- The Euler equation (t – to)?y" + alt - to)y' + By = 0 is known to have the general solution y(t) = Cit? cos(in |t|) + cat sin(In \tſ), t = 0 = What are the constants to, a and B?

First, let's check why V1, which is the vector [..)-(-3) 1:1, spans R2 but does not form a basis. We can see that V1 has two linearly independent components, which means it can span R2. However, V1 is not a basis because it is not linearly independent.

To express 15 -3 as a linear combination of V1, V2, V3, we need to solve the equation aV1 + bV2 + cV3 = 15 -3, where a, b, and c are coefficients. We can rewrite this equation as a system of linear equations:

a + 2b - 3c = 15
-3b + c = -3

Solving this system of linear equations, we get:

a = -1
b = -6
c = -15

Therefore, we can express 15 -3 as a linear combination of V1, V2, V3 as:

-1V1 - 6V2 - 15V3 = 15 -3

Now, let's find another way to express 15 -3 as a linear combination of V1, V2, V3 when the coefficient of V3 is 0. This means we need to solve the equation aV1 + bV2 = 15 -3, where a and b are coefficients. We can rewrite this equation as a system of linear equations:

a + 2b = 15
-3b = -3

Solving this system of linear equations, we get:

a = 3
b = 1

Therefore, we can express 15 -3 as a linear combination of V1, V2, V3 when the coefficient of V3 is 0 as:

3V1 + V2 + 0V3 = 15 -3

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Answer:

Step-by-step explanation:

It is the point in the middle which means it is the bisector it stays the same

Answer:

1361.88ft³

Step-by-step explanation:

Unrounded answer

1361.88042ft³

Answer:

The answer to your problem is, 5447.5

Step-by-step explanation:

The formula we will be using to answer our problem is:

V = π [tex]r^{2}[/tex] h

17 replacing the “ r “

6 replacing the “ h “

Now using in expression:

V = π [tex]r^{2}[/tex] h = π × [tex]17^2[/tex] × 6 ≈ 5447.52166

Rounded: 5447.5

Thus the answer to your problem is, 5447.5

The calculated value of the number of people who had slept for less than 6 hours is 20

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

The density plot

From the plot, we have

The cumulative frequency (CF) of people who had slept for less than 6 hours to be

CF = 20

This means that the number of people who had slept for less than 6 hours is 20

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a) The volume of the region E bounded by the paraboloid is 56π

b)  The centroid of E is (0,0, 32/3)

What is paraboloid?

In geometry, a paraboloid is described as a quadric surface which has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from the term parabola which is a part of conic section.

a) Given that,

the region E bounded by the paraboloids z = x² + y² and z = 28 − 6x² − 6y²

Here we will use the  polar coordinates.

At first in the xy plane we locate the bounds on (r, ∅)

The x and y coordinate in the solid region must lie in the disk which is of radius 2.

So, 0≤r≤2 and 0≤∅≤2π

Let, x= r cos∅ and y= r sin∅

where sin²∅ + cos²∅ = 1

Therefore the above paraboloid becomes

z= r²

and z= 28- 6r²

To find the surface now we subtract r² (lower surface) from that of

28- 6r²(upper surface)

we get, 28- 7r²

and integrate the expression over r and ∅ with the limits 0≤r≤2 and 0≤∅≤2π,

∫∫(28-7r²)r dr d∅ -----(1)

= ∫∫ (28r- 7r³) dr d∅

now integrating for r at first we get,

[tex]14r^{2} - 7\frac{r^{4} }{4}[/tex]

Putting the limit for r we get,

56-28

= 28

Now from (1) we get,

∫ 28 d∅

integrating we get,

28∅

Putting the limit for ∅ we get,

56π

Hence, the volume of the region bounded by the paraboloid is 56π.

b) Here both curve is symmetric about x-axis and y - axis.

So, the x- coordinate and y-coordinate of the centroid is zero.

