A researcher predicts that a new pain medication will increase levels of flexibility in patients. Thirty- one chronic pain patients are recruited and each is given the normal dose of the medicine. Twenty-four hours later, each patient's activity level of flexibility is measured. The scores for the sample averaged M = 5.2 with SS -170 after treatment. Assuming that flexibility levels in the chronic pain population averages mu = 4.5 are the data sufficient to conclude that the medication significantly increased flexibility? Use a one-tailed test and a .01 level of significance. If applicable, find Cohen's d. State your hypotheses in symbols, not words, and show your work for the standard error and obtained statistic!

All of the expressions are not a factor of the polynomial function 4x³ − 7x²+x+6

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

The polynomial function 4x³ − 7x²+x+6

To check the expression that is not a factor, we set the expression to 0, solve for x and calculate the value of the polynomial at this x value

If the result is not zero (0), then it is not a factor of the polynomial

Using the above as a guide, we have the following:

x - 1 gives x = 1

So, we have

4(1)³ − 7(1)² + (1) + 6 = 4

x - 2 gives x = 2

So, we have

4(2)³ − 7(2)² + (2) + 6 = 12

x + 1 gives x = -1

So, we have

4(-1)³ − 7(-1)² + (-1) + 6 = -6

x + 2 gives x = -2

So, we have

4(-2)³ − 7(-2)² + (-2) + 6 = -56

None of the expressions give a solution of 0

Hence, all of the expressions are not a factor of the polynomial function

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The  maximum allowable slope for a wheelchair ramp is 1/12 or approximately 0.0833.

a) The maximum allowable slope for a wheelchair ramp can be calculated using the ratio of the rise to the horizontal distance. According to federal building codes, the maximum rise is 1 inch for every 12 inches of horizontal distance. Therefore, the maximum allowable slope is:

Maximum allowable slope = Rise / Horizontal distance
= 1 inch / 12 inches
= 1/12

So, the maximum allowable slope for a wheelchair ramp is 1/12 or approximately 0.0833.

b) Let's assume that the maximum rise of the ramp is h and the corresponding horizontal distance is x. We can use the slope formula to find the slope of the ramp:

Slope = rise / run
= h / x

According to federal building codes, the maximum allowable slope is 1/12. Therefore, we can set up an equation to represent this:

h / x <= 1/12

Multiplying both sides by x, we get:

h <= x/12

So, the height of the ramp above the ground cannot exceed x/12. Therefore, the linear function that models the height of the ramp above the ground as a function of the horizontal distance x is:

h(x) = kx, where k is a constant that represents the slope of the ramp.

However, we know that the maximum allowable slope is 1/12. So, k must be less than or equal to 1/12. Therefore, the linear function that models the height of the ramp above the ground as a function of the horizontal distance x is:

h(x) = (1/12)x.

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The time taken by the train to pass the man is 54 seconds

To find the time taken by the train to pass a man standing near the railway line, we need to convert the train's speed to meters per second and then use the formula time = distance/speed.

1. Convert the speed of the train from km/hr to m/s: 40 km/hr * (1000 m/km) / (3600 s/hr) = 40000/3600 = 40/3.6 = 10/0.9 = 100/9 m/s.

2. Now, use the formula: time = distance/speed. The distance is the length of the train (600 m) and the speed is 100/9 m/s.

time = 600 m / (100/9 m/s) = 600 * 9 / 100 = 54 seconds.

Therefore, the time taken by the train to pass the man is 54 seconds (Option A).

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Answer:

The area of the pool is 10*20 = 200 square meters. Let's assume the width of the path is x. Then the dimensions of the entire region would be (10+2x) by (20+2x). The area of the entire region would be (10+2x)*(20+2x) = 400 + 60x + 4x^2. We know that the area of the pool and the path combined is 1200 square meters. So we can set up the equation as follows:

200 + 1200 = 400 + 60x + 4x^2

Simplifying the equation, we get:

4x^2 + 60x - 1000 = 0

Dividing both sides by 4, we get:

x^2 + 15x - 250 = 0

Factoring the equation, we get:

(x + 25)(x - 10) = 0

x = 10 or x = -25

Since the width of the path can't be negative, the width of the path is 10 meters.

