
z2 = (148.2 - 119.8) / 12.5 = 2.25
Next, we look up the area between these two z-scores using a standard normal table or calculator. Using a calculator, we can use the normalcy function:
normalcy(-2.05, 2.25, 0, 1) = 0.9032

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Answer:

x = 7.25

Step-by-step explanation:

2×(2x + 5) = 39

4x + 10 = 39

4x = 29

x = 7.25

The inverse function is a scaling of the original function by a factor of 1/6.

What is function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

The inverse of f(x) = 6x can be found by solving for x in terms of y and then replacing y with x:

y = 6x

x = y/6

Therefore, the inverse function is:

f^-1(x) = x/6

Alternatively, we can write it as:

f^-1(x) = (1/6) x

Note that the inverse function is a scaling of the original function by a factor of 1/6.

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The maximum revenue is $6,075.

The elasticity of demand when the hourly instruction rate is $65 can be determined using the formula E = (dq/dp)*(p/q). To increase your revenue, you should lower the hourly instruction rate. To maximize revenues, you should change the instruction rate to $45.


1. Calculate q when p is $65: q = 270(65) - 65d => q = 17550 - 65d
2. Calculate derivative dq/dp: dq/dp = -d
3. Calculate E: E = (-d)*(65/(17550-65d))
4. Set E = -1 (for maximum revenue) and solve for d: -1 = (-d)*(65/(17550-65d))
5. Solve for d: d = 3
6. Substitute d in the demand equation to find p: 270p - 3p = 17550 => p = $45
7. Calculate the maximum revenue: q = 270(45) - 3(45) => q = 135
8. Maximum revenue: $45 * 135 = $6,075

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The probability that a Democrat opposes stronger gun control laws is 0.16.

To find the probability that a Democrat opposes stronger gun control laws, we need to look at the table provided. In the table, the row for Democrats shows two values: 0.25 for "Favor" and 0.16 for "Oppose." These values represent the proportion of Democrats who favor and oppose stronger gun control laws, respectively.

Since the question asks for the probability that a Democrat opposes stronger gun control laws, we focus on the value of 0.16, which represents the proportion of Democrats who oppose stronger gun control laws.

Therefore, the probability that a Democrat opposes stronger gun control laws is 0.16.

Therefore, the probability that a Democrat opposes stronger gun control laws is 0.16.

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Since Σk=1[infinity] 1/√n^3 converges by the p-series test, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 converges as well.

We will use the Limit Comparison Test to determine the convergence or divergence of the series Σk=1[infinity] 1/ √n^3 +5.

Let a_n = 1/√(n^3 + 5)

Then, we need to find a series b_n such that:

b_n > 0 for all n

The limit of (a_n/b_n) as n approaches infinity is a positive, finite number.

To find such a series b_n, we can compare a_n to a simpler series that we know converges or diverges. One such series is the series:

Σk=1[infinity] 1/√n^3

which converges by the p-series test with p=3/2.

We know that 0 < a_n < 1/√n^3 for all n, so we can use the inequality:

1/√n^3 + 5 < 1/√n^3

Multiplying both sides by 1/n, we get:

1/n√n^3 + 5/n < 1/n√n^3

1/n^(5/2) + 5/n < 1/n^(5/2)

Let b_n = 1/n^(5/2)

Then, we have:

0 < a_n/b_n < (1/n^(5/2) + 5/n)/1/n^(5/2) = 1 + 5/n^(3/2)

Taking the limit as n approaches infinity, we get:

lim (a_n/b_n) = lim [1/(n^(5/2)√(n^3 + 5))] / (1/n^(5/2))

= lim [(n^(5/2))/(√(n^3 + 5))] = 1

Since 0 < a_n/b_n < 1 + 5/n^(3/2) and lim (a_n/b_n) = 1, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 and Σk=1[infinity] 1/√n^3 have the same behavior, meaning they both converge or both diverge. Since Σk=1[infinity] 1/√n^3 converges by the p-series test, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 converges as well.

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1. The derivative of f(x) = 4 is 0.

2. The derivative of f(x) =63x is 63

3. The derivative of f(x)=7 is 0.

4. The derivative of f(x) is 12x+3

To find the derivatives of f(x) = 4, f(x) = 63x, f(x) = 7, and f(x) = 3(2x²+x) using the limit definition, follow these steps:

1. For f(x) = 4, the derivative, f'(x), is 0 since it is a constant function.

2. For f(x) = 63x, use the limit definition: f'(x) = lim(h→0) [(f(x+h)-f(x))/h]. Plug in f(x) = 63x and simplify: f'(x) = lim(h→0) [(63(x+h)-63x)/h] = lim(h→0) [63h/h] = 63.