The z coordinate of centroid is given by,

Let c be the centroid of z-axis

c= (1/v)∫∫∫ z dv where v denotes the volume

= (1/2v) ∫∫((28-6r²)² - (r²)²) rdr d∅----- (2)

Multiplying the integration part we get,

=784r-336r³+ 35r⁵

here the limits of r is 0≤r≤2 and 0≤∅≤2π

At first taking integration for r we get,

(392r² - 84r⁴+ (35/6)r⁶)

Putting the limits for r we get,

1568-1344+ (1120/3)

= 1792/3

Now integrating for ∅ we get,

1792∅/3

Taking limit for ∅ we get,

3584π/ 3

Now from equation (2)

(1/(2×56π)×((1792×2π)/3)

= 32/3

Hence, the centroid is (0,0, 32/3)

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The area of the square t inscribed in square s of perimeter 40 cm is 50 sq cm.

If a square is inscribed in a square then the square is formed by joining the midpoints of the square of edges. This is the only square thus the square with the minimum possible area that can be inscribed in a square. Thus we can calculate the side of the inscribed square t as we following:

In right-angled triangle APS, right-angled at A,

By Pythagoras' theorem,

[tex]a^2=b^2+c^2[/tex]

where a is the hypotenuse

b is the base

c is the height

[tex]PS^2=AP^2+AS^2[/tex]

Since P is the midpoint, the length of AP and AS is 5 cm.

[tex]PS^2[/tex] = 25 + 25

PS = [tex]5\sqrt{2}[/tex] cm

Thus, the side of the square t is [tex]5\sqrt{2}[/tex] cm

The area of square t is [tex]side^2[/tex]

= [tex](5\sqrt{2})^2[/tex]

= 50 sq cm.

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To estimate the proportion of all new cell phone batteries that fail to last through a claimed number of charges, a consumer group will use a random sample and construct a percent confidence interval based on the proportion of batteries that fail to last through the charges in the sample.

To construct a confidence interval to estimate the proportion of all such batteries that fail to last through charges, the following steps can be followed:

Determine the sample size:

The consumer group should select a random sample of cell phones with the new battery and use the phones through charges of the battery.

The sample size should be determined based on the desired level of precision and confidence level.

A larger sample size will provide a more precise estimate.

Calculate the sample proportion:

The consumer group should record the proportion of batteries that fail to last through charges in the sample.

Calculate the standard error:

The standard error can be calculated using the formula:

[tex]SE = \sqrt{(p_hat * (1 - p_hat) / n) }[/tex]

where [tex]p_hat[/tex] is the sample proportion and n is the sample size.

Calculate the margin of error:

The margin of error can be calculated using the formula:

ME = z * SE

where z is the critical value from the standard normal distribution corresponding to the desired confidence level.

For example, if the desired confidence level is 95%, then z = 1.96.

Calculate the confidence interval: The confidence interval can be calculated using the formula:

[tex]CI = (p_hat - ME, p_hat + ME)[/tex]

This interval represents the range of values within which the true proportion of batteries that fail to last through charges is expected to fall with the desired level of confidence.

For example, suppose a random sample of 100 cell phones with the new battery is selected, and the proportion of batteries that fail to last through charges is found to be 0.10. If a 95% confidence level is desired, the standard error can be calculated as:

SE = [tex]\sqrt{(0.10 * 0.90 / 100)}[/tex] = 0.03

The margin of error can be calculated as:

ME = 1.96 * 0.03 = 0.06

The 95% confidence interval can be calculated as:

CI = (0.10 - 0.06, 0.10 + 0.06) = (0.04, 0.16)

Therefore, we can say with 95% confidence that the proportion of all such batteries that fail to last through charges is expected to be between 0.04 and 0.16.

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The expected value of the winnings is -2.

To find the Expected value of winning Multiply payout to probability

So, sum of all values

= -6 x 0.34 + (-4) x 0.13 + (-2) x 0.06 + 0 x0.13 + 2 x 0.34

= -2.04 - 0.52 - 0.12 + 0 + 0.68

= -2

Thus, the expected value is $3.1

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The differential equation dx/dt = x^2/14 with the initial condition x(0) = 2 has a particular solution given by x(t) = -1/(t/14 - 1/2).

