for time, , in hours, 0≤≤1, a bug is crawling at a velocity, , in meters/hour given by 4 / 2 t.Use Δt=0.2 to estimate the distance that the bug crawls during this hour. Use left- and right-hand Riemann sums to find an overestimate and an underestimate. Then average the two to get a new estimate.

The joint PDF of X_1 and X_2 is fx_1,x_2(x_1,x_2) = x_1×x_2/4 for 0 ≤ x_1 ≤ 2 and 0 ≤ x_2 ≤ 2.

To determine whether X and Y are independent, we need to check if their joint PMF can be expressed as the product of their marginal PMFs. Similarly, for Q and G, we need to check if their joint PMF can be expressed as the product of their marginal PMFs.

If the joint PMF of X and Y is not expressible in terms of the marginal PMFs, then we can conclude that X and Y are dependent. Similarly, if the joint PMF of Q and G is not expressible in terms of the marginal PMFs, then we can conclude that Q and G are dependent.

As for the joint PDF of X_1 and X_2, since they are independent and identically distributed, we can write:

fx_1,x_2(x_1,x_2) = fx(x_1) × fx(x_2)

= {x_1/2, 0 ≤ x_1 ≤ 2} × {x_2/2, 0 ≤ x_2 ≤ 2}

= {x_1×x_2/4, 0 ≤ x_1 ≤ 2, 0 ≤ x_2 ≤ 2}

Therefore, the joint PDF of X_1 and X_2 is fx_1,x_2(x_1,x_2) = x_1×x_2/4 for 0 ≤ x_1 ≤ 2 and 0 ≤ x_2 ≤ 2.

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The function "most_corr" takes in a DataFrame "df" that contains columns listed in "y" and "xes", where "xes" is a list of column names in "df" and "y" is the name of a column in "df".

The function returns the column name and Pearson's R correlation coefficient from "xes" that has the highest absolute correlation with "y". In other words, the function calculates the correlation coefficient between each column in "xes" and "y" and returns the name of the column with the highest absolute correlation coefficient.

The term "column" refers to the individual columns within the DataFrame, while "coefficient" refers to the Pearson's R correlation coefficient used to measure the strength of the correlation between two variables.

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we would use a one tail test. However, based on the information given in the question, it seems that a two-tail test would be more appropriate.

To determine whether to use a one-tail or two tail test in this scenario, we need to consider the directionality of the hypothesis. If we are simply testing whether the sample mean of visual memory for art majors is different from the population mean, without specifying a direction, then we should use a two-tail test. This is because the alternative hypothesis would be that the sample mean is either significantly higher or significantly lower than the population means. On the other hand, if we had a specific directional hypothesis (e.g. that art majors have better visual memory than the population mean), then we would use a one tail test. However, based on the information given in the question, it seems that a two tail test would be more appropriate.

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First, let's compute the 6-point DFT of x = [1, 2, 3, 4, 5, 6]:

[tex]X[k] = ∑_{n=0}^{5} x[n] exp(-i 2πnk/6)[/tex]

For k = 0:

[tex]X[0] = ∑_{n=0}^{5} x[n] exp(-i 2πn(0)/6)\\= ∑_{n=0}^{5} x[n]\\= 1 + 2 + 3 + 4 + 5 + 6\\= 21[/tex]

For k = 1:

[tex]X[1] = ∑_{n=0}^{5} x[n] exp(-i 2πn(1)/6)\\= ∑_{n=0}^{5} x[n] exp(-i πn/3)\\= x[0] + x[1] exp(-i π/3) + x[2] exp(-i 2π/3) + x[3] exp(-i π) + x[4] exp(-i 4π/3) + x[5] exp(-i 5π/3)\\= 1 + 2 exp(-i π/3) + 3 exp(-i 2π/3) + 4 exp(-i π) + 5 exp(-i 4π/3) + 6 exp(-i 5π/3)[/tex]

For k = 2:

X[2] = ∑_{n=0}^{5} x[n] exp(-i 2πn(2)/6)