3. For f(x) = 7, the derivative, f'(x), is 0 since it is a constant function.

4. For f(x) = 3(2x²+x), apply the limit definition and simplify:

f'(x) = lim(h→0) [(f(x+h)-f(x))/h] = lim(h→0) [(3(2(x+h)²+(x+h))-3(2x²+x))/h] = lim(h→0) [(6x²+6xh+6h²+3h)/h] = lim(h→0) [6x+6x+3] = 12x+3.

In summary, the derivatives are: f'(x) = 0, f'(x) = 63, f'(x) = 0, and f'(x) = 6x+3.

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The maximum number of packages  of skittles Tom can buy at a cost $0. 48 per package is equal to 7.

Total amount of money to spend with Tom = $40

Amount of money Tom spent on a light saber = $21.40  

And amount of money Tom  set aside for a Yoda t-shirt = $15

Cost of skittles per package = $0. 48

Amount of money left to spend on skittles is,

= $40 - $21.40 - $15

= $3.60

Maximum number of packages Tom can buy of skittles

= Amount of money Tom has left  / The cost per package of skittles

Substitute the values we get,

⇒ Maximum number of packages Tom can buy of skittles

= $3.60 ÷ $0.48 per package

= 7.5 packages

Since Tom cannot buy a fraction of a package,.

This implies Tom can buy a maximum of 7 packages of skittles with the money he has left.

Therefore, maximum of 7 packages of skittles with the money after buying the light saber and setting aside money for the Yoda t-shirt.

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a) The coefficient a of the term [tex]x^4[/tex] in the expansion of[tex](x^2 + 4)^8[/tex]is 17920.

b) The coefficient a of the term [tex]x^8y^2[/tex] in the expansion of [tex](x - 4y)^{10[/tex] is

2949120.

We can use the Binomial Theorem, which states that the coefficient of

the term[tex]x^r[/tex] in the expansion of[tex](a + b)^n[/tex] is given by the expression:

[tex]C(n, r) \times a^{(n-r)} \times b^r[/tex]

where C(n, r) is the binomial coefficient, given by:

C(n, r) = n! / (r! × (n-r)!)

So in our case, we have:

n = 8

r = 4

a =[tex]x^2[/tex]

b = 4

Plugging these values into the formula, we get:

[tex]C(8, 4) \times (x^2)^{(8-4)} \times4^4\\= C(8, 4) \times x^8 \times 256\\= 70 \times x^8 \times 256\\= 17920x^8[/tex]

b.) We can again use the Binomial Theorem. This time, we have:

n = 10

r = 8

a = x

b = -4y

(Note that we use -4y for b, since the term involves a negative power of y.)

Plugging these values into the formula, we get:

[tex]C(10, 8) \times x^{(10-8)} \times (-4y)^8\\= C(10, 8) \times x^2 \times 65536y^8\\= 45 \times x^2 \times 65536y^8\\= 2949120x^2y^8[/tex]

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Answer:

7

Step-by-step explanation:

subtract greatest number (7) by smallest number (0)

7-0=7

The resulting Cartesian equation in the form x = f(y) is x = (y + 7)².

To eliminate the parameter t and find a Cartesian equation in the form x = f(y), we need to solve for t from one of the given equations and then substitute it into the other equation. We'll use the y(t) equation for this purpose:

y(t) = -7 + 2t

Now, solve for t:
t = (y + 7) / 2

Next, substitute this value of t into the x(t) equation:
x(t) = 2t²
x = 2((y + 7) / 2)²

Simplify the equation:
x = (y + 7)²

So, the resulting Cartesian equation in the form x = f(y) is x = (y + 7)².

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The dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

Let's start by setting up some notation for the dimensions of the cylindrical container. Let the height of the container be h, and let the radius of the top and bottom be r. Then, the volume of the container is given by:

[tex]V =\pi r^2h[/tex]

We want to minimize the cost of the material used to make the container. The cost is composed of two parts: the cost of the material used for the top and bottom, and the cost of the material used for the side. Let's compute these separately.