To solve the separable differential equation dx/dt = x^2/14 and find the particular solution satisfying the initial condition x(0) = 2, follow these steps,

1. Identify the differential equation: dx/dt = x^2/14
2. Separate the variables: dx/x^2 = dt/14
3. Integrate both sides: ∫(1/x^2) dx = ∫(1/14) dt
4. Evaluate the integrals: -1/x = t/14 + C₁
5. Solve for x: x = -1/(t/14 + C₁)
6. Apply the initial condition x(0) = 2: 2 = -1/(0 + C₁)
7. Solve for C₁: C₁ = -1/2
8. Substitute C₁ back into the equation for x: x(t) = -1/(t/14 - 1/2)

The particular solution to the differential equation dx/dt = x^2/14 with the initial condition x(0) = 2 is x(t) = -1/(t/14 - 1/2).

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the given statement is true.div(curl f) = ∇⋅(∇×f) = 0, For a vector field f of class C^2 (meaning it has continuous second partial derivatives) in ℝ^3, the divergence of the curl of f (div(curl f)) is always equal to 0.

The following statement is true or false:

The statement is true.


1. Start with a vector field f of class C^2 in ℝ^3.
2. Calculate the curl of the vector field f, which is denoted as ∇×f.
3. Compute the divergence of the curl, represented by ∇⋅(∇×f).
4. According to the vector calculus identity, the divergence of the curl of any vector field is always equal to 0. This is known as the "curl of the gradient" theorem.

Therefore, div(curl f) = ∇⋅(∇×f) = 0, which makes the statement true.

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The coordinates of the vertices of triangle A'B'C' are:

A'(-2, 1), B'(0, -2), and C'(1, 2)

From inspection of the given diagram, the coordinates of the vertices of triangle ABC are:

A = (-2, 1)

B = (2, -5)

C = (4, 3)

If the figure is translated left 2 units and up 1 unit, then the mapping rule of the translation is:

(x,y) ---> (x-2, y +1)

If a figure is dilated by scale factor k with the origin as the center of dilation, the mapping rule is:

(x,y) ---> (kx, ky)

Therefore, given the scale factor is 0.5, the final mapping rule that translates and dilates triangle ABC is:

(x,y) ---> (0.5 (x-2), 0.5 (y +1 ))

To find the coordinates of the vertices of triangle A'B'C', substitute the coordinates of the vertices of triangle ABC into the final mapping rule:

A'=(-2,1)

B'= (0, -2)

C'= (1,2)

Therefore, the coordinates of the vertices of triangle A'B'C' are:

A'(-2, 1), B'(0, -2), and C'(1, 2)

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The sum of the numbers 1, 2, 3 and 4 when added is 10.

Given numbers are :

1, 2, 3 and 4.

We have to add these numbers.

The addition of the numbers is defined as the process of adding up to get a bigger number. It is also called finding the sum.

1 + 2 + 3 + 4 = 3 + 3 + 4

                    = 6 + 4

                    = 10

Hence the addition of the given numbers is 10.

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Yes, a weighted analysis table can be a helpful tool in determining which business function is the most critical to an organization. By assigning weights to different factors or criteria that contribute to the success of each business function.

This can be especially useful when an organization is deciding where to allocate resources or focus efforts. The weighted analysis table helps to ensure that decisions are based on a thorough and systematic analysis of all relevant factors, rather than subjective opinions or assumptions.

A weighted analysis table can indeed be useful in determining the most critical business function to an organization. Here's a step-by-step explanation of how to use a weighted analysis table for this purpose:

1. Identify the business functions: List all the key business functions within the organization, such as marketing, finance, human resources, operations, and IT.

2. Determine criteria for evaluation: Define the criteria that will be used to evaluate the importance of each business function. Examples could be revenue generation, cost reduction, customer satisfaction, or employee productivity.

3. Assign weights to criteria: Assign a weight to each criterion based on its relative importance to the organization's success. The sum of the weights should equal 100%.