Step-by-step explanation:

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Area measures the size of a closed curve in square units. It is the degree of the measure of a two-dimensional region enclosed \by a closed bend. It is solved in square units.

The equation for the areas of diverse plane figures are:

Therefore, for composite figures, which are made up of two or more basic figures, the zone can be found by including the ranges of the person figures. Some of the time, it may be essential to subtract ranges that are numbered twice.

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The correct answer is d. The significance level tests for the coefficients are not valid.

Multicollinearity is a statistical term that refers to the presence of high correlation among predictor variables in a regression model. This can cause issues in the model, such as unstable or unreliable coefficients, and can lead to incorrect conclusions about the relationships between the predictors and the response variable.

When multicollinearity is present, the R-squared value of the model can become inflated because the model is able to explain more of the variation in the response variable due to the high correlation among the predictor variables. This can give the impression that the model is a good predictor when in fact it may not be. Additionally, multicollinearity can cause the F-test significance level to be low, indicating that the model is a good fit, even though the individual coefficients may not be statistically significant.

Multicollinearity can cause inflated R-squared values and low F-test significance levels. However, it does not necessarily mean that the model is a poor predictor. The interpretation of coefficients may also be affected by multicollinearity.

However, the most significant issue with multicollinearity is that it can lead to unreliable significance tests for individual coefficients, making it difficult to determine which predictors are contributing significantly to the model.

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The point V(3, 5) is reflected over the x axis, the new point is V'(3, -5)

Reflection is a type of transformation. Transformation is the movement of a point either up, left, right or down in the coordinate plane.

Reflection is a rigid transformation because it conserves the size and shape of the figure.

If a point A(x, y) is reflected over the x axis, the new point is A'(x, -y)

The coordinate of point V is (3, 5). If the point is reflected over the x axis, the new point is V'(3, -5)

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The total area is given as 1600 sq. ft

The formula used to calculate the area of a square is A = s^2, where "A" depicts the area and "s" represents the length of one side of the square.

This essentially means that the area obtained through this formula is equivalent to the outcome produced by squaring the measure for one of its sides.

To clarify, if an object in the shape of a square has a sidereal extension of 5 spatial units, then it's overall measurement will equal 25 quadrangular units (5 x 5 = 25).

It is particularly pertinent to note that expressions for physical space, such as those associated with centimeters, meters or even feet are invariably employed when referring to areas corresponding to squares.

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No, it is not possible for a binary relation to be both symmetric and antisymmetric because a relation cannot be both symmetric and antisymmetric unless it is the empty relation, as the properties of symmetry and antisymmetry contradict each other for non-empty sets.

A relation is symmetric if for any elements x and y in the set, if (x,y) is in the relation, then (y,x) is also in the relation. This means that the relation is symmetric around the diagonal.

On the other hand, a relation is antisymmetric if for any elements x and y in the set, if (x,y) is in the relation and (y,x) is in the relation, then x=y. This means that the relation is asymmetric around the diagonal.

If a relation is both symmetric and antisymmetric, then it means that for any elements x and y in the set, if (x,y) is in the relation, then (y,x) is also in the relation, and x=y. This implies that the relation is only defined on pairs of identical elements, and therefore it is the diagonal relation, which is reflexive, symmetric, and antisymmetric.

In summary, the only binary relation that is both symmetric and antisymmetric is the diagonal relation, which is defined as {(x,x) : x is an element of the set}.

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The distance of a circular path can be found using the formula for the circumference of a circle, which is C = πd.

The distance of a circular path can be calculated using the formula for the circumference of a circle. The circumference of a circle is the distance around the circle and can be found by multiplying the diameter of the circle by pi (π), which is a mathematical constant equal to approximately 3.14159.

The formula for the circumference of a circle is:

C = πd

where C represents the circumference of the circle, and d represents the diameter of the circle.