= ∑_{n=0}^{5} x[n] exp(-i 2πn/3)

= x[0] + x[1] exp(-i 2π/3) + x[2] exp(-i 4π/3) + x[3] exp(-i 2π) + x[4] exp(-i 8π/3) + x[5] exp(-i 10π/3)

= 1 + 2 exp(-i 2π/3) + 3 exp(-i 4π/3) + 4 + 5 exp(-i 8π/3) + 6 exp(-i 10π/3)

For k = 3:

X[3] = ∑_{n=0}^{5} x[n] exp(-i 2πn(3)/6)

= ∑_{n=0}^{5} x[n] exp(-i πn)

= x[0] + x[1] exp(-i π) + x[2] exp(-i 2π) + x[3] exp(-i 3π) + x[4] exp(-i 4π) + x[5] exp(-i 5π)

= 1 - 2 + 3 - 4 + 5 - 6

= -3

For k = 4:

X[4] = ∑_{n=0}^{5} x[n] exp(-i 2πn(4)/6)

= ∑_{n=0}^{5} x[n] exp(-i 4πn/3)

= x[0] + x[1] exp(-i 4π/3) + x[2] exp(-i 8π/3) + x[3] exp(-i 4π) + x

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Answer:

$1.10

Step-by-step explanation:

Given: 5 pens cost a total of $2.75.

Divide 2.75 by 5 to find how much one pen costs.

2.75/5 = 0.55

To find how much two pens cost we take 0.55 and multiply by 2.

0.55*2 = $1.10

It will cost $1.10 for two pens.

Answer:

$1.1

Step-by-step explanation:

5 pens = $2.75

2 pens =      x

Let x by the unknown price of the 2 pens

x =  $2.75 × 2 pens    =   $5.5      =    $1.1

               5 pens                 5

Answer:

4.5 centimeters

Step-by-step explanation:

For the triangle with area 144 square cm:

144 = (1/2)(6)x

144 = 3x, so x = 48 cm

So for the triangle with area 81 square cm:

(1/2)(x)(8x) = 81

4x^2 = 81

x^2 = 81/4 so x = 9/2 = 4.5 cm and

8x = 36 cm

Answer:

4.5

Step-by-step explanation:

The probability that a randomly selected child has an IQ above 115 is approximately 0.1056.

You've asked for the probability that a randomly selected child (Ages 6-12) has an IQ above 115, given that children's IQs follow a normal distribution with a mean of 100 and a standard deviation of 12. Here's a step-by-step explanation:

1. Calculate the z-score by using the formula: z = (X - μ) / σ
  Where X = 115 (the IQ value), μ = 100 (mean), and σ = 12 (standard deviation).
  z = (115 - 100) / 12 = 15 / 12 = 1.25

2. Use a standard normal distribution table (also known as a z-table) to find the probability associated with the z-score of 1.25. The table shows that the probability of a z-score being less than 1.25 is approximately 0.8944.

3. Since we need to find the probability of a child having an IQ above 115, we need to find the probability of having a z-score greater than 1.25. This can be calculated as:

1 - P(z ≤ 1.25) = 1 - 0.8944 = 0.1056.

So, the probability that a randomly selected child has an IQ above 115 is approximately 0.1056.

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Answer:

the answer is 2400

Step-by-step explanation:

There is precisely one local minimum point in the two-dimensional space of function f(x,y), and it is located at the critical point (1,1).

To find the local minimum points of f(x,y) in R², we need to find the critical points where the gradient of f is zero or does not exist,

and then test these points using the second partial derivative test or another appropriate method to determine whether they are local minima, maxima, or saddle points.

The gradient of f(x,y) is given by:

∇f(x,y) = (y - y², x - x²)

To find the critical points, we need to solve the system of equations:

This gives us two possible critical points: (0,0) and (1,1).

To test these points, we can use the second partial derivative test.

The Hessian matrix of f is:

Evaluating the Hessian matrix at each critical point gives:

which has eigenvalues λ1 = -1 and λ2 = 1, indicating a saddle point.

which has eigenvalues λ1 = λ2 = -1, indicating a local maximum.

