What method is best for solving for (m+8)^2=72?

The relative minimum is at t = 1/√2, the relative maximum is at t = -1/√2, the absolute minimum is at t = ±2, and the absolute maximum is at t = 1/√2.

To find the relative and absolute extrema of the function f(t) = t^2 - (1/t^2) over the domain [-2, 2], we first find the critical points by setting the derivative equal to zero:

f'(t) = 2t + 2/t^3 = 0

Solving for t, we get:

t = ±(1/√2)

We now check the second derivative to classify the critical points:

f''(t) = 2 - 6/t^4

At t = 1/√2, f''(t) > 0, so we have a relative minimum at t = 1/√2.

At t = -1/√2, f''(t) < 0, so we have a relative maximum at t = -1/√2.

To determine if there are any absolute extrema, we evaluate the function at the endpoints of the domain:

f(-2) = 4 - 1/4 = 15/4

f(2) = 4 - 1/4 = 15/4

Since f(t) is a continuous function over the closed interval [-2, 2], and the critical points and endpoints are finite, the extreme value theorem tells us that f(t) must have an absolute minimum and an absolute maximum on the interval.

Since f(-1/√2) is greater than both endpoints, the absolute minimum must occur at one of the endpoints.

Therefore, the absolute minimum of f(t) over the domain [-2, 2] is 15/4, which occurs at t = ±2.

Since f(1/√2) is less than both endpoints, the absolute maximum must occur at t = 1/√2.

Therefore, the absolute maximum of f(t) over the domain [-2, 2] is 7, which occurs at t = 1/√2.

In summary, the relative minimum is at t = 1/√2, the relative maximum is at t = -1/√2, the absolute minimum is at t = ±2, and the absolute maximum is at t = 1/√2.

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The defined sets are as follows:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10....25}

A = {3, 6, 9, 12, 15, 18}

A' = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25}

The universal set contains natural numbers that are at most 25. Natural numbers are counting numbers.

Therefore,

universal set = U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10....25}

A is the set of multiples of 3 that are less than 20. Therefore,

A = {3, 6, 9, 12, 15, 18}

A'(A prime) is the number that is not in the A set but is present in the universal set.

Therefore,

A' = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25}

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The overall main text structure of "How Archaeologists Found the Lost City of Troy" is best described as a description. The Option D is correct.

This article titted "How Archaeologists Found the Lost City of Troy" is focused on providing a detailed description of how archaeologists were able to locate the ancient city of Troy.

The author uses descriptive language and provides a chronological account of the excavation process as well as detailing the various methods and techniques used by archaeologists to uncover the city's remains.

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In polar coordinates, the equation for (a) is r = 10sinθ, the equation for (b) is r = √10, the equation for (c) is r = 6cosθ, and the equation for (d) is r = 10sinθ / (cos^2θ - sin^2θ).

In polar coordinates, a point (r, θ) represents the distance r from the origin and the angle θ measured counterclockwise from the positive x-axis. To express each equation in polar coordinates, we can substitute x = rcosθ and y = rsinθ, then simplify using trigonometric identities and algebraic manipulations.

(a) y = 10

Substituting y = rsinθ, we get rsinθ = 10, or r = 10sinθ.

(b) x^2 + y^2 = 10

Substituting x = rcosθ and y = rsinθ, we get r^2 = 10, or r = √10.

(c) x^2 + y^2 - 6x = 0

Substituting x = rcosθ and y = rsinθ, we get r^2 - 6rcosθ = 0, or r = 6cosθ.

(d) x^2(x^2 + y) = 10y^2

Substituting x = rcosθ and y = rsinθ, we get r^2cos^2θ(r^2cos^2θ + rsinθ) = 10r^2sin^2θ, or r = 10sinθ / (cos^2θ - sin^2θ).

Therefore, in polar coordinates, the equation for (a) is r = 10sinθ, the equation for (b) is r = √10, the equation for (c) is r = 6cosθ, and the equation for (d) is r = 10sinθ / (cos^2θ - sin^2θ).