To calculate the distance of a circular path, we first need to know the circumference of the circle. If we know the radius of the circle, we can find the diameter by multiplying the radius by 2. Once we have the diameter, we can use the formula above to find the circumference.

Alternatively, if we know the length of the circular path or the angle through which we have traversed, we can use trigonometric functions to calculate the radius and then use the formula above to find the circumference.

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Complete question:

"What formula or equation can be used to calculate the distance of a circular path?"

Answer:

69

Step-by-step explanation:

180-45=135

69 is the nearest angle degree.

The slope of the line given above which passes through points (-2, -1) and (0, -2) is calculated as: m = -1/2.

To find the slope of the line given above, apply the slope formula, then use the coordinates of any two points on the line to calculate the slope.

Slope of a line (m) = change in y / change in x = y2 - y1 / x2 - x1

We have:

(-2, -1) = (x1, y1)

(0, -2) = (x2, y2)

Plug in the values:

Slope (m) = (-2 -(-1)) / (0 - (-2))

m = -1 / 2

Slope of the line (m) = -1/2

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a) Using the Law of Cosines for this triangle, we can write the equation:
d² = (36)² + (3s)² - 2(36)(3s)cos(60°)

b) The second ship must travel at approximately 4.24 miles per hour to be 42 miles apart from the first ship at noon.

(a) To write an equation that relates the speed s and the distance d between the two ships at noon using the Law of Cosines, we first need to determine the distance each ship has traveled by noon. Since they leave at 9 a.m. and we're interested in the distance at noon, they travel for 3 hours.

Ship 1:
Speed: 12 miles per hour
Distance traveled: 12 miles/hour * 3 hours = 36 miles

Ship 2:
Speed: s miles per hour
Distance traveled: s miles/hour * 3 hours = 3s miles

Now, we can form a triangle where Ship 1 travels 36 miles, Ship 2 travels 3s miles, and the distance between them (d) is the third side. The angle between Ship 1 and Ship 2 is 180° - (53° + 67°) = 60°.

Using the Law of Cosines for this triangle, we can write the equation:
d² = (36)² + (3s)² - 2(36)(3s)cos(60°)

(b) To find the speed s that the second ship must travel so that the ships are 42 miles apart at noon, we can plug d = 42 into our equation from part (a) and solve for s.

42² = (36)² + (3s)² - 2(36)(3s)cos(60°)

Solving for s, we get:
s ≈ 4.24 miles per hour (rounded to two decimal places)

So, the second ship must travel at approximately 4.24 miles per hour to be 42 miles apart from the first ship at noon.

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You need to deposit at least $74 into your bank account to avoid an overdraft fee.

We have,

Start with the initial balance = $120

Subtract the first expense, the electricity bill = $120 - $67 = $53

Subtract the second expense, the night out with friends = $53 - $44 = $9

Subtract the third expense, the gas purchase = $9 - $35 = -$26

Now,

Since the remaining balance is negative, you would receive an overdraft fee if you left it at this amount.

To avoid the overdraft fee, you need to deposit enough money to bring your account balance back to $0 or higher.

To do this, you need to add the absolute value of the negative balance to your desired minimum balance, which in this case is:

$0 = |-26| + $0 = $26

Therefore,

You need to deposit at least $74 into your bank account to avoid an overdraft fee.

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The solution to the system is [tex]$\$\left(x_{-} 1, x_{-} 2\right)=(4,-3) \$$[/tex].

To solve the system, we can use the method of elimination or Gaussian elimination.