The cost of the material used for the top and bottom is given by the area of two circles with radius r, multiplied by the cost per square meter:

[tex]C1 = 2\pi r^2 * 20[/tex]

The cost of the material used for the side is given by the area of the side of the cylinder, which is a rectangle with height h and length equal to the circumference of the base (which is 2πr), multiplied by the cost per square meter:

C2 = 2πrh * 15

The total cost is the sum of these two costs:

[tex]C = C1 + C2 = 2\pi r^2 * 20 + 2\pi rh * 15[/tex]

We want to minimize this cost subject to the constraint that the volume is 10 [tex]m^3[/tex]:

[tex]V = \pi r^2h = 10[/tex]

We can use the volume equation to eliminate h, obtaining:

[tex]h = 10/(\pi r^2)[/tex]

Substituting this expression for h into the cost equation, we obtain:

[tex]C = 2\pi r^2 * 20 + 2\pi r * 15 * 10/(\pi r^2)[/tex]

Simplifying, we have:

[tex]C = 40\pi r^2 + 300/r[/tex]

To minimize this function, we take its derivative with respect to r and set it equal to zero:

[tex]dC/dr = 80\pi r - 300/r^2 = 0[/tex]

Solving for r, we obtain:

[tex]r = (300/(80\pi ))^{(1/3)} = 0.508 m[/tex]

To find the corresponding value of h, we can use the volume equation:

[tex]h = 10/(\pi r^2)[/tex] ≈ 3.132 m

Therefore, the dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

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Equation {−7(4x−2) +7x} has the simplified form is (−21x + 14)

A simple equation is a mathematical statement that equates two expressions using an equal sign, and which can be solved in a straightforward manner without using complex mathematical operations. In other words, it is an equation that involves only one variable and has a degree of 1 (linear).

To simplify −7(4x−2) +7x, we can first use the distributive property to remove the parentheses:

⇒ −7(4x−2) +7x

⇒ −28x + 14 + 7x

Next, we can combine like terms by adding the x terms and the constant terms separately:

⇒ −28x + 14 + 7x

⇒ (−28x + 7x) + 14

⇒ −21x + 14

Therefore, the simplified form of {−7(4x−2) +7x} is {−21x + 14}.

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Answer: -1

Step-by-step explanation:

when looking at a graph, count the distance between 2 points. Divide it by 2 then count that many and you have your answer

This is integral to find the length of the arc of the parabola y = x² + 1 inscribed in the disc of equation x² + (y – 1)² = 1.

To set up an integral to find the length of the arc of parabola y = x2 + 1 inscribed in the disc of equation x2 + (y – 1)2 = 1, we can use the formula for arc length:

[tex]L =\int_{[a,b]} \sqrt{[1 + (dy/dx)2] }dx[/tex]

where a and b are the x-coordinates of the points where the parabola intersects the circle, and dy/dx is the derivative of y with respect to x.

First, we need to find the x-coordinates of the points of intersection. We can solve for x in the equation of the circle:

[tex]x^2 + (y - 1)^2 = 1\\x^2 + y^2 - 2y + 1 = 1\\x^2 + y^2 - 2y = 0\\x^2 + (y - 1)^2 - 1 = 0\\x^2 + (x^2 + 2y - 1) - 1 = 0\\2x^2 + 2y -2 = 0x^2 + y - 1 = 0[/tex]

Substituting y = x² + 1, we get:

x² + x² + 1 – 1 = 0
2x² = 0
x = 0

So the parabola intersects the circle at (0,1) and (0,-1).

Next, we need to find the derivative of y with respect to x:

dy/dx = 2x

Substituting into the formula for arc length, we get:

[tex]L = \int_{[-1,1]} \sqrt{[1 + (2x)^2}dx[/tex]

This is integral to find the length of the arc of the parabola y = x² + 1 inscribed in the disc of equation x² + (y – 1)² = 1. We do not ask to evaluate this integral.

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The probability that the average tail length of the sample is between 3.8 and 3.9 feet is approximately 0.2417.

We have,

First, we need to find the mean and standard deviation of the sample distribution of the mean tail length:

The mean of the sample distribution is equal to the mean of the population, which is 3.75 feet.

The standard deviation of the sample distribution is equal to the standard deviation of the population divided by the square root of the sample size:

σ/√n = 0.6/√36 = 0.1 feet

Now we can use the standard normal distribution to find the probability:

z1 = (3.8 - 3.75) / 0.1 = 0.5

z2 = (3.9 - 3.75) / 0.1 = 1.5

Using a standard normal table or calculator, we can find the probability that z is between 0.5 and 1.5:

P(0.5 ≤ z ≤ 1.5) = P(z ≤ 1.5) - P(z ≤ 0.5) = 0.9332 - 0.6915 = 0.2417

Therefore,

The probability that the average tail length of the sample is between 3.8 and 3.9 feet is approximately 0.2417.