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Percentage of Mrs. smith's students scored at or above a 92% for given mean and standard deviation using standard normal distribution is equal to 15.87%.

Mean = 80

Standard deviation = 12

To find the percentage of Mrs. Smith's students who scored at or above a 92%,

Find the area under the normal distribution curve to the right of 92%.

Use a standard normal distribution table or a calculator ,

Find the corresponding z-score for a raw score of 92 in a normal distribution with mean 80 and standard deviation 12.

z = (92 - 80) / 12

  = 1

This means that a score of 92 is one standard deviation above the mean.

From a standard normal distribution table,

Attached table.

Find that the area to the right of a z-score of 1 is approximately 0.1587.

This means that approximately 15.87% of Mrs. Smith's students scored at or above a 92%.

Therefore, about 15.87% of Mrs. Smith's students scored at or above a 92% on the psychology test using standard normal distribution .

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Answer:

B

Step-by-step explanation:

using the converse of Pythagoras' identity

if the longest of the 3 sides is equal to the sum of the squares on the other 2 sides then the triangle is right.

A

longest = 20 → 20² = 400

11² + 16² = 121 + 256 = 377

400 ≠ 377 , so triangle is not right

B

longest = 17 → 17² = 289

8² + 15² = 64 + 225 = 289

then this triangle is right

C

longest = 9 → 9² = 81

6² + 8² = 36 + 64 = 100

81 ≠ 100 , so this triangle is not right

D

longest = 17 → 17² = 289

6² + 15² = 36 + 225 = 261

289 ≠ 261 , so this triangle is not right

Climatologists have determined that the number of hurricanes hitting the East Coast each year follows a Poisson distribution. They arrived at this conclusion by:

1. Collect historical hurricane data: Climatologists gather data on the number of hurricanes that have hit the East Coast each year over a long period of time.

2. Analyze the data: They then analyze this data to identify patterns and trends in the frequency of hurricanes.

3. Determine distribution: Based on their analysis, climatologists found that the number of hurricanes follows a Poisson distribution. This means that the probability of a certain number of hurricanes occurring in a given year is determined by a single parameter, λ (lambda), which represents the average number of hurricanes per year.

4. Calculate probabilities: Using the Poisson distribution formula, climatologists can calculate the probability of different numbers of hurricanes hitting the East Coast in a given year. The formula is:

P(x) = (e^(-λ) * λ^x) / x!

Where P(x) is the probability of x hurricanes occurring in a year, e is the base of the natural logarithm (approximately 2.718), λ is the average number of hurricanes per year, and x! is the factorial of x.

By applying this distribution, climatologists can estimate the likelihood of various hurricane scenarios and better understand the risks associated with hurricanes on the East Coast.

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Answer:

0

Step-by-step explanation:

Given

f ( x ) = - x² - 10x - 9

To find : f ( - 1 )

x = 1

f ( 1 ) = - ( - 1 )² - 10 ( - 1 ) - 9

       = - 1 + 10 - 9

       = 10 - 9 - 1

       = 10 - 10

f ( 1 ) = 0

Answer:

118/3x - 59

Step-by-step explanation:

Hope this helps! :)

In the triangle, value of n is,

n = 16

We have to given that;

In triangle KMN,

MN = 24 yards

KL = 30 yards

LM = 15 yards

Hence, By proportionality theorem we get;

⇒ MN / KM = OL / KL

⇒ 24 / 45 = n / 30

Solve for n;

⇒ 24 × 30 / 45 = n

⇒ n = 16

Thus, In the triangle, value of n is,

n = 16

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The height of the trapezoid is H = 10 units

Given data ,

A trapezoid has bases of lengths 30 and 50

Now , the area of the trapezoid is A = 400 units²

where Area of Trapezoid = ( ( a + b ) h ) / 2

On simplifying , we get

400 = ( 30 + 50 ) ( H ) / 2

Multiply by 2 on both sides , we get

800 = 80 H

Divide by 80 on both sides , we get

H = 10 units

Hence , the height is H = 10 units

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The probability of choosing a green tile and then a blue tile is given as follows:

1/7.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is then calculated as the division of the number of desired outcomes by the number of total outcomes.

