yall please help me :((

To evaluate 2∫1[tex]x^{2}[/tex]f([tex]x^{3}[/tex]) dx, we can use a substitution where u = [tex]x^{3}[/tex]. Then, du/dx = 3[tex]x^{2}[/tex] and dx = du/(3[tex]x^{2}[/tex]). Substituting these into the integral, we get:

2∫1 [tex]x^{2}[/tex] f([tex]x^{3}[/tex]) dx = 2∫1 ([tex]u^{2/3}[/tex])/3 f(u) du

Next, we can use the given information that 8∫1 f(t)dt = 1. Solving for ∫1 f(t)dt, we get:

∫1 f(t)dt = 1/8

Substituting this into our integral, we get:

2∫1 [tex]x^{2}[/tex] f([tex]x^{3}[/tex]) dx = 2∫1 ([tex]u^{2/3}[/tex])/3 f(u) du
= 2∫1 ([tex]u^{2/3}[/tex])/3 (1/8) du
= ∫1 ([tex]u^{2/3}[/tex])/12 du
= (3/5) [tex]u^{5/3}[/tex] evaluated from 1 to 2
= (3/5) ([tex]2^{5/3}[/tex] - 1)

Therefore, the value of 2∫1 [tex]x^{2}[/tex] f[tex]x^{3}[/tex]) dx is (3/5) ([tex]2^{5/3}[/tex] - 1).

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The evaluated integral 2∫(from 1 to x^2) x^2 f(x^3) dx is equal to 1/12.

First, we can use the given information to find the value of the constant in front of the integral:

8∫1 f(t)dt = 1

Dividing both sides by 8:

∫1 f(t)dt = 1/8

Now we can use this to evaluate the second integral:

2∫1 x^2 f(x^3) dx

Let u = x^3, then du/dx = 3x^2 and dx = du/3x^2

Substituting:

2∫1 x^2 f(x^3) dx = 2∫1 (u^(2/3))(1/3u^(1/3))f(u) du

Simplifying:

2/3 ∫1 u^(5/3) f(u) du

Now we can use the fact that f(t) satisfies the given equation to solve:

∫1 f(t)dt = 1/8

Letting t = u^(1/3):

∫1 u^(1/3) f(u) du = 1/8

Multiplying both sides by u^(2/3):

∫1 u^(5/3) f(u) du = 1/8

So we can substitute this in:

2/3 ∫1 u^(5/3) f(u) du = 2/3 (1/8) = 1/12

So, the evaluated integral 2∫(from 1 to x^2) x^2 f(x^3) dx is equal to 1/12.

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The work done in pumping the entire contents of the fuel tank into the tractor is 7,021,796 joules.

What is volume of cylinder?

The volume of a cylinder V = πr²h where r is the radius of the tank and h is the height of the tank.

Here given that r = 1.5 meters and h = 3 meters, so:

V = π(1.5)²(3) = 21.2 cubic meters

Next, we can calculate the weight of the gasoline using its density and volume,

W = ρVg

where ρ is the density of gasoline (6600 N/m³), g is the acceleration due to gravity (9.81 m/s²), and W is the weight of the gasoline.

So,

W = (6600)(21.2)(9.81) = 1,404,359.2 newtons

Now we can calculate the work done in lifting this weight from the level of the truck bed to the level of the tractor tank opening. This is given by

Work = Force x Distance

where Force is the weight of the gasoline, and Distance is the vertical distance it is lifted.

The distance is given as 5 meters in the problem,

Work = 1,404,359.2 x 5 = 7,021,796 joules

Therefore, the work done in pumping the entire contents of the fuel tank into the tractor is 7,021,796 joules.

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Answer:

The answer in the simplest form is 9³ or 729

Step-by-step explanation:

√81⁶

=9³=729

Answer:

Rewrite 81 as 92. is√92y^6 Pull terms out from under the radical, assuming positive real numbers. is⋅9y^6 Move 9 to the left of its. 9isy^6

The interval does not contain zero, which means that there is a statistically significant difference between the two proportions. With 99% probability, we can say that the preference change is not due to chance and is likely due to an actual change in the population.