Therefore, f(x,y) has exactly one local minimum point in R², at the critical point (1,1).

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Answer:

12. John consumed 4 servings of pasta in this one meal. (2 cups / 0.5 cups per serving = 4 servings)

13. The clinic has a total of 2,500 Band-Aids in stock. (2,000 + 53 + 250 + 197 = 2,500 Band-Aids)

14. The three doctors evaluated a total of 65 patients. (23 + 25 + 17 = 65 patients)

15. The chiropractor spent a total of 1,170 minutes (19.5 hours) on direct patient care. (90 + 30 + 15 + 60 + 30 + 60 + 45 + 60 + 30 + 60 + 45 + 60 + 30 + 60 + 45 + 30 + 60 + 30 + 60 + 30 + 60 + 45 + 30 + 60 + 30 + 60 + 45 + 30 + 60 + 30 + 60 = 1,170 minutes; 1,170 / 60 = 19.5 hours)

Step-by-step explanation:

Mark Brainliest!!

The contour lines closer to the origin

Let's start by understanding the equation given: y = f(x,t) = cos(t)sin(x)

Here, t represents time in milliseconds and x represents the distance in millimeters from the left end of the string. The function f(x,t) gives the displacement of the string at a given point (x,t) from its equilibrium position.

To graph the traces f(x,0) and f(x,pi/2), we need to fix the value of t and plot the function against x.

f(x,0) = cos(0)sin(x) = 0, as the displacement of the string is zero when t = 0.

f(x,pi/2) = cos(pi/2)sin(x) = sin(x), which gives us the displacement of the string at time t = pi/2 milliseconds.

The x-axis represents the distance from the left end of the string in millimeters, and the y-axis represents the displacement of the string in millimeters.

The trace f(x,0) represents the initial position of the string when it is at rest. The trace is a straight line at y=0, indicating that all points on the string are in their equilibrium positions.

The trace f(x,pi/2) represents the displacement of the string at time t = pi/2 milliseconds. It shows the shape of the string when it has completed a quarter of its vibration cycle. The curve starts at 0 when x = 0 and reaches a maximum displacement of 1 at x = pi/2. The curve then goes back to 0 at x = pi, indicating that the string has completed one cycle of vibration.

Now, let's graph the traces f(0,t) and f(pi/2,t):

f(0,t) = cos(t)sin(0) = 0, as the displacement of the string at x=0 is zero.

f(pi/2,t) = cos(t)sin(pi/2) = cos(t), which gives us the displacement of the string at time t for all points x = pi/2.

The x-axis represents time in milliseconds, and the y-axis represents the displacement of the string in millimeters.

The trace f(0,t) represents the displacement of the left end of the string, which is fixed at x=0. As expected, the trace is a straight line at y=0, indicating that the left end of the string remains stationary throughout the vibration cycle.

The trace f(pi/2,t) represents the displacement of the midpoint of the string, which is x=pi/2. The trace is a cosine curve, which indicates that the midpoint of the string oscillates back and forth between positive and negative displacements with a frequency of one cycle per millisecond.

The x-axis represents the distance from the left end of the string in millimeters, and the y-axis represents time in milliseconds. The contours represent the displacement of the string at a given point (x,t) from its equilibrium position.

The contour lines are labeled with the displacement values in millimeters. The contour lines closer to the origin

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In order to double the error margin, the new sample size should be four times smaller than the original sample size. Therefore, the correct answer is: C. One-fourth of the original sample size.

To determine how big of a sample size should be used to double the error margin compared to the original sample size, we need to understand the relationship between error margin, sample size, and original sample size.

Error margin is inversely proportional to the square root of the sample size. This means that when you increase the sample size, the error margin decreases, and vice versa. The formula for this relationship is:

Error Margin = Constant / √(Sample Size)

To double the error margin, we can set up the following equation:
2 * (Constant / √(Original Sample Size)) = Constant / √(New Sample Size)

Now, we can solve for the New Sample Size:
2 * √(Original Sample Size) = √(New Sample Size)

Square both sides of the equation:
4 * Original Sample Size = New Sample Size

Based on this equation, the new sample size should be four times smaller than the original sample size. Therefore, the correct answer is One-fourth of the original sample size.