For each outcome, the probabilities are given as follows:

Hence the probability is given as follows:

2/7 x 1/2 = 1/7.

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Answer:

(10,13)

Step-by-step explanation:

the distance from A to D is 8. you can see that from subtracting the x coordinates 10-2=8.

This is a square so all the sides are equal. To find C, add 8 to the Y coordinate from point B.

5+ 8 = 13

so c is at (10,13)

The length of the curve is:

[tex]=\frac{(68)^\frac{3}{2}-8 }{3}[/tex]

Using the arc length formula in terms of polar coordinates [tex]\int\limits\sqrt{r^2+(\frac{dr}{d\theta})^2 }[/tex]

To find the length of the curve we will use the formula:

[tex]\int\limits\sqrt{r^2+(\frac{dr}{d\theta})^2 }[/tex]

Now, Let us put it in the expression:

[tex]r = \theta^2\\\\\frac{dr}{d\thera} =2\theta[/tex]

Now the integral becomes:

[tex]=\int\limits^8_0 \sqrt{(\theta)^4+(2\theta)^2} \, d\theta\\ \\=\int\limits^8_0\theta \sqrt{(\theta)^2+4} \, d\theta\\[/tex]

Now using the substitution method:

[tex]\theta^2+4=t\\\\2\thetad\theta=dt\\\\=\int\limits\frac{\sqrt{t}dt }{2}\\ \\=\frac{t^\frac{3}{2} }{3}\\ \\=\frac{(\theta^2+4)^\frac{3}{2} }{3}[/tex]

Now, Let us plug in the values:

[tex]=\frac{(68)^\frac{3}{2}-8 }{3}[/tex]

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Answer:c

Step-by-step explanation::Add all the absolute deviations and divide by number of swimmers. Step-by-step explanation:

Neil is growing sugar crystals for his science fair experiment. He wants to find out if the amount of sugar affects the size of the crystal. Neil adds 18 tablespoons of sugar to his first bowl of water. He plans to add less sugar to the second bowl.

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Simulate the process of drawing 10 papers from the bag repeatedly, and count the number of times exactly 4 of the 10 papers are juniors. Divide the count by the total number of simulations to estimate the probability.

To implement the simulation, we can write a program that randomly generates 10 papers from the bag based on the known proportion of juniors and seniors. We can repeat this process for a large number of times, such as 10,000, and count the number of times exactly 4 of the 10 papers are juniors. The estimated probability is then the count divided by the total number of simulations. We can also calculate a confidence interval for the estimate to assess the uncertainty of the simulation result.

To ensure the simulation accurately reflects the actual process, we need to make sure that the program generates the papers randomly and without replacement, and that the proportion of juniors and seniors in the bag matches the known percentage. We also need to use a large enough number of simulations to reduce the sampling error and obtain a stable estimate.

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The best method for solving X² - 8x + 16=0 is the factor method. This is because the given quadratic equation has two identical factors.

Using the factor method, we can factorize the quadratic expression as (X-4)² =0. This can be solved easily by taking the square root of both sides, which gives us X - 4 = 0 or X = 4.

The square root method would also work in this case since the quadratic expression is a perfect square trinomial. However, it is more efficient to use the factor method since it only involves factoring and solving a linear equation, while the square root method requires taking square roots of both sides of the equation.

The quadratic formula method would also work for solving this equation, but it is more complicated and time-consuming compared to the factor method.

Therefore, the factor method is the best method for solving X²- 8x + 16=0

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The linear regression equation for the data is y=0.47x+23.3 and the projected test grade for a student with a homework grade of 62 is 63.

The linear regression equation for the data is y=0.47x+23.3.