4. Evaluate business functions against criteria: Rate each business function on how well it meets each criterion on a scale (e.g., 1-5 or 1-10). Be objective and consistent when assigning these ratings.

5. Calculate weighted scores: Multiply the rating for each criterion by its respective weight, and then sum up the resulting values to get the weighted score for each business function.

6. Rank the business functions: Rank the business functions based on their weighted scores, from highest to lowest. The business function with the highest weighted score will be considered the most critical to the organization.

By following these steps, a weighted analysis table can help you effectively identify the most critical business function within your organization, allowing you to allocate resources and prioritize efforts accordingly.

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Given that the perimeter of the rhombus ABCD is 52 meters, each side has a length of 52/4 = 13 meters. Since AC is diagonal with a length of 24 meters, the area of rhombus ABCD is 120 square meters.

We know that a rhombus has all sides equal in length, so each side of $abcd$ must have a length of $13$ meters ($\frac{52}{4}=13$).

We also know that the diagonal $\overline{ac}$ splits the rhombus into two congruent triangles, each with base $13$ meters and height (or length of the other diagonal) $12$ meters (half of the length of $\overline{ac}$).

Since AC is a diagonal with a length of 24 meters, we can find the other diagonal BD by using the Pythagorean theorem in the right-angled triangles formed by the diagonals. Let BD = x meters.

In triangle ABD (right-angled at B): (13^2) = (x/2)^2 + (24/2)^2 169 = (x/2)^2 + 144 25 = (x/2)^2 x = 10 meters The area of a rhombus can be found using the formula: Area = (diagonal1 * diagonal2) / 2 Area = (24 * 10) / 2 Area = 120 square meters

To find the area of one of these triangles, we use the formula for the area of a triangle: $A = \frac{1}{2}bh = \frac{1}{2}(13)(12) = 78$ square meters

Since the rhombus is made up of two congruent triangles, the area of the entire rhombus is twice this amount: $A_{\text{rhombus}} = 2A_{\text{triangle}} = 2(78) = \boxed{156}$ square meters.

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If the boxplot for one set of data is much wider than the boxplot for a second set of data, Option d, "none of the above need to be true" is the correct answer.

The width of a boxplot is determined by the range of the data, as well as the spread of the data within the interquartile range (IQR). A wider boxplot indicates a greater range and/or more spread in the data.

However, the mean and median are measures of central tendency that describe where the "middle" of the data is located. The spread of the data does not necessarily provide any information about the mean or median.

Therefore, right answer is option d "none of the above need to be true". As the width of the boxplot alone cannot tell us anything about the means or medians of the two sets of data, nor can it tell us whether one set of data contains outliers.

We need to examine additional measures, such as the mean and median, to make any conclusions about the data.

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Yes, the machine needs to be recalibrated as the hypothesis test at the 5% level showed that the standard deviation of the weight of the cereal boxes filled by the machine is greater than 0.5 oz.

To determine if the machine needs to be recalibrated, we need to test the hypothesis:

Null hypothesis:

The standard deviation of the weight of the cereal boxes filled by the machine is at most 0.5 oz. (i.e., σ ≤ 0.5)

Alternative hypothesis:

The standard deviation of the weight of the cereal boxes filled by the machine is greater than 0.5 oz. (i.e., σ > 0.5)

We will conduct a hypothesis test at the 5% level of significance using the critical value method.

First, we need to calculate the test statistic:

[tex]t = [(n - 1) * s^2] / \sigma ^2[/tex]

where n is the sample size (n = 86), s is the sample standard deviation (s = 0.52), and σ is the hypothesized population standard deviation (σ = 0.5).

[tex]t = [(86 - 1) * 0.52^2] / 0.5^2 = 6.53[/tex]

Next, we need to find the critical value for a one-tailed test with a significance level of 5% and 85 degrees of freedom (df = n - 1).

Using a t-distribution table or calculator, the critical value is approximately 1.66.

Since the test statistic (t = 6.53) is greater than the critical value (1.66), we reject the null hypothesis and conclude that the standard deviation of the weight of the cereal boxes filled by the machine is greater than 0.5 oz. Therefore, the machine needs to be recalibrated.