We start by writing the system in augmented matrix form:

[tex]$$\left[\begin{array}{cc|c}2 & 4 & -4 \\5 & 7 & 11\end{array}\right]$$[/tex]

We can eliminate the [tex]$\$ x_{-} 1 \$$[/tex] variable from the second equation by subtracting 5 times the first equation from the second:

[tex]$$\left[\begin{array}{cc|c}2 & 4 & -4 \\5-5(2) & 7-5(4) & 11-5(-4)\end{array}\right] \Rightarrow\left[\begin{array}{cc|c}2 & 4 & -4 \\-3 & -13 & 31\end{array}\right]$$[/tex]

Next, we can eliminate the [tex]$\$ x_{-} 2 \$[/tex]$ variable from the first equation by subtracting twice the second equation from the first:

[tex]$$\left[\begin{array}{cc|c}2-2(-13) & 4-2(7) & -4-2(31) \\-3 & -13 & 31\end{array}\right] \Rightarrow\left[\begin{array}{cc|c}28 & -10 & -66 \\-3 & -13 & 31\end{array}\right]$$[/tex]

We can simplify this further by dividing the first row by 2 :

[tex]$$\left[\begin{array}{cc|c}14 & -5 & -33 \\-3 & -13 & 31\end{array}\right]$$[/tex]

Now we can solve for [tex]$\$ x_{-} 2 \$$[/tex] in terms of [tex]$\$ x_{-} 1 \$$[/tex] by multiplying the first equation by 13 and adding it to the second equation:

[tex]$$13(14) x_1-13(5) x_2-13(33)-3(-13) x_1-3(-13) x_2=13(31)-3(14) x_1$$[/tex]

Simplifying:

[tex]$$\begin{aligned}& 169 x_1-91 x_2-429+39 x_1+39 x_2=403 \\& 208 x_1=832 \\& x_1=4\end{aligned}$$[/tex]

Substituting back into the first equation, we get:

[tex]$$2(4)+4 x_2=-4 \Rightarrow x_2=-3$$[/tex]

Therefore, the solution to the system is [tex]$\$\left(x_{-} 1, x_{-} 2\right)=(4,-3) \$$[/tex].

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The total radius of a cone with a radius of 4in and a height of 12in is 8in.

The total radius of a cone is the sum of the radius of the base and the slant height of the cone. The slant height of a cone can be found using the Pythagorean theorem, which states that the square of the slant height is equal to the sum of the square of the height and the square of the radius of the base.

So, to find the total radius of the cone, we need to calculate the slant height and add it to the radius of the base.

Slant height = sqrt(radius² + height²)

= √(4² + 12²)

= √(160)

= 12.65in (rounded to two decimal places)

Total radius = radius + slant height

= 4in + 12.65in

= 16.65in

≈ 8in (rounded to two decimal places)

Therefore, the total radius of the cone is approximately 8 inches.

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The time for which plane B will take to refuel plane A is equals to 10 minutes. The two points who satisfy the equation, y = x + 1, are (0, 1), (-1,0). The two points who satisfy the equation, y = -x + 21, are (0,21), (21,0).

We have a fuel left in Plane A = 1 ton

fuel left in Plane B = 21 tons

Fuel transfer rate = 1 ton per minute

In order that for them to have the same amount of fuel, We add up the fuel left in Plane A and Plane B = 21 + 1 = 22 tons. This implies each plane will have fuel of 11 tons. Time that plane B will take to refuel plane A until both planes have the same amount of fuel is calculated by : Plane B will transfer 10 tons of fuel to A.

Plan A has a total of 11 tons. Since, the transfer rate = 1 ton per minute

=> 1 ton will transfer in 1 minute

So, 10 tons fuel will need 10 minutes. Hence, required time value is 10 minutes. Now, The equation for quantity of fuel with respect to time in plane A is, y = x + 1 --(1). If x = 0 => y = 1

and y = 0 => x = -1. So, (0, 1) and (-1,0).

The equation for quantity of fuel with respect to time in plane B is, y = -x + 21 --(2). For it, x = 0 => y = 21 and y= 0 => x = 21. Hence, two points that satisfy the equation(1) and equation(2) are (0, 1), (-1,0) and (0,21), (21,0) respectively.

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The area of the rectangle is 62.4 square centimeters.

The area of a rectangle is a measure of the amount of space enclosed by the rectangle in two-dimensional (2D) space. It is the product of the length and width of the rectangle, and is usually expressed in square units.