(a) The sample proportion for the first poll is 0.70 (98/140) and the sample proportion for the second poll is 0.80 (80/100).
(b) The difference between the two sample proportions is 0.10 (0.80 - 0.70).
(c) The standard error of the difference in proportions is 0.0791 (sqrt((0.70*(1-0.70)/140) + (0.80*(1-0.80)/100))).
(d) The critical z value for a confidence level of 99% is 2.576.
(e) The null hypothesis is that there is no significant difference between the proportion of registered voters who supported Eric Adams in the first poll and the proportion of registered voters who supported him in the second poll. Mathematically, this can be represented as H0: p1 = p2.
(f) The 99% confidence interval for the difference in population proportions is (0.0079, 0.1921).
(g) The confidence interval does imply that the change in registered voters' preferences is significant.

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The cylindrical coordinates are (r, θ, z) = (6√2, 20°, -9).

The given rectangular coordinates are (6, 2√3, -9)

To convert to cylindrical coordinates (r, θ, z), we must find r, θ, and z.

r = √(6^2 + (2√3)^2)

r = √(36 + 12)

r = √48

r = 6√2

Now,

tanθ = 2√3/6

θ = tan^-1(2√3/6)

θ = tan^-1(1/3)

θ = 20°

z: z = -9

Therefore, the cylindrical coordinates are (r, θ, z) = (6√2, 20°, -9).

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Answer:

Bus 18 is more consistent than Bus 47 based on the IQR, which is smaller for Bus 18 (16) compared to Bus 47 (24), indicating less spread of data between the 25th and 75th percentiles.

Step-by-step explanation:

Answer: The correct option regarding which bus has the least spread among the travel times is given as follows:

Bus 14, with an IQR of 6.

How to obtain the measures of spread?

First, we consider the dot plot, which shows the number of times that each observation appears in the data set.

Then we consider the interquartile range, which gives the difference between the third quartile and the first quartile of the data set.

The interquartile range is a better measure of spread compared to the range of a data set, as it does not consider outliers.

For groups of 15 students, we have that:

The first half is composed of the first seven students, hence the first quartile is the fourth dot, which is the median of the first half.

The second half is composed of the last seven students, hence the first quartile is the eleventh dot, which is the median of the first half.

The quartiles for Bus 14 are given as follows:

Q1 = 12.

Q3 = 18.

Hence the IQR is of:

IQR = Q3 - Q1 = 18 - 12 = 6.

The quartiles for Bus 18 are given as follows:

Q1 = 9.

Q3 = 16.

Hence the IQR is of:

IQR = Q3 - Q1 = 16 - 9 = 7.

Step-by-step explanation:

The total number of sand bags added to the beach over the interval of 3 hrs from 7 AM to 10 AM is 255.368 tons, under the given condition that  sand being added at a rate of s(t) = 65+24 sin (0.3) tons/hr. Then the required correct option is Option A.

Let us look at  s(t) = 65+24 sin (0.3) tons per hour, here t = hours since 5:00 A.M.

Then the amount of sand added to the beach over the 3-hour period from 7:00 A.M. to 10:00 AM can be evaluated by performing definite integral s(t) from t=7 to t=10.

Then,

∫(7 to 10) [65+24 sin (0.3)] dt = [65t - (80/3) cos(0.3t)] from t=7 to t=10

=[tex][65(10) - (80/3) cos(0.3*10)] - [65(7) - (80/3) cos(0.3*7)][/tex]

= 650 - (80/3)[cos(3) - cos(2.1)]

= 255.368 tons

The total number of sand bags added to the beach over the interval of 3 hrs from 7 AM to 10 AM is 255.368 tons, under the given condition that  sand being added at a rate of s(t) = 65+24 sin (0.3) tons/hr.

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Therefore, the dimensions of the box that minimize the amount of material used are a base length of 90 cm and a height of 450 cm.

To minimize the amount of material used, we need to minimize the surface area of the box. Since the base is a square, we can let the length of one side be x. The height of the box can then be expressed as (364500/x^2).

The surface area of the box can be found by adding the area of the base (x^2) to the area of the four sides (4xh).