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Answer:

(7, -1)

Step-by-step explanation:

3x + 7y = 14

y = x - 8

3x + 7(x - 8) = 14

3x + 7x - 56 = 14

10x - 56 = 14

Add 56 to both sides.

10x = 70

Divide both sides by 10.

x = 7

3(7) + 7y = 14

21 + 7y = 14

Subtract 21 from both sides.

7y = -7

Divide both sides by 7.

y = -1

(7, -1)

The volume of the solid obtained by rotating the region enclosed by [tex]$\$ y=1-x^{\wedge} 2 \$$[/tex] and [tex]$\$ y=7 \$$[/tex] about the line [tex]$a=2$[/tex] using the method of disks or washers is [tex]$\$ \backslash f r a c\{64 \backslash p i\}\{15\} \$$[/tex].

To use the method of disks or washers, we need to first graph the region enclosed by the equations [tex]$y=1-x^2$[/tex] and [tex]$y=7$[/tex].

Let's find the x-intercepts of [tex]$y=1-x^2$[/tex]:

[tex]$$\begin{aligned}& 0=1-x^2 \\& x= \pm 1\end{aligned}$$[/tex]

So the region enclosed by the two equations is a parabolic shape with [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(1,0)$[/tex] and a vertex at [tex]$(0,1)$[/tex]. The line [tex]$a=2$[/tex] is a vertical line passing through the point [tex]$(2,0)$[/tex].

To use the method of disks or washers, we need to integrate along the axis of rotation. Since the line of rotation is vertical, we need to integrate with respect to [tex]$x$[/tex].

We need to find the area of the region enclosed by [tex]$\$ y=1-x^{\wedge} 2 \$$[/tex] and [tex]$\$ y=7 \$$[/tex] as a function of [tex]$x$[/tex]. This can be found by subtracting the equations of the two curves:

[tex]$$\begin{aligned}& A(x)=\pi\left(\left(2-\left(1-x^2\right)\right)^2-2^2\right) \\& A(x)=\pi\left(\left(3-x^2\right)^2-4\right)\end{aligned}$$[/tex]

The volume of the solid obtained by rotating this region about the line [tex]$a=2$[/tex]is given by the integral:

[tex]$$V=\int_{-1}^1 \pi\left(\left(3-x^2\right)^2-4\right) d x$$[/tex]

Evaluating this integral, we get:

[tex]$V=\frac{64 \pi}{15}$[/tex]

Therefore, the volume of the solid obtained by rotating the region enclosed by [tex]$\$ y=1-x^{\wedge} 2 \$$[/tex] and [tex]$\$ y=7 \$$[/tex] about the line [tex]$a=2$[/tex] using the method of disks or washers is [tex]$\$ \backslash f r a c\{64 \backslash p i\}\{15\} \$$[/tex].

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The width of the rectangle is 0.9 cm to 1 d.p.

A triangle is a three-sided polygon or a three-dimensional object composed of three flat surfaces that intersect at three vertices. Triangles can be classified based on their sides and angles. Equilateral triangles have all three sides equal, isosceles triangles have two sides equal, and scalene triangles have all three sides of different lengths. Triangles can also be classified based on their angles. Right triangles have one 90-degree angle, obtuse triangles have one angle greater than 90-degrees, and acute triangles have all angles less than 90-degrees.

The circumference of the shaded face can be calculated using the formula for circumference of a circle, C = 2πr, where r is the radius of the circle. The radius of the shaded face can be found by subtracting the height of the net (h) from the radius of the cylinder (R). Therefore, the circumference of the shaded face can be calculated as follows:

C = 2π(R-h)
= 2π(2-1)
= 2π
= 6.28

The circumference of the shaded face is 6.28 cm to 1 d.p.

b) To calculate the width, w, of the rectangle, we can use the formula for area of a rectangle, A = lw, where l is the length of the rectangle. The area of the rectangle can be found by adding the area of the two semicircles (πr2) and subtracting the area of the triangular part (½bh). Therefore, the width of the rectangle can be calculated as follows:

A = lw
w = A/l
w = (2πr2+2πr2-½bh)/2(2πr)
w = (4πr2-½bh)/(4πr)
w = (8-1)/(8)
w = 7/8

The width of the rectangle is 0.9 cm to 1 d.p.

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