The area of a rectangle will be given by the formula;

Area = Length × Width

where "Length" represents the length of one side of the rectangle, and "Width" represents the length of the other side of the rectangle.

Given that the length of the rectangle is 9.6 cm and the width is 6.5 cm, we can substitute these values into the formula;

Area = 9.6 cm × 6.5 cm

Calculating the area using these values;

Area = 62.4 cm²

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The area of the shaded region will be 8 square unit as per the given figure.

The rectangle has dimensions 2 x 4, so its area is:

Area of rectangle = length x width = 2 x 4 = 8 square units

The triangle has dimensions 4 x 8, so its area is:

Area of triangle = (1/2) x base x height = (1/2) x 4 x 8 = 16 square units

To find the area of the shaded region, we need to subtract the area of the triangle from the area of the rectangle.

Area of shaded region = Area of the triangle - Area of rectangle

Area of shaded region = 16-8

Area of shaded region = 8

The area of the shaded region will be 8 square units.

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Complete question:

Answer:

D) The MEAN is the average of the numbers in the data set.

The correct statement is:

D) The MEAN is the average of the numbers in the data set.

In mathematics and statistics, the term "mean" refers to the average of a set of variables. There are various ways to determine the mean, including geometric means, harmonic means, and simple arithmetic means (putting the numbers together and dividing the result by the quantity of observations).

The statement about the mean provides a definition of a measure of central tendency and is a useful starting point in deciding which measure to use.

The other statements are not relevant to deciding which measure of central tendency to use.

A) The mode can be used for numerical data as well as non-numerical data.

B) The range is not a measure of central tendency, it is a measure of dispersion or variability.

C) The median is often used to avoid the effect of extreme values, but it is not specific to data sets with extreme values.

D) The Mean is the average of the numbers in the data set.

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The proportion of shoppers who can identify the highly advertised trademark is significantly higher than 40% at the 0.01 level of significance.

Let's break it down step-by-step using the provided information:

1. Hypotheses: - Null hypothesis (H0): p = 0.40 (40% of all shoppers can identify the trademark) - Alternative hypothesis (Ha): p > 0.40 (more than 40% of all shoppers can identify the trademark)

2. Level of significance: - α = 0.01

3. Sample information: - n (sample size) = 18 - x (number of successful identifications) = 13 - p-hat (sample proportion) = x / n = 13 / 18 = 0.7222 4. Test statistic

calculation: - We'll use a one-sample z-test for proportions. - z = (p-hat - p) / sqrt((p * (1 - p)) / n) - z = (0.7222 - 0.40) / sqrt((0.40 * (1 - 0.40)) / 18) - z ≈ 2.88 5. Decision: - Since α = 0.01, we'll compare our test statistic to the critical value from the z-table, which is 2.33 for a one-tailed test. - Our test statistic, z ≈ 2.88, is greater than the critical value of 2.33.

Conclusion: Since our test statistic is greater than the critical value at the 0.01 level of significance, we have enough evidence to reject the null hypothesis (H0). This means that there is sufficient evidence to support the claim that more than 40% of all shoppers can identify a highly advertised trademark.

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If you pick 3 fruits at random, the mean weight will be greater than 757 grams approximately 1.26% of the time.


To solve this problem, we need to use the properties of the normal distribution. We know that the weights of the fruit are normally distributed with a mean of 753 grams and a standard deviation of 9 grams.
The mean of the sample of 3 fruits will also be normally distributed with a mean of 753 grams and a standard deviation of 3 grams (since we are dividing by the square root of the sample size).

To find the probability that the mean weight of the sample of 3 fruits will be greater than a certain amount, we need to convert this amount to a z-score using the formula:
z = (x - μ) / (σ / √n)

where x is the amount we are interested in, μ is the mean, σ is the standard deviation, and n is the sample size (in this case, 3).