Surface Area[tex]= x^2 + 4x(364500/x^2)[/tex]
Surface Area =[tex]x^2 + 1458000/x[/tex]

To minimize the surface area, we can take the derivative of the surface area function and set it equal to zero:

[tex]\frac{d}{dx} (Surface\ Area) = 2x - 1458000/x^2 = 0\\2x = 1458000/x^2\\x^3 = 729000\\x = 90 cm[/tex]

Therefore, the base length of the box is 90 cm. The height can be found using the equation we derived earlier:

Height =[tex]364500/90^2[/tex]
Height = 450 cm

Therefore, the dimensions of the box that minimize the amount of material used are a base length of 90 cm and a height of 450 cm.

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The answer is that [tex]$\bar{X}-\bar{Y}$[/tex] is normal with mean 0 and standard deviation [tex]$5 / 9$[/tex]. None of the given options match this result exactly, but the closest one is "normal with mean 0 and standard deviation [tex]$5 / 6^{\prime \prime}$[/tex].

The mean of [tex]$\bar{X}$[/tex] and [tex]$\bar{Y}$[/tex] are:

[tex]E(\bar{X})=E\left(\frac{1}{16} \sum_{i=1}^{16} X_i\right)=\frac{1}{16} \sum_{i=1}^{16} E\left(X_i\right)=\frac{1}{16}(16 \times 15)=15[/tex]

and

[tex]$$E(\bar{Y})=E\left(\frac{1}{9} \sum_{i=1}^9 Y_i\right)=\frac{1}{9} \sum_{i=1}^9 E\left(Y_i\right)=\frac{1}{9}(9 \times 15)=15$$[/tex]

The variance of [tex]$\bar{X}$[/tex] and [tex]$\bar{Y}$[/tex] are:

[tex]$$\{Var}(\bar{X})=\{Var}\left(\frac{1}{16} \sum_{i=1}^{16} X_i\right)=\frac{1}{16^2} \sum_{i=1}^{16} \{Var}\left(X_i\right)=\frac{1}{16^2}(16 \times 4)=\frac{1}{4}$$[/tex]

and

[tex]$$\{Var}(\bar{Y})=\{Var}\left(\frac{1}{9} \sum_{i=1}^9 Y_i\right)=\frac{1}{9^2} \sum_{i=1}^9 \{Var}\left(Y_i\right)=\frac{1}{9^2}(9 \times 4)=\frac{4}{81}$$[/tex]

Now, we have:

[tex]E(\bar{X}-\bar{Y})=E(\bar{X})-E(\bar{Y})=0[/tex]

and

[tex]\{Var}(\bar{X}-\bar{Y})=\{Var}(\bar{X})+\{Var}(\bar{Y})=\frac{1}{4}+\frac{4}{81}=\frac{25}{81}[/tex]

Therefore, [tex]$\bar{X}-\bar{Y}$[/tex] follows a normal distribution with a mean 0 and a standard deviation:

[tex]$$\sqrt{{Var}(\bar{X}-\bar{Y})}=\sqrt{\frac{25}{81}}=\frac{5}{9}$$[/tex]

So, the answer is that [tex]$\bar{X}-\bar{Y}$[/tex] is normal with mean 0 and standard deviation [tex]$5 / 9$[/tex]. None of the given options match this result exactly, but the closest one is "normal with a mean 0 and standard deviation [tex]$5 / 6^{\prime \prime}$[/tex].

Definition: To distribute a product is to make it available to a wide audience so that they can purchase it. These actions are involved in distribution: 1. A reliable transportation system to deliver the commodities to various locations.

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The Cartesian equation of the curve with parametric equations x = t² + t + 1 and y = t² - t + 1 is y = x - 2t + 1.

To find the Cartesian equation, follow these steps:
1. Solve one of the parametric equations for t.
2. Substitute the expression for t found in step 1 into the other parametric equation.
3. Simplify the equation to obtain the Cartesian equation.