The projected test grade for a student with a homework grade of 62 is:

y = 0.47(62)+23.3

y = 63.

The correlation coefficient is r=0.78, which indicates a strong positive correlation between homework grade and test grade. This means that students who do well on their homework tend to also do well on their tests.

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To decrease the margin of error by a factor of 1/7, the sample size needs to be increased by a factor of 49.

To understand why this is the case, recall that the margin of error is a function of the sample size, the standard deviation of the population, and the confidence level. The margin of error is proportional to the inverse square root of the sample size, which means that if we want to decrease the margin of error by a factor of 1/7, we need to increase the sample size by a factor of 7.

However, increasing the sample size by a factor of 7 only decreases the margin of error by a factor of the square root of 7 (approximately 2.65). To decrease the margin of error by a factor of 1/7, we need to increase the sample size by a factor of 7 times the square of 2.65, which is approximately 49.

In other words, to decrease the margin of error by a factor of 1/7, we need to increase the sample size by a factor of 49. This means that we need to collect almost 50 times more data than we currently have in order to achieve the desired level of precision.

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Answer:

11.5

Step-by-step explanation:

65=10x-50
+50.     +50
=115=10x

Step-by-step explanation:

From distance formula

(x

2

−x

1

)+(y

2

−y

1

)+(z

2

−z

1

)

AB=

1+4+1

=

6

AC=

36+5+9

=

56

=3

6

AC

AB

=

EC

BE

⇒

3

6

6

=

EC

BE

⇒

EC

BE

=

3

1

So now we can find point E

E=(

m

1

+m

2

x

1

x

2

+m

2

m

1

),(

m

1

+m

2

m

1

y

2

+m

2

y

1

),(

m

1

m

2

m

1

z

2

+m

2

z

1

)

E=(

4

−5+6

,

4

2+3

,

4

−6−6

)=E(

4

1

,

4

5

,−3)

Now we can find distance AE

AE=

(1−

4

1

)

2

+(1−

4

5

)

2

+(−3+3)

2

AE=

16

9

+8

16

1

+0

AE=

4

3

10

Check the picture below.

[tex]\sin(A )=\cfrac{\stackrel{opposite}{2\sqrt{2}}}{\underset{hypotenuse}{4}}\implies \sin(A )=\cfrac{\sqrt{2}}{2} \\\\\\ \cos(A )=\cfrac{\stackrel{adjacent}{2\sqrt{2}}}{\underset{hypotenuse}{4}}\implies \cos(A )=\cfrac{\sqrt{2}}{2} \\\\\\ \tan(A )=\cfrac{\stackrel{opposite}{2\sqrt{2}}}{\underset{adjacent}{2\sqrt{2}}}\implies \tan(A)=1[/tex]

The  arc length of the curve on the interval 0 ≤ t ≤ π/2 is approximately 1.28 units.

To find the arc length of the curve given by the parametric equations x = e^(-t) cos(t), y = e^(-t) sin(t) on the interval 0 ≤ t ≤ π/2, we use the formula for arc length:

L = ∫(a to b) √[dx/dt)^2 + (dy/dt)^2] dt

where a and b are the starting and ending values of t, respectively.

In this case, we have:

x = e^-t cos(t)
y = e^-t sin(t)

Taking the derivatives with respect to t, we get:

dx/dt = -e^-t cos(t) - e^-t sin(t)
dy/dt = -e^-t sin(t) + e^-t cos(t)

So, we have:

(dx/dt)^2 = e^(-2t) cos^2(t) + 2e^(-2t) sin(t) cos(t) + e^(-2t) sin^2(t)
(dy/dt)^2 = e^(-2t) sin^2(t) - 2e^(-2t) sin(t) cos(t) + e^(-2t) cos^2(t)

Adding the two together and taking the square root, we obtain:

√[(dx/dt)^2 + (dy/dt)^2] = e^(-t)

Therefore, the arc length of the curve on the interval 0 ≤ t ≤ π/2 is given by:

L = ∫(0 to π/2) e^-t dt

Integrating, we get:

L = [-e^-t] from 0 to π/2

L = [-e^(-π/2) + 1]

L ≈ 1.28

Therefore, the arc length of the curve on the interval 0 ≤ t ≤ π/2 is approximately 1.28 units.