In this case, we want to find the z-score for a mean weight of 757 grams:

[tex]z = \frac{(757 - 753)}{\frac{9}{\sqrt{3}} }} =1.26[/tex]

We can use a standard normal distribution table or calculator to find that the probability of getting a z-score greater than 1.26 is approximately 0.0985, or 9.85%. However, since we are interested in the probability that the mean weight will be greater than 757 grams (not just greater than the mean), we need to add half of the probability of getting exactly 757 grams (which is the mode of the distribution) to this value.
Since the normal distribution is symmetrical, the probability of getting exactly 757 grams is the same as the probability of getting exactly 749 grams (which is the mean minus one standard deviation). Using the same formula as before, we can find the z-score for a weight of 749 grams:

z =  \frac{(749 - 753)}{\frac{9}{\sqrt{3}} }} =-1.26[/tex]

The probability of getting a z-score less than -1.26 is also approximately 0.0985, so the probability of getting exactly 757 grams is approximately 0.197. Half of this value is 0.0985, which we add to the probability of getting a z-score greater than 1.26 to get the final answer:
0.0985 + 0.0985 = 0.197
So the probability of getting a mean weight greater than 757 grams is approximately 0.0985 + 0.197 = 0.2965, or 29.65%.

To convert this probability to weight, we can use a standard normal distribution table or calculator to find the z-score corresponding to a probability of 29.65%. This is approximately 0.56. Using the same formula as before, we can solve for x:

\frac{(x - 753)}{\frac{9}{\sqrt{3}} }} =0.56[/tex]

x ≈ 757.23

So if you pick 3 fruits at random, their mean weight will be greater than 757 grams approximately 1.26% of the time.

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Answer:

I think its 42

Step-by-step explanation:

Hope u get the right answer!

The reflection from green triangle to blue triangle is a reflection over the x-axis

Reflecting a two-dimensional shape over the x-axis is a geometric transformation that represents an image flipping or mirroring itself across the fixed x-axis.

This axis appears as the horizontal marker in a Cartesian coordinate system, providing the reference line to then split the plane into its top and bottom elements.

When which this action is fullfilled, all the y-coordinates of each point within the figure will be reversed while the x-coordinate remains unchanged;

The image of the reflection is attached and the reflection line labeled

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The appropriate area of the composite figure with given dimensions is given by option b.  71.77cm² (nearest hundredth).

Composite figure consist of a triangle and rectangle from which semicircle is cut.

Diameter of the semicircle = 7cm

Radius of the semicircle 'r' = 3.5 cm

Area of the semicircle = ( 1/2) πr²

                                     = ( 1/2) × 3.14 × (3.5)²

                                     = 19.2325cm²

length of the rectangle = 10cm

Width of the rectangle = 7cm

Area of the rectangle = length × width

                                    = 10 × 7

                                     = 70cm²

base of the triangle = 7cm

height of the triangle = 6cm

Area of the triangle = ( 1/2) × base × height

                                = ( 1/2) × 7 × 6

                                = 21 cm²

Appropriate area  of the composite figure

= Area of the triangle + area of the rectangle - area of semicircle

= 21 + 70 - 19.2325cm²

= 71.7675cm²

= 71.77cm² ( nearest hundredth )

Therefore, the area of the composite figure is equal to option b.  71.77cm².

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Using Table B at line 133, the consumers assigned to treatment 1 are those numbered 4, 5, 7, 1, and 8.

The table displaying the treatments in a design with two factors would be: | | Ambient scent | |----------|--------------| | Product density | Seashore (A) | Enclosed space (B) | No scent | | Jam-packed | 2 | 1 | 3 | | Minimally stocked | 4 | $ | 6 |

To randomly assign the 30 subjects to the six treatments, we would first label the subjects from 01 through 30. Then, we would use Table B starting at line 133 to randomly select the appropriate number of subjects for each treatment. For example, to randomly assign 5 consumers to treatment 1, we would use Table B to select 5 numbers from 01 through 30, and label those subjects as treatment 1. We would then repeat this process for each of the six treatments. An example of this would be: Assign = 5 consumers to each of the six treatments. Label the subjects from 01 through 30. Randomly select 5 numbers for treatment 1, then 5 of the remaining consumers for treatment 2, and so on. Using Table B at line 133, the consumers assigned to treatment 1 are those numbered 4, 5, 7, 1, and 8.