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The third quartile of the given distribution is -0.5.

To find the third quartile of the given distribution, we need to find the value of x such that the cumulative distribution function (CDF) is equal to 0.75.

The CDF of the distribution is given as:

F(x) = {0, x < 0

1 - 10x²/100, 0 ≤ x < 10

1, x ≥ 10}

We can see that the CDF is defined piecewise, with different expressions for different ranges of x.

To find the third quartile, we need to find the value of x such that F(x) = 0.75.

For 0 ≤ x < 10, we have:

1 - 10x²/100 = 0.75

10x²/100 = 0.25

x² = 0.025

x = ±0.5

Since x<10, the only valid solution is x = -0.5.

Therefore, the third quartile of the given distribution is -0.5.

In summary, the third quartile of the given distribution is -0.5, and we found this by solving the equation F(x) = 0.75, where F(x) is the cumulative distribution function of the distribution.

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The statement, "alternative hypothesis is the hypo-thesis that an "analyst" is trying to prove" is True, because it is the hypothesis that an analyst is trying to prove through their research.

The "Alternative-hypothesis" is defined as the hypothesis which an analyst is trying to prove or support through their research or analysis.

It is the opposite of the "null-hypothesis", and it suggests the presence of a relationship or effect between variables being studied.

In statistical hypothesis testing, the analyst generally formulates both a "null-hypothesis" and an "alternative-hypothesis", and collects data to determine which hypothesis is supported by the evidence.

Therefore, the statement is True.

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The given question is incomplete, the complete question is

The alternative hypothesis is the hypothesis that an analyst is trying to prove. True or False

The values of the first six terms of the sequence are

1) a₀ = -1

2) a₁ = 2

3) a₂ = -4

4) a₃ = 8

5) a₄ = -16

6) a₅ = 32

The given sequence is defined recursively as follows

aₙ = -2aₙ-1

where a₀ = -1.

This means that each term in the sequence is equal to twice the negative of the previous term. To find the first few terms of the sequence, we can start with the given value of a0 and apply the recursive formula repeatedly to generate the next terms.

Using this approach, we get

a₁ = -2a₀ = -2(-1) = 2

a₂ = -2a₁ = -2(2) = -4

a₃ = -2a₂ = -2(-4) = 8

a₄ = -2a₃ = -2(8) = -16

a₅ = -2a₄ = -2(-16) = 32

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Answer: B. 4 inches

Step-by-step explanation:

The width of the stage can be found by multiplying the width of the scale drawing by the scale factor. Since the scale is 1 inch: 7 feet, you can set up a proportion:

1 inch / 7 feet = x inches / 28 feet

To solve for x (the width of the scale drawing), cross-multiply and simplify:

7 feet * x inches = 1 inch * 28 feet

7x = 28

x = 4

Therefore, the width of the stage is 4 inches.

There are 120 people in a theatre. 72 are female and 48 are male. 12 females purchase an ice cream and 31 males purchase an ice cream.

Frequency trees display the real frequency of certain events. They can display the same data as a two-way table, but frequency trees are more readable since they illustrate the frequency hierarchy. Probability trees depict the likelihood of a series of occurrences.

Solution:

From the question, we can see the there are 120 people in the theatre.

Since, there are 72 females in the theatre we can find the total number of males by 120-72 = 48

Also, the male who purchased Ice Cream were 36 therefore, the males who did not purchased Ice Cream are 48-36 = 12

And, the females who purchased Ice Cream are 41. So, the female who did not purchased Ice Cream are 72 - 41 = 31

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complete question"

there are 120 people in a theatre 72 are female of these 41 purchase an ice cream 36 males purchase and ice cream use this information to compete the frequency tree

The sum of the given infinite geometric series is 5/6 and is convergent.