Step 1: From the x equation (x = t² + t + 1), solve for t:
t² + t = x - 1
t(t + 1) = x - 1

Step 2: Since it's challenging to solve for t directly, use the y equation to eliminate t:
y = t² - t + 1

Step 3: Notice that t² is present in both the x and y equations, so substitute x - 1 for t(t + 1) in the y equation:
y = (x - 1) - (t + 1) + 1
y = x - 2t + 1

Thus, the Cartesian equation of the curve is y = x - 2t + 1.

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The number that is rational in the given options is C. 0.333...

A rational number is a given number which can be expressed as a fraction, or which has a series of recurring digits on expressing it in decimal. Such that it can be rounded off to a required number of decimal places or significant figure of the recurring digits.

In the given question, comparing the values of the given options, it can be observed that only 0.333... is the rational number. Therefore, the required number that is rational is option C. 0.333...

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Answer:

C. 0.333 is the answer.

Step-by-step explanation:

The pitch of the roof or the angle of elevation is 59°.

First, let's find the length of the center beam AC. We can use the Pythagorean theorem:

AC² = AD² + CD²

AC² = 20² + 40²

AC² = 1600

AC = 40

Next, let's find the coordinates of point E, the midpoint of AC.

Since A and C have coordinates (0,0,0) and (20,40,20), respectively, the coordinates of E are:

E = [(0+20)/2, (0+40)/2, (0+20)/2] =

E = (10,20,10)

Now, let's find the distance from E to the top of the garage. We can use the Pythagorean theorem again:

BE² = BD² + DE²

BE² = 20² + 30²

BE² = 1300

BE = √1300 = 10√13

Finally, let's find the angle of elevation of the new beam. We can use trigonometry, specifically the tangent function:

Recall that,

tanθ = opposite/adjacent

tanθ = BE/CE

where CE is the distance from E to the ground.

Since CE is just the height of the garage, which is 20 feet, we have:

tanθ = BE/20

Solving for angle:

θ = tan⁻¹(BE/20)

        = tan⁻¹(10√13/20)

        = tan⁻¹(√13/2)

θ = 59°

Therefore, the pitch of the roof is approximately 59°.

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The derivative of the given function is [√(r²+4)]/2 + (r²/2)[ln(√(r²+4)+r)-ln(2)] under the condition the given derivative is g(r) = Sr 0 (√x²+4)dx.

Following the principles of performing a derivative let us proceed towards the given function, g(r) = Sr 0 (√x²+4)dx

Then, placing the function on the calculation side and performing derivate

g'(r) = S r 0 (√x²+4)' dx

g'(r) = S r 0 (1/2)(x²+4)^(-1/2)(2x) dx

g'(r) = S r 0 x/(√x²+4) dx

g'(r) = [√(r²+4)]/2 + (r²/2)[ln(√(r²+4)+r)-ln(2)]

The derivative of the given function is [√(r²+4)]/2 + (r²/2)[ln(√(r²+4)+r)-ln(2)] under the condition the given derivative is g(r) = Sr 0 (√x²+4)dx.

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The post split market cap would be $951000000

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

This means that

2/5 of x = 634,000,000

Where x is the total market

So, we have

x = 634,000,000 * 5/2

Evaluate

x = 1585000000

For the post split market, we have

Post split market = 3/5 * 1585000000

Evaluate

Post split market = 951000000

Hence, thepost split market is $951000000

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The number of registered voters in the voting districts in a county where the districts are drawn fairly. We would be more interested in looking at the mean because it will help indicate how fairly the districts are drawn.

In evaluating the number of registered voters in voting districts in a county where the districts are drawn fairly, you would be more interested in looking at the mean.

The mean is the average number of registered voters per district, which can provide a general idea of the distribution of voters across all districts. This is helpful in understanding if the districts are drawn fairly because, in a fair system, the average number of voters should be relatively similar across districts.

To calculate the mean, you would sum the total number of registered voters in all districts and then divide by the total number of districts. This will give you the average number of registered voters per district, which can help indicate how fairly the districts are drawn.