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We can proceed with finding the specific solution using either method mentioned above.

The order of the differential equation is 2.

Since the auxiliary equation has complex roots (±3i), we know that the general solution to the differential equation will involve sine and cosine functions.

To find the specific solution, we can use the method of undetermined coefficients or variation of parameters. However, we first need to check for any singular points or irregular singular points in the equation.

Since the coefficient of y is a polynomial in x and the coefficient of y" is also a polynomial in x, there are no singular points or irregular singular points in the equation.

Therefore, we can proceed with finding the specific solution using either method mentioned above.

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Answer:

To find the slope of the tangent line to the path of the particle at the point where t = 2, we first need to find the values of x(2) and y(2), as well as their derivatives x'(2) and y'(2).

Using the given parametric functions, we can find:

x(2) = ∫ x'(t) dt = ∫ t sin(t) dt = -t cos(t) + sin(t) + C

where C is the constant of integration.

Since we want x(2), we can evaluate the above expression at t = 2:

x(2) = -2 cos(2) + sin(2) + C

Similarly, we can find:

y(2) = ∫ y'(t) dt = ∫ (5e^(-3t) + 2) dt = (-5/3)e^(-3t) + 2t + C'

where C' is the constant of integration.

Again, since we want y(2), we can evaluate the above expression at t = 2:

y(2) = (-5/3)e^(-6) + 4 + C'

Now we can find the derivatives x'(2) and y'(2) by taking the derivative of x(t) and y(t), respectively, and evaluating them at t = 2:

x'(2) = 2 sin(2) - cos(2)

y'(2) = (5/3)e^(-6)

Therefore, at t = 2, the particle is at the point (x(2), y(2)) = (-2 cos(2) + sin(2) + C, (-5/3)e^(-6) + 4 + C'), and the slope of the tangent line to the path of the particle at this point is given by:

dy/dx = (dy/dt)/(dx/dt) = y'(2)/x'(2)

Substituting the values we found:

dy/dx = [(5/3)e^(-6) + 4 + C']/(2 sin(2) - cos(2))

Since we don't have enough information to find the value of C', we cannot find an exact value for the slope. However, we can simplify the expression by using the trigonometric identities:

sin(2) = 2 sin(1) cos(1)

cos(2) = cos^2(1) - sin^2(1)

where we let t = 1 for simplicity. Then, we can substitute these expressions and simplify:

dy/dx = [(5/3)e^(-6) + 4 + C']/(4 sin(1) cos(1) - cos^2(1) + sin^2(1))

dy/dx = [(5/3)e^(-6) + 4 + C')/(4 sin(1) cos(1) - 1)

Therefore, the slope of the tangent line to the path of the particle at the point where t = 2 is given by the above expression.

Step-by-step explanation:

The slope of the tangent line to the path of the particle at the point where t=2 is approximately 1.55. To find the slope of the tangent line to the path of the particle at the point where t=2,

we need to use the derivatives of x(t) and y(t).

First, we can find the slope of the tangent line by using the formula:

slope = dy/dx = (dy/dt)/(dx/dt)

So, we need to find both dy/dt and dx/dt.

Given that x′(t)=t sin t, we can find dx/dt by taking the derivative of x(t):

dx/dt = x′(t) = t sin t

Given that y′(t)=5e−3t+2, we can find dy/dt by taking the derivative of y(t):

dy/dt = y′(t) = 5e−3t+2

Now, we can find the slope of the tangent line at t=2 by plugging in these values:

slope = (dy/dt)/(dx/dt) = (5e−3t+2)/(t sin t) = (5e−6+2)/(2 sin 2)

Therefore, the slope of the tangent line to the path of the particle at the point where t=2 is approximately 1.55.

Learn more about tangent line here:

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