The common ratio between any two consecutive terms in the series is -1/5. Since the absolute value of the common ratio is less than 1, the infinite geometric series is convergent.

for sum calculation use the formula below;

sum = a / (1 - r)

where a is the first term and r is the common ratio.

In this case, a = 1 and r = -1/5. So, the sum is:

sum = 1 / (1 - (-1/5)) = 1 / (6/5) = 5/6

Therefore, the sum of the given infinite geometric series is 5/6.

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To find the derivative of y, we need to use the power rule and the summation rule for derivatives. The power rule states that if y = cx^n, then y' = ncx^(n-1).

Applying this rule to each term in the summation, we get:

y' = (1*1)x^(1-1) + (2*1)x^(2-1) + (3*1)x^(3-1) + ...

Simplifying this expression, we get:

y' = 1 + 2x + 3x^2 + 4x^3 + ...

Therefore, if y = sum from k=0 to infinity of (k 1)x^(k 3), then y' = 1 + 2x + 3x^2 + 4x^3 + ...

If we have the function y given by the sum from k=0 to infinity of (k+1)x^(k+3), we can find the derivative y' as follows:

y' = d/dx (sum from k=0 to infinity of (k+1)x^(k+3))

To find the derivative, we can differentiate term by term within the sum:

y' = sum from k=0 to infinity of d/dx((k+1)x^(k+3))

Using the power rule for differentiation, we get:

y' = sum from k=0 to infinity of (k+1)(k+3)x^(k+2)

So, the derivative y' is the sum from k=0 to infinity of (k+1)(k+3)x^(k+2).

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Hydrogen bonding is a unique type of intermolecular force that occurs when a hydrogen atom is bonded directly to a highly electronegative atom such as fluorine, oxygen, or nitrogen.

These highly electronegative atoms have a strong attraction for electrons, which causes the hydrogen bonding atom to take on a partial positive charge. The resulting electrostatic attraction between the positively charged hydrogen atom and the negatively charged atom creates a hydrogen bond.

This type of bonding is responsible for many of the unique properties of water, including its high boiling and melting points, as well as its ability to dissolve a wide range of substances.

Hydrogen bonding is also important in biological processes, such as protein folding and DNA structure. Without hydrogen bonding, many of the structures and functions that we observe in nature would not be possible.

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since LHS = -16x + 16C - 4, and RHS = [tex]4|x|^4[/tex], so they are not equal. Therefore, 4x is not a solution of the differential equation y' + 4y = 4ex

To show that [tex]y = 4e^{(-4x)}  - 4x + C[/tex] is a solution of the differential equation y' + 4y = 4ex, we need to verify that when y is substituted into the differential equation, both sides are equal.

First, let's find y' by taking the derivative of y:
[tex]y' = -16e^{(-4x)}  - 4[/tex]

Now, we can substitute y and y' into the differential equation and simplify:
[tex]y' + 4y = (-16e^{(-4x)}  - 4) + 4(4 e^{(-4x)}  - 4x + C)[/tex]

[tex]= -16e^{(-4x)}  + 16e^{(-4x)}  - 16x + 16C - 4[/tex]
= -16x + 16C - 4

Next, we need to find the right-hand side of the differential equation by substituting 4ex:
[tex]4ex = 4e^{(4ln|x|)}   = 4e^{(ln|x|^{4}) }    = 4|x|^{4}[/tex]

Finally, we can compare the left-hand side and right-hand side:
LHS = y' + 4y = -16x + 16C - 4
RHS = [tex]4|x|^4[/tex]

We can see that LHS = -16x + 16C - 4, and RHS = [tex]4|x|^4[/tex], so they are not equal. Therefore, 4x is not a solution of the differential equation y' + 4y = 4ex.

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Answer:

  (A)  10°

Step-by-step explanation:

You want the measures of the marked angles in the figure showing vertical angles marked (4x-8)° and (2x+1)°.

The angles marked are vertical angles, which means they are congruent.

  4x -8 = 2x +1

  2x -8 = 1 . . . . . . . . subtract 2x

  2x +1 = 10 . . . . . . . add 9

The angle measures are 10°, choice A.