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The exponential function that models the population growth over the 30-year period is [tex]P(t) = 227 * e^{(0.025t)}.[/tex]

A number that increases or decreases over time at a constant percentage rate is described by an exponential function, which is a sort of mathematical function. Population expansion, compound interest, radioactive decay, and other natural processes that display exponential behaviour are frequently modelled using exponential functions. The base of the natural logarithm of exponential functions is frequently the mathematical constant e, which is roughly equal to 2.71828.

The population growth that corresponds to the exponential growth is given as:

[tex]P(t) = P_0 x e^{(rt)}[/tex]

Now, for [tex]P_0 = 227[/tex] (thousands), t = 30 we have:

[tex]309 = 227 * e^{(r x 30)}\\e^{(r x 30)} = 309/227\\r x 30 = ln(309/227)\\r = ln(309/227)/30[/tex]

r ≈ 0.025

Substituting the value of r we have:

[tex]P(t) = 227 * e^{(0.025t)}[/tex]

Hence, the exponential function that models the population growth over the 30-year period is [tex]P(t) = 227 x e^{(0.025t)}.[/tex]

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Dominic's statement is incorrect, and the difference between the rounded values of two decimal numbers can be the same as the difference between the original values.

Daniel's mistake is that he incorrectly rounded the decimal 1.725 to 172.5 instead of 1.7, which is the correct rounded value to one decimal place. When rounding decimals, we must look at the digit in the place immediately to the right of the required decimal place. If this digit is 5 or greater, we round up by increasing the digit in the required decimal place by one. If the digit is less than 5, we round down by leaving the digit in the required decimal place as it is. In this case, the digit immediately to the right of the required decimal place is 2, which is less than 5. Therefore, the correct rounded value is 1.7, not 172.5.

Nicole could have rounded any of the following numbers to one decimal place to get 9.2:

9.15, 9.24, 9.19, 9.21, 9.16, 9.25, 9.23, 9.27, 9.18, 9.22.

When rounding decimals, there are many possible numbers that could have been rounded to a specific value, which is why it is important to understand the context and the significance of the numbers being rounded.

Dominic's statement that the difference between A and B cannot be the same as the difference between C and D is incorrect. Let's consider an example:

Suppose A = 2.33 and B = 1.77. The difference between A and B is 2.33 - 1.77 = 0.56.

Now, let's round A to one decimal place. The rounded value of A is 2.3, and let's call this value C.

Similarly, let's round B to one decimal place. The rounded value of B is 1.8, and let's call this value D.

The difference between C and D is 2.3 - 1.8 = 0.5, which is different from the difference between A and B.

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a. The coefficient of determination[tex](r^2)[/tex] is 0.0777, meaning that approximately 7.77% of the variation in the cost of a restaurant meal can be explained by the summated rating of the 10 restaurants.

The coefficient of determination, denoted as [tex]r^2[/tex], is a statistical measure that represents the proportion of the total variation in the dependent variable that is explained by the independent variable(s).

In other words, [tex]r^2[/tex]indicates how well the independent variable(s) can predict the dependent variable.

To determine the coefficient of determination  [tex]r^2[/tex] , follow these steps:

Identify the values of SSR and SST.
SSR = 133,146.39
SST = 144,376.47
Use the formula [tex]r^2 = 1 - (SSR/SST)[/tex]
[tex]r^2 = 1 - (133,146.39/144,376.47)[/tex]

Calculate the value of[tex]r^2.[/tex]
[tex]r^2 = 1 - 0.9223[/tex] (rounded to four decimal places)
Subtract to get the final result.
[tex]r^2 = 0.0777[/tex] (rounded to four decimal places).

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question 3.

Option A sin45 = s/x ; s=s will show how to find x.

Therefore option A is correct.

question 4.

To find the length of hypotenuse C, we use option B sin45 = s/c ; s√2

Therefore option B is correct.

We apply Pythagoras theorem to find the side of a triangle.

The Pythagoras theorem sates that In a right triangle, if hypotenuse, perpendicular and base are its sides, then as per the theorem, the square of hypotenuse side is equal to the sum of the square of base and square of perpendicular.

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The value of c for which P(X > c) = 0.0592 is approximately 31.225  where  X is normally distributed with a mean of 23 and a standard deviation of 5.