In summary, based on the hypothesis test conducted at the 5% level of significance, we conclude that the plant manager should recalibrate the machine as the standard deviation of the weight of the cereal boxes filled by the machine is greater than 0.5 oz.

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There are a total of 10 letters in the word REPETITION. To determine the number of arrangements where the first E occurs before the first T, we can treat the first E and the first T as distinct entities and count the number of arrangements where the first E appears before the first T.

There are 3 possible scenarios:
1. The first E is in the first position, and the first T is in one of the positions 3 through 10.
2. The first E is in the second position, and the first T is in one of the positions 4 through 10.
3. The first E is in the third position, and the first T is in one of the positions 5 through 10.

For each scenario, we can count the number of arrangements of the remaining letters. There are 6 distinct letters left, with 2 Es and 2 Ts. Therefore, the number of arrangements for each scenario is 6!/2!2!, or 180.

Multiplying the number of arrangements for each scenario by the number of possible positions for the first E and first T yields a total of 3 x 180 = 540 arrangements. However, we must divide by 2!4! to account for the fact that there are two sets of identical letters (2 Es and 4 Ts).

Therefore, the final answer is 3 x (10!/2!4!) = 226800.

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Identify the two points on the scatter plot that correspond to the teams you want to compare. Let's call these points A and B.

Find the y-coordinate (average ticket price) of point A and subtract the y-coordinate of point B from it. This will give you the difference in average ticket price between the two teams.

The formula for finding the difference in average ticket price between two teams can be written as:

Difference in average ticket price = Average ticket price of team A - Average ticket price of team B

So, to find the difference in average ticket price between a team with "x" wins and a team with 3 wins, you would need to identify the points on the scatter plot that correspond to these teams, find their respective y-coordinates (average ticket prices), and then subtract the y-coordinate of the team with 3 wins from the y-coordinate of the other team.

Thus, once you have done that, round the difference to the nearest dollar if necessary to get the final answer.

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Two different possible values for height h and radius r for the cones that can hold 9 pi cubic inches of popcorn are h=3.08 inches, r=3.08 inch and h=2.55 inches, r= 5.11 inches.

The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height of the cone.

If we assume the height and radius are equal, then

(1/3)πr²h = 9

(1/3)πr³ = 9

πr³ = 27

r³ = 27/π

r ≈ 3.08

h ≈ 3.08

Other way, we can assume the height is twice the radius, then

(1/3)πr²h = 9

(1/3)πr²(2r) = 9

(2/3)πr³ = 9

πr³ = 27/2

r³ = (27/2)/π

r ≈ 2.55

h ≈ 5.11

Therefore, two different possible values for height and radius are: (3.08, 3.08) and (2.55, 5.11).

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The gardener used 26.5 gallons of gasoline in his lawn mowers in one month. To answer your question, we know that the gardener used a total of 61.5 gallons of gasoline in one month, and 35 of those gallons were used in his lawn mowers.

Therefore, to find out how many gallons of gasoline the gardener used in his lawn mowers in one month, we simply subtract 35 from 61.5.

61.5 - 35 = 26.5

So the gardener used 26.5 gallons of gasoline in his lawn mowers in one month. It's important for gardeners and anyone using gasoline-powered equipment to be mindful of their usage and try to conserve whenever possible. This not only saves money, but also helps reduce emissions and environmental impact.

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The solution is :

a.

The type of mirror needed is a convex mirror with a radius of curvature of approximately 57.7 cm.

b.

The magnification of the image is equal to 2.26.

c.

The image is a virtual

We have,

The radius of curvature, R, is described as the reciprocal of the curvature.

The image is a virtual and erect image with a height of 48.1 cm, and  is greater than the height of the object.

The magnification of the image can be calculated as the ratio of the height of the image to the height of the object and is equal to 2.26.