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The area of the trapezoid that is given in the image below is calculated as: 62 square centimeters.

A trapezoid, like the one given in the image above, is a four-sided flat shape with one pair of parallel sides, and the parallel sides are called bases, while the other two sides are called legs. The area of trapezoid is given as:

A = 1/2 * (sum of the bases) * height

Given the following:

sum of bases = 10.4 + 7.9 + 6.5 = 24.8 cm

Height of trapezoid = 5 cm

Plug in the values:

Area of trapezoid (A) = 1/2 * 24.8 * 5

Area of trapezoid (A) = 62 square centimeters.

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Answer:

[tex]\textsf{C)}\quad \dfrac{5}{16}\; \sf yd^2[/tex]

Step-by-step explanation:

The net of a square-based pyramid is made up of:

From inspection of the given net:

The area of a square is the square of one of its side lengths.

The area of a triangle is half the product of its base and height.

The total surface area of the pyramid is the sum of the area of the square base and the area of 4 congruent triangles.

Therefore:

[tex]\begin{aligned}\sf Total\;surface\;area&=\sf Area_{square}+4 \cdot Area_{triangle}\\\\&=s^2+4 \cdot \dfrac{1}{2}bh\\\\&=\left(\frac{1}{4}\right)^2+4 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{2}\\\\&=\dfrac{1^2}{4^2}+\dfrac{4 \cdot 1 \cdot 1 \cdot 1}{2 \cdot 4 \cdot 2}\\\\&=\dfrac{1}{16}+\dfrac{4}{16}\\\\&=\dfrac{1+4}{16}\\\\&=\dfrac{5}{16}\; \sf yd^2\end{aligned}[/tex]

Therefore, the surface area of the pyramid is 5/16 yd².

Geometry leading the eye from a specific place on a drawing to a block of text.  By incorporating geometry in your design, you can effectively lead the viewer's eye from a specific place on a drawing to a block of text, ensuring they take in the information you wish to convey.

In art and design, geometry can be used to create visual pathways that guide the viewer's eye from one part of a composition to another. To lead the eye from a specific place on a drawing to a block of text, you can use geometric shapes, lines, and angles strategically.
Here's a step-by-step explanation:
Step:1. Identify the starting point on the drawing and the block of text you want to direct the viewer's attention to.
Step2. Use geometric shapes such as rectangles, triangles, or circles to create a visual connection between the two points. Place these shapes along a path that connects the drawing and the text.
Step3. Utilize lines or angles to reinforce this connection. Lines can be straight, curved, or diagonal, while angles can be acute, obtuse, or right. Experiment with different combinations to create a visually appealing pathway.
Step4. Use color, contrast, or size to emphasize the geometric elements and enhance the overall composition.

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To find the correct expression for f(t) given the Laplace transform function f(s) = 320/s^2(s+8), you will need to perform an inverse Laplace transform. The inverse Laplace transform of f(s) is denoted as L^(-1){f(s)} = f(t).

For f(s) = 320/s^2(s+8), you can rewrite it as a sum of partial fractions. After finding the partial fraction decomposition, you can then apply the inverse Laplace transform to each term individually. I highly recommend consulting a table of Laplace transforms for this process.

Once you've applied the inverse Laplace transform to each term, you can sum up the resulting terms to find the final expression for f(t).

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The solution of the equation m=h²kt²x is t = √(m/h²kx) for t>0.

A value or values which, when substituted for a variable in an equation, makes the equation true is known as a solution.

Also, to solve for some variable in an equation, just isolate that variable on one side of the equation.

To solve for t, we need to isolate it on one side of the equation m=h²kt²x.

We can start by dividing both sides by h²kx:

m/h²kx = t²

To solve for t, we need to take the square root of both sides.

However, we also know that t>0, so we need to take the positive square root:

t = √(m/h²kx)

Therefore, the solution for the indicated variable t is t = √(m/h²kx) for t>0.