We know that X follows a normal distribution with a mean of 23 and a standard deviation of 5. We need to find the value of c such that P(X > c) = 0.0592.

To find the value of c, you can use a standard normal distribution table or a calculator that can calculate the inverse normal probability.

A standard normal distribution table can be used to find the Z-score corresponding to a given probability. In this case, find the Z-score such that the area to the right of the Z-score is 0.0592. From the standard normal distribution table, we can see that the z-score corresponding to a region of 0.0592 to the right is approximately 1.645.

So it looks like this:

z = (c - μ) / σ

where μ = 23 and σ = 5.

Inserting the given value will result in:

1.645 = (c - 23) / 5

Multiplying both sides by 5 gives:

c-23 = 8.225

Adding 23 to both sides gives:

c = 31.225

Therefore, the value of c for which P(X > c) = 0.0592 is approximately 31.225.  

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Yes, there is statistically convincing evidence that electricity production by the windmill is related to wind velocity.

The study conducted a regression analysis on the data collected from twenty-five randomly selected days, which allows for the examination of the relationship between wind velocity (mph) and electricity production (amperes). Since the regression model assumptions were checked and satisfied, the results of the regression analysis can be considered reliable and indicate a statistically significant relationship between the two variables.

Based on the regression model, which examined the relationship between wind velocity in miles per hour and electricity production in amperes for a particular windmill, there is statistically convincing evidence that electricity production is related to wind velocity. This conclusion was made after checking and satisfying the regression model assumptions. Therefore, it can be inferred that as wind velocity increases, so does electricity production.

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The determining of one or two samples by it's available data. The standard error is √[(P₁ * (1 - P₁) / n₁) + (P₂ * (1 - P₂) / n₂)], using the pooled proportion as it gives true population proportion, formula is (x₁ + x₂) / (n₁ + n₂). we use proportions for numerator in formula. we need same proportions of success of two population to write Null Hypothesis. The alternative hypotheses are Ha: p₁ < p₂, Ha: p₁ > p₂, Ha: p₁ ≠ p₂. For normal distribution, sample must independent.

We determine if we are comparing two samples if we have data from two different groups or populations, and only have one sample if we have data from only one group or population.

The formula to find the Standard Error for the Confidence Interval for a difference of two proportions is

SE = √[(P₁ * (1 - P₁) / n₁) + (P₂ * (1 - P₂) / n₂)], where P₁, and P₂ are the sample proportions and n₁ and n₂ are the sample sizes.

We use the pooled proportion to find the Standard Error for a Hypothesis Test for a difference of two proportions because it provides a more accurate estimate of the true population proportion.

The formula to find the pooled proportion is

Pp = (x₁ + x₂) / (n₁ + n₂), where x₁ and x₂ are the number of successes in each sample and n₁ and n₂ are the sample sizes.

We use proportions in the numerator of the pooled proportion formula. The Null Hypothesis for a difference of two proportions is that the two populations have the same proportion of successes. The three ways we can write the Alternative Hypothesis for a Difference of Two Proportions are

Ha: p₁ < p₂ (the proportion of successes in population 1 is less than the proportion of successes in population 2)

Ha: p₁ > p₂ (the proportion of successes in population 1 is greater than the proportion of successes in population 2)

Ha: p₁ ≠ p₂ (the proportion of successes in population 1 is different than the proportion of successes in population 2)

In order to use the normal distribution for a difference of two proportions, the sample sizes for each group must be sufficiently large (at least 10 successes and failures in each group) and the samples must be independent.

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Answer:

The answer is ≈3 to the nearest whole number

Step-by-step explanation:

using SOH CAH TOA

BCA

sin0=opp/hyp

sin0=7.9/11

0=sin‐¹(7.9/11)

0=46 to the nearest degree

<A=<D

so,<EDF=46°

tan0=adj/hyp

tan46=x/3.3

x=tan46×3.3

x=3 to the nearest whole number

Using the method of undetermined coefficients the solution of the given Initial Value Problem is y(x) = ln(x) + x² + x - 1,

We assume a particular solution of the form yp = a ln(x) + b. Taking the first and second derivatives, we get y'p = a/x and y"p = -a/x². Substituting these into the differential equation and simplifying, we get:

a = 1/3 and b = 2/3

Therefore, the particular solution is yp = (1/3)ln(x) + (2/3).