To calculate the magnification = height of image / height of object = 48.1 cm / 21.3 cm = 2.26

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The circumference of the circle is given by C = πd, where d is the diameter of the circle. For the beverage area, d = 50 feet, so the circumference is C = π(50) = 157.08 feet (rounded to two decimal places). Each booth needs 10.5 feet (arc length) between its center and the center of the booth next to it. To determine how many booths can fit, we need to subtract the arc length between booths from the circumference of the circle and then divide by the arc length of each booth:

Number of booths = (C - nL) / L

where n is the number of spaces between the booths and L is the arc length of each booth. We can rearrange this equation to solve for n:

n = (C - L x Number of booths) / L

Substituting the values, we get:

n = (157.08 - 10.5x) / 10.5

where x is the number of booths. To find the maximum value of x, we need to round down to the nearest whole number:

x = floor(n) = floor((157.08 - 10.5x) / 10.5) = 14

Therefore, 14 booths can fit in the beverage area.

The circumference of the circle is C = πd = π(100) = 314.16 feet (rounded to two decimal places). The angle between each pole is 15 degrees, so the angle between the centers of each booth is also 15 degrees. To find the arc length between the centers of each booth, we need to multiply the circumference of the circle by the ratio of the angle between the centers of each booth to the angle between the poles:

Arc length between centers of each booth = C x (15/360) = 13.09 feet (rounded to two decimal places)

Therefore, there will be approximately 13.1 feet between the centers of each booth.

The circumference of the circle is C = πd = π(150) = 471.24 feet (rounded to two decimal places). Each food booth needs 30 feet between its center and the center of the booth next to it. To determine how many booths can fit, we can use the same equation as in part 1:

Number of booths = (C - nL) / L

where L = 30 feet. Substituting the values, we get:

x = floor((471.24 - 30x) / 30) = 15

Therefore, there will be approximately 15 food booths.

I hope this helps!

Answer:

Step-by-step explanation:

Expressions consist of basic mathematical operators. The product of 1/2x-1/4 & 5x^2-2x+6 is not equal to the product of 1/4x-1/2 & 5x^2-2x+6.

What is an Expression?

In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.

A.) We need to find the product of the two given expressions,

In order to multiply the two given expressions, we will first multiply the first term of the first bracket with the entire expression in the second term, and then we will do the same with the second term in the first bracket.

Now simplify the entire equation,

Thus, the product of the two expressions is .

B.) To compare the two expressions we will find the value of the two given expressions,

For the value of

we will simplify this,

As we can see the result of the product of both expressions is different.

Hence, the product of 1/2x-1/4 & 5x^2-2x+6 is not equal to the product of 1/4x-1/2 & 5x^2-2x+6.

The 95% confidence interval for the difference in population proportions of students consuming protein-rich food in 2000 and 2010 is (-0.105, -0.035).

To construct a 95% confidence interval for the difference in population proportions of students who were consuming protein-rich food in 2000 and 2010, we can use the formula:

( p1 - p2 ) ± zα/2 * sqrt( p1(1-p1)/n1 + p2(1-p2)/n2 )

where:

p1 and p2 are the sample proportions of students consuming protein-rich food in 2000 and 2010, respectively.

n1 and n2 are the sample sizes of the two years.

zα/2 is the critical value of the standard normal distribution corresponding to a 95% confidence level, which is 1.96.

Using the given data, we have:

p1 = 0.75, n1 = 700

p2 = 0.82, n2 = 850

Substituting these values into the formula, we get:

(0.75 - 0.82) ± 1.96 * sqrt( 0.75(1-0.75)/700 + 0.82(1-0.82)/850 )

Simplifying, we get:

-0.07 ± 0.035

Rounded to three decimal places, the lower bound is -0.105 and the upper bound is -0.035.

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Answer:

There are three types of angles that are outside a circle: an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants

Step-by-step explanation:

There are 420 rabbits in the park.

Given that, on Monday 20 rabbits were tagged on Tuesday 12 out of 42 rabbits were found tagged,

So,

Tuesday = 12 / 42 = 2/7

2/7 of the total rabbits on the farm = 20 tagged on Monday

Let r represent the total rabbits on the farm

then (2/7)r = 20

r = 120(7/2)

r ≈ 420

Hence, there are 420 rabbits in the park.

