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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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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:

D: (2,0) and (0,-4)

Step-by-step explanation:

The solutions to the graphs are where the 2 seperate graphs intersect with each other

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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Answer:

7

Step-by-step explanation:

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

7-0=7

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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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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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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The p-value is very small, likely less than 0.0001, providing strong evidence against the null hypothesis that the proportion of fathers who take no responsibility for childcare in Littleton is the same as for American fathers.

What is null hypothesis?

The null hypothesis is a statement that assumes there is no significant difference or relationship between two variables in a population, and any observed difference is due to chance.

what is proportion?

A proportion is a ratio of two quantities that represent a part of a whole, typically expressed as a fraction or a percentage. It measures the relative size of one quantity compared to another.

According to the give information:

To find the p-value for the hypothesis test, we need to follow these steps:

State the null and alternative hypotheses:

Null hypothesis (H0): The proportion of fathers who take no responsibility for childcare in Littleton is the same as the proportion for American fathers, which is 0.34.

Alternative hypothesis (Ha): The proportion of fathers who take no responsibility for childcare in Littleton is higher than the proportion for American fathers, which is greater than 0.34.

Determine the test statistic, which follows a normal distribution under the null hypothesis:

z = (p - P) / √[P(1-P) / n]

where p is the sample proportion, P is the hypothesized proportion under the null hypothesis, and n is the sample size.

In this case, we have:

p = 97/225 = 0.4311

P = 0.34

n = 225

So, the test statistic is:

z = (0.4311 - 0.34) / √[(0.34)(0.66) / 225] = 3.583

Calculate the p-value using the test statistic:

The p-value is the probability of observing a test statistic as extreme or more extreme than the one obtained, assuming the null hypothesis is true.

Since this is a one-tailed test in the upper tail (Ha: proportion is greater than 0.34), we need to find the area to the right of the test statistic in the standard normal distribution.

Using a standard normal distribution table or calculator, we find that the area to the right of z = 3.583 is very close to 0.

Therefore, the p-value is very small, likely less than 0.0001 (the exact value depends on the level of precision used in the standard normal distribution table).

In conclusion, the p-value is very small, which provides strong evidence against the null hypothesis and suggests that the proportion of fathers who take no responsibility for childcare in Littleton is higher than the proportion for American fathers.

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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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The length of an arc is 12.5 cm.

To find the length of an arc, we need to first find the measure of each central angle. Since the circle is divided into nine equal central angles, we can use the formula:

measure of each central angle = 360 degrees / number of central angles

measure of each central angle = 360 degrees / 9

measure of each central angle = 40 degrees

Now, we can use the formula for the length of an arc:

length of an arc = (central angle in degrees / 360 degrees) x (circumference of the circle)

We know that the diameter of the circle is 35.7 cm, so the radius is half of that, or 17.85 cm. The circumference of the circle is:

circumference = 2 x pi x radius

circumference = 2 x 3.14 x 17.85

circumference = 112.15 cm

Now we can plug in the values:

length of an arc = (40 degrees / 360 degrees) x 112.15 cm

length of an arc = 0.1111 x 112.15 cm

length of an arc = 12.46 cm

Rounding to tenths, the length of an arc is 12.5 cm.

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The slope of a line perpendicular to the line whose equation is 3x − 3y=45 is equal to -1.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

Since the line is perpendicular to 3x - 3y = 45, the slope is given by;

3x - 3y = 45

3y = 3x - 45

y = 3x/3 - 45/3

y = x - 45

In Mathematics and Geometry, a condition that is true for two lines to be perpendicular is given by:

m₁ × m₂ = -1

1 × m₂ = -1

m₂ = -1

Slope, m₂ = -1

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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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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

No, the events A and B are not mutually exclusive.

Mutually exclusive events are events that cannot occur simultaneously, meaning that if one event happens, the other cannot happen at the same time. In this case, events A and B are not mutually exclusive because a student can be both at most 24 years old (event A) and at least 40 years old (event B) at the same time. It is possible for a student to fall into both categories if they are exactly 24 or 40 years old.

Therefore, events A and B are not mutually exclusive.

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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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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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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)

On solving the provided question ,we can say that By combining related phrases, the following expression results: -3.6x + 24

A sequence is a grouping of "terms," or integers. Term examples are 2, 5, and 8. Some sequences can be extended indefinitely by taking advantage of a specific pattern that they exhibit. Use the sequence 2, 5, 8, and then add 3 to make it longer. Formulas exist that show where to seek for words in a sequence. A sequence (or event) in mathematics is a group of things that are arranged in some way. In that it has components (also known as elements or words), it is similar to a set. The length of the sequence is the set of all, possibly infinite, ordered items. the action of arranging two or more things in a sensible sequence.

Start by placing the negative sign outside of the brackets to simplify the calculation 3(1.2x 3.7) + 12.9:

-3(1.2x - 3.7) + 12.9 = -3(1.2x) + 3(3.7) + 12.9

The concepts included in brackets can then be clarified:

-3(1.2x) = -3.6x

Likewise, clarify the other words:

3(3.7) = 11.1

Finally, you may reintroduce the original phrase using these simplifications:

-3(1.2x - 3.7) + 12.9 = -3.6x + 11.1 + 12.9

By combining related phrases, the following expression results:

-3.6x + 24

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The estimated cost of the same goods and services on January 1, 2010 is approximately $715.07.

To estimate the cost of the same goods and services on January 1, 2010, we will use the exponential growth formula:

[tex]Future Value (FV) = Present Value (PV) * (1 + growth rate)^number of years[/tex]

1. Determine the growth rate:
Initial cost in 1995 (PV) = $600
Final cost in 2007 (FV) = $689.64
Number of years from 1995 to 2007 = 12 years

[tex]$689.64 = $600 * (1 + growth rate)^12[/tex]

Divide both sides by $600:
[tex]1.1494 = (1 + growth rate)^12[/tex]

Take the 12th root of both sides to find the annual growth rate:
1.0123 = 1 + growth rate

Subtract 1 from both sides to find the growth rate:
0.0123 = growth rate (or 1.23% per year)

2. Estimate the cost in 2010:
Number of years from 2007 to 2010 = 3 years

[tex]FV_2010 = $689.64 * (1 + 0.0123)^3[/tex]
[tex]FV_2010 = $689.64 * (1.0123)^3[/tex]
FV_2010 = $689.64 * 1.0373
FV_2010 ≈ $715.07

Therefore, the estimated cost of the same goods and services on January 1, 2010 is approximately $715.07.

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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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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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To construct a confidence interval for the population mean, we can use the following formula:

Confidence interval = sample mean ± (t-value * standard error)

Where the standard error is calculated as the population standard deviation divided by the square root of the sample size.

In this case, the sample size is 39, the sample mean is 38, and the population standard deviation is 4.4.

First, we need to find the t-value for an 80% confidence level with 38 degrees of freedom (n-1). Using a t-table or calculator, we find that the t-value is 1.303.

Next, we can calculate the standard error as:

standard error = 4.4 / sqrt(39) = 0.703

Finally, we can plug in the values to the formula and get:

Confidence interval = 38 ± (1.303 * 0.703)

Confidence interval = 38 ± 0.916

The 80% confidence interval for the population mean is, therefore (37.084, 38.916). This means that we can be 80% confident that the true population mean falls within this range based on the sample of 39 tasks that were analyzed.

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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 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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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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Answer:

The y-intercept is (0, 2).

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

Answer:

x = 7.25

Step-by-step explanation:

2×(2x + 5) = 39

4x + 10 = 39

4x = 29

x = 7.25