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a. The sample points are the number of houses sold per day, which are: 0, 1, 2, 3, and 4. So there are a total of 5 sample points.

a. There are five sample points, corresponding to the number of houses sold on each of the 200 days.

b. To assign probabilities to the sample points, we need to count how many times each outcome occurred in the 200 days:

c. The probability of not selling any houses on a given day is the same as the probability of 0 houses sold, which is 0.3.

d. To find the probability of selling at least 2 houses, we need to add up the probabilities of selling 2, 3, or 4 houses:

P(selling at least 2 houses) = P(2 houses) + P(3 houses) + P(4 houses)

= 0.2 + 0.08 + 0.02

= 0.3

e. To find the probability of selling 1 or 2 houses, we need to add up the probabilities of selling 1 or 2 houses:

P(selling 1 or 2 houses) = P(1 house) + P(2 houses)

= 0.4 + 0.2

= 0.6

f. To find the probability of selling less than 3 houses, we need to add up the probabilities of selling 0, 1, or 2 houses:

P(selling less than 3 houses) = P(0 houses) + P(1 house) + P(2 houses)

= 0.3 + 0.4 + 0.2

= 0.9

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If two square matrices of order n, namely a and b, have the same determinant (det(a) = det(b)), then the determinant of their product ab, denoted as det(ab), is equal to the determinant of the square of matrix a, denoted as det(a²).

The determinant of a matrix is a scalar value that can be computed using various methods, such as cofactor expansion or row reduction. The determinant of a product of two matrices is equal to the product of their determinants, i.e., det(ab) = det(a) × det(b).

Given that det(a) = det(b), we can substitute this equality into the determinant of the product of a and b, i.e., det(ab) = det(a) × det(b).

Since we are trying to prove that det(ab) = det(a²), we need to find the determinant of a². The square of a matrix a, denoted as a², is the product of matrix a with itself, i.e., a² = a × a.

Using the determinant property for the product of two matrices, we have det(a²) = det(a) × det(a).

Now, substituting det(a) = det(b) into the equation for det(a²), we get det(a²) = det(a) × det(a) = det(a) × det(b).

Comparing this with the earlier equation for det(ab), we see that det(ab) = det(a²), as both equations are equal.

Therefore, we can conclude that if a and b are square matrices of order n, and det(a) = det(b), then the determinant of their product ab, denoted as det(ab), is equal to the determinant of the square of matrix a, denoted as det(a²).

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A sample size of at least 150 is needed to achieve a 98% confidence interval with a margin of error of ±55 vehicles per hour.

We can use the t-distribution to construct a confidence interval for the population mean improvement in traffic flow. With a sample size of 50, the degrees of freedom are 50 - 1 = 49. Using a 95% confidence level, the critical value of t is 2.009. Therefore, the 95% confidence interval is:

653.5 ± 2.009 * (311.7 / sqrt(50))

= 653.5 ± 89.09

= (564.41, 742.59)

So, the 95% confidence interval for the improvement in traffic flow is (564.41, 742.59) vehicles per hour.

Using a 98% confidence level, the critical value of t for 49 degrees of freedom is 2.681. Therefore, the 98% confidence interval is:

653.5 ± 2.681 * (311.7 / sqrt(50))

= 653.5 ± 119.66

= (533.84, 773.16)

So, the 98% confidence interval for the improvement in traffic flow is (533.84, 773.16) vehicles per hour.

To find the necessary sample size, we can use the formula:

n = (z * σ / E)^2

where z is the critical value of the standard normal distribution, σ is the standard deviation of the sample, and E is the margin of error. For a 95% confidence interval with a margin of error of ±55, the value of z is 1.96. Substituting the given values, we get:

n = (1.96 * 311.7 / 55)^2

= 97.22

So, a sample size of at least 98 is needed to achieve a 95% confidence interval with a margin of error of ±55 vehicles per hour.

Using a 98% confidence level and a margin of error of ±55, the value of z is 2.33. Substituting the given values, we get:

n = (2.33 * 311.7 / 55)^2

= 149.33

So, a sample size of at least 150 is needed to achieve a 98% confidence interval with a margin of error of ±55 vehicles per hour.

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