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The function f(x, y) is not bounded above or below as x and y approach infinity, so it does not have a global maximum or minimum. Thus, f(0,0) is neither a global maximum nor a global minimum of f(x, y)

Let's analyze the function f(x, y) = x^3 - xy + y^2.

(i) To find the critical points, we first need to find the partial derivatives of f with respect to x and y:

f_x = ∂f/∂x = 3x^2 - y
f_y = ∂f/∂y = -x + 2y

A critical point occurs when both partial derivatives are zero:

3x^2 - y = 0
-x + 2y = 0

Solving this system of equations, we find two critical points: (0, 0) and (2, 1).

(ii) To classify the critical points, we compute the second partial derivatives:

f_xx = ∂²f/∂x² = 6x
f_yy = ∂²f/∂y² = 2
f_xy = ∂²f/∂x∂y = -1

Now we evaluate the second derivative test by computing the determinant of the Hessian matrix D = (f_xx * f_yy) - (f_xy)^2:

For (0,0): D = (6*0 * 2) - (-1)^2 = 0 - 1 = -1
For (2,1): D = (6*2 * 2) - (-1)^2 = 24 - 1 = 23

Since D is negative at (0, 0), it is a saddle point. Since D is positive and f_yy > 0 at (2, 1), it is a local minimum.

(iii) The function f(x, y) is not bounded above or below as x and y approach infinity, so it does not have a global maximum or minimum. Thus, f(0,0) is neither a global maximum nor a global minimum of f(x, y).

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To find the area of the ellipse cut from the plane z=9x by the cylinder x2 y2=4, we need to first visualize the situation. The cylinder x2 y2=4 is a circular cylinder with a radius of 2 in the xy-plane. The plane z=9x is a tilted plane passing through the origin. The intersection of these two surfaces will be an ellipse.

To find the equation of the ellipse, we need to substitute z=9x into the equation of the cylinder:

x2 y2 = 4 becomes 9x2 y2 = 36

Dividing both sides by 36, we get:

x2/4 + y2/4 = 1

This is the equation of an ellipse centered at the origin with a semi-major axis of length 2 and a semi-minor axis of length 2. The area of an ellipse is given by the formula A = πab, where a and b are the lengths of the semi-major and semi-minor axes, respectively.

In this case, a = 2 and b = 2, so the area of the ellipse is:

A = π(2)(2) = 4π

Therefore, the area of the ellipse cut from the plane z=9x by the cylinder x2 y2=4 is 4π. To find the area of the ellipse cut from the plane z=9x by the cylinder x^2 + y^2 = 4, follow these steps:

1. Express the equation of the cylinder in terms of x: x^2 = 4 - y^2.
2. Substitute this expression into the equation of the plane: z = 9(4 - y^2).
3. Find the range of y-values within the cylinder: -2 ≤ y ≤ 2.
4. Use the formula for the area of an ellipse: A = πab, where a and b are the semi-major and semi-minor axes, respectively.
5. Determine the lengths of the semi-major and semi-minor axes using the equation for z (found in step 2) at the endpoints of the range of y-values (found in step 3). For y = -2, a = 9(4 - 4) = 0, and for y = 2, b = 9(4 - 0) = 36.

So, the area of the ellipse cut from the plane z=9x by the cylinder x^2 + y^2 = 4 is A = π(0)(36) = 0.

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The annual worth (AW) of an alternative for a given interest rate (i) is calculated by:
AW = [Σ (n)(t=0) At(1+i)^n-1](P/A i%, n)

In this formula, AW represents the annual worth, At represents the cash flow at time t, i is the interest rate, and n is the number of periods. The summation symbol (Σ) indicates that you need to sum the product of the cash flow at each time period t, multiplied by the present worth factor for each period, which is (1+i)^n-1.

Finally, you multiply the result by the uniform series present worth factor (P/A i%, n) to find the annual worth of the alternative which is AW = [Σ (n)(t=0) At(1+i)^n-1](P/A i%, n)

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