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We can make some assumptions of a possible probability distribution. However, it's important that this probability distribution is based on a set of assumptions

Let's assume that the ages of children in the population of interest are integers between 1 and 10, and that each age has an equal probability of being selected. In this case, the probability distribution of the age of a child would be:

x P(X=x)

1 1/10

2 1/10

3 1/10

4 1/10

5 1/10

6 1/10

7 1/10

8 1/10

9 1/10

10 1/10

This probability distribution assigns a probability of 1/10 to each age between 1 and 10, indicating that each age is equally likely to be selected. Note that the sum of all probabilities equals 1, as it should be for any probability distribution.

However, it's important to note that this probability distribution is based on a set of assumptions, and in reality, the ages of children may not follow this exact distribution. It's always important to carefully consider the assumptions underlying any probability model and to gather as much relevant data as possible to inform the model.

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Answer: 30/40

Step-by-step explanation: 10 blue marbles - 40 = 30

The probability that both Lucy and Zaki miss the target is 1/4, or 0.25.

Let's say that the probability of Lucy missing the target is P(L) and the probability of Zaki missing the target is P(Z). Then, the probability that both of them miss the target is:

P(L and Z) = P(L) x P(Z)

This is thus because the likelihood that two separate occurrences will occur simultaneously is the product of their respective probabilities.

If we assume that Lucy and Zaki are equally skilled at throwing the ball and have the same chance of missing the target, then we can say:

P(L) = P(Z) = 1/2

So, the probability that both Lucy and Zaki miss the target is:

P(L and Z) = P(L) x P(Z)

= (1/2) x (1/2)

= 1/4

Therefore, the probability that both Lucy and Zaki miss the target is 1/4, or 0.25.

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In order to construct a confidence interval for the population mean when the population standard deviation is known, two assumptions must be met. Firstly, the sample data must be a simple random sample from the population being studied.

Secondly, the variable being measured must have a normal distribution or the sample size must be sufficiently large. The first assumption of a simple random sample ensures that the sample is representative of the population being studied, which is necessary for valid statistical inference. The second assumption is necessary because the confidence interval calculation relies on the assumption that the sample mean follows a normal distribution. When the variable being measured has a normal distribution, the sample mean will also have a normal distribution regardless of the sample size. When the variable being measured does not have a normal distribution, the sample size must be large enough (typically at least 30) in order for the sample mean to follow a normal distribution due to the central limit theorem. By meeting these two assumptions, we can be confident in the accuracy of the resulting confidence interval for the population mean.

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The complement of the event "both of the marbles are blue" is the event "one or more of the marbles are not blue".

The complement of an event refers to all the outcomes that are not part of that event. In this case, the complement of the event "both of the marbles are blue" would include all outcomes where at least one of the marbles is not blue.

Let's analyze the given options:

A. Both of the marbles are green:

This option describes an event that is different from the complement.

It implies that both marbles are green, which is not the opposite of "both marbles are blue."

B. One or more of the marbles are blue:

This option correctly describes the complement of the event.

It includes all outcomes where at least one marble is not blue, which covers situations where one or both marbles are green.

C. One or more of the marbles are green:

This option is the same as option B.

It describes the complement of the event "both marbles are blue" accurately.

D. Exactly one marble is blue and one marble is green:

This option describes a specific outcome rather than the complement of the event.

It specifies a scenario where one marble is blue and the other is green, but it does not cover other possibilities.

Based on the analysis, the statement that best describes the complement of the event "both of the marbles are blue" is B.

One or more of the marbles are blue (or equivalently,

C. One or more of the marbles are green).

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