The complementary solution is found by solving the homogeneous equation x²y" + xy' + 9y = 0. This can be done by assuming a solution of the form yc = [tex]e^{(mx)}[/tex], which gives the characteristic equation m² + (1/x)m + 9 = 0. Solving for m, we get m = (-1/2x) ± (√(-35)/2x)i. Therefore, the complementary solution is yc = c₁[tex]e^{((-1/2x) + (\sqrt(-35)/2x)i)}[/tex] + c₂[tex]e^{((-1/2x)[/tex] - [tex](\sqrt(-35)/2x)i)}[/tex].

The general solution is the sum of the particular and complementary solutions:

y = yp + yc = (1/3)ln(x) + (2/3) + c₁[tex]e^{((-1/2x)}[/tex] + (√(-35)/2x)i) + c₂[tex]e^{((-1/2x)}[/tex] - (√(-35)/2x)i).

Using the initial conditions, we get:

y(1) = (1/3)(0) + (2/3) + c₁ + c₂ = 2, which gives c₁ + c₂ = 4/3.

y'(1) = (1/3)(1) + c₁((-1/2) + (√(-35)/2)i) + c₂((-1/2) - (√(-35)/2)i) = 4, which gives c₁ - c₂ = (-2/3) - ((√(-35))/3)i.

Solving these two equations simultaneously, we get c₁ = (2 - (√(-35))/3)i and c₂ = (2 + (√(-35))/3)i.

Therefore, the solution to the I.V.P is:

y = (1/3)ln(x) + (2/3) + (2 - (√(-35))/3)i([tex]e^{((-1/2x)}[/tex] + (√(-35)/2x)i)) + (2 + (√(-35))/3)i([tex]e^{((-1/2x)}[/tex] - (√(-35)/2x)i)).

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The question is -

Solve the I.V.P: x²y" + xy' + 9y = 9ln(X), y(1) = 2, y'(1) = 4.

Mary, who finished in 4th position, received $1,600.

Let x be the amount of money received by the 5th position.

Then, the amount received by each position is:

5th position: x

4th position: x + d

3rd position: x + 2d

2nd position: x + 3d

1st position: 4000

The total amount of money awarded is:

x + (x + d) + (x + 2d) + (x + 3d) + 4000 = 12000

4x + 6d = 8000

2x + 3d = 4000 --- Equation (1)

We also know that the average award from 5th to 1st position is:

(x + 4000)/2 = (4000 + 2(x + d) + (x + 3d))/5

10x + 20d = 16000 + 6x + 12d

4x + 8d = 3200

2x + 4d = 1600 --- Equation (2)

Solving equations (1) and (2), we get:

x = 800

d = 800

So, Mary, who finished in 4th position, received $1,600.

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To calculate the projected PGI (Potential Gross Income) per square foot, we need to take into account both the renewing tenant and the possibility of a new tenant.

If the rent for renewing tenants is $25/SF, and the probability of them renewing their space is .75, then the effective rent for that space would be:

Adding the effective rents for both tenants gives us the projected PGI per square foot:
Projected PGI = Effective Rent Renewing Tenant + Effective Rent New Tenant
Projected PGI = $18.75/sf + $7/sf
Projected PGI = $25.75/sf

Therefore, the projected PGI per square foot is $25.75/sf.
1. Multiply the rent for a renewing tenant by the probability of the current tenant renewing their space: $25/sf * 0.75 = $18.75/sf
2. Calculate the probability of a new tenant leasing the space, which is the complement of the current tenant renewing: 1 - 0.75 = 0.25
3. Multiply the rent for a new tenant by the probability of a new tenant leasing the space: $28/sf * 0.25 = $7/sf
4. Add the two results together to find the projected PGI per square foot: $18.75/sf + $7/sf = $25.75/sf the projected PGI per square foot is $25.75/sf.

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We can say with 90% confidence that the true population mean number of gallons sold to customers at Nale's Quick Fill lies between 5.27 and 5.53 gallons.

a. The point estimate of the population mean is simply the sample mean, which in this case is 5.40 gallons.

b. To develop a 90% confidence interval for the population mean, we need to first find the critical value of z from the z-distribution table. Since we want a 90% confidence interval, the level of significance is α = 0.10, which means we need to split this α/2 = 0.05 between the two tails of the distribution. From the table, the corresponding z-value for a 0.05 tail area is 1.645. We can use the formula: Confidence interval = sample mean ± (z-value x standard error), where the standard error is the population standard deviation divided by the square root of the sample size, or

[tex]2.50 / √55 = 0.336[/tex]

Plugging in the values, we get:

Confidence interval =

[tex]5.40 ± (1.645 \times 0.336)[/tex]

Confidence interval = (5.27, 5.53)

If we were to repeatedly take samples of size 55 from the population and compute the 90% confidence interval for each sample, we can expect 90% of these intervals to contain the true population mean.

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The cosecant function of expression cscâ¹(â2) is undefined. The value of cosâ¹(1/â2) = 2π/3 radians.

The expression cscâ¹(â2), since the cosecant function is only defined for angles between -π/2 and π/2, we cannot find an angle with a cosecant of -2. Therefore, the expression is undefined.

The expression cosâ¹(1/â2) is asking "what angle has a cosine of 1/(-2) = -1/2?" Since the cosine function is negative for angles between π/2 and 3π/2, we know that the angle we are looking for is in the second or third quadrant.

To find the angle, we can use the inverse cosine function, which gives us the angle whose cosine is equal to the given value. Therefore, we have

cosθ = -1/2

Taking the inverse cosine of both sides, we get

θ = cos⁻¹(-1/2)

Using the unit circle or trigonometric identities, we can find that cos⁻¹(-1/2) = 2π/3 or 4π/3. Since the cosine function is negative in the second quadrant and also in the third quadrant, we choose the solution in the second quadrant, which is θ = 2π/3.

Therefore, cosâ¹(1/â2) = 2π/3 radians.

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Rosa will cover 1056 yards in 8 minutes and 1716 yards in 13 minutes, while Peng will cover 880 yards in 8 minutes and 1430 yards in 13 minutes.

We can start by finding the speed of each person in yards per minute, and then use these speed to find the distance covered in 8 minutes and 13 minutes.

Rosa's speed is:

396 yards in 3 minutes = 132 yards per minute

Distance covered by Rosa in 8 minutes is:

8 minutes × 132 yards per minute = 1056 yards

Distance covered by Rosa in 13 minutes is:

13 minutes × 132 yards per minute = 1716 yards

Peng's speed is:

330 yards in 3 minutes = 110 yards per minute

Distance covered by Peng in 8 minutes is:

8 minutes × 110 yards per minute = 880 yards

Distance covered by Peng in 13 minutes is:

13 minutes × 110 yards per minute = 1430 yards

Therefore, Rosa will cover 1056 yards in 8 minutes and 1716 yards in 13 minutes, while Peng will cover 880 yards in 8 minutes and 1430 yards in 13 minutes.

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The estimated difference between 18,000, 2,351, 1,987, and 2,416, rounded to the nearest thousand, is about 12,000.

The four numbers given are 18,000, 2,351, 1,987, and 2,416. To round each number to the nearest thousand, we look at the digit in the hundreds place. If it is less than 500, we round down to the nearest thousand, and if it is 500 or greater, we round up to the nearest thousand.

So, rounding 18,000 to the nearest thousand gives us 18,000. Rounding 2,351 to the nearest thousand gives us 2,000 (since the hundreds digit is less than 500). Rounding 1,987 to the nearest thousand gives us 2,000 (since the hundreds digit is also less than 500). Finally, rounding 2,416 to the nearest thousand gives us 2,000 (since the hundreds digit is less than 500).

Now we can find the difference between these rounded numbers. The difference between 18,000 and 2,000 is 16,000. The difference between 16,000 and 2,000 is 14,000. The difference between 14,000 and 2,000 is 12,000.

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