what is the resulting expression when (7x - 4/3) is subtracted from -3/5x + 5/3?

Answer:

Who is the current world number one in men’s tennis?

What is the name of the sport that combines skiing and shooting?

How many countries are in the European Union?

What is the largest animal that ever lived on Earth?

What is the most spoken language in the world?

Step-by-step explanation:

is that ok?

The given question is in Spanish, English translation of given question is below.

The perimeter of a rectangle is 54 inches. its length is twice as wide. find the length and width of the rectangle find the area of ​​the rectangle.

The width of the rectangle is 9 inches, the length of the rectangle is 18 inches, and the area of the rectangle is 162 square inches.

Let us assume the width of the rectangle w and the length l.

Given, the perimeter of the rectangle is 54 inches:

We know that

Perimeter = 2(length + width) = 54

2(l + w) = 54

Given, length is twice as width i.e. l=2w

2(2w + w) = 54

2(3w) = 52

6w = 54

w = 54/6

w = 9

l = 2(9)

l = 18

We know that

Area = length x width

Area = 18 x 9

= 162

Therefore, the width of the rectangle is 9 inches, the length of the rectangle is 18 inches, and the area of the rectangle is 162 square inches.

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the solution to the differential equation is: x = (1/36) y² - (13/36) Note that this equation represents a parabolic curve in the (x,y)-plane, opening upwards and with its vertex at (-13/36,0).

We can start by separating the variables and integrating both sides of the equation:

1/9 y dy = 2 dx

Integrating both sides with respect to their respective variables, we get:

(1/18) y² = 2x + C

where C is the constant of integration.

Using the initial condition y(9) = 7, we can substitute x=9 and y=7 to solve for C:

(1/18) (7²) = 2(9) + C

C = 49/2 - 18 = 13/2

Substituting this value of C back into the general solution, we get:

(1/18) y² = 2x + 13/2

Simplifying and solving for x, we get:

x = (1/36) y² - (13/36)

Therefore, the solution to the differential equation is:

x = (1/36) y² - (13/36)

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To find the volume of the cylinder, we need to use the formula:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

If we have a cylinder with radius r and height h, and another cylinder with the same radius r but double the height (2h), the volume of the second cylinder is:

V' = πr^2(2h) = 2πr^2h

So, to answer the questions:

The volume of the cylinder is V = π(5 cm)^2(8 cm) = 100π cubic cm, which is approximately 314.16 cubic cm rounded to two decimal places.

The volume of the cylinder with the same radius and double the height is V' = 2π(5 cm)^2(8 cm) = 200π cubic cm, which is approximately 628.32 cubic cm rounded to two decimal place

Answer is: manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548

We can use the normal approximation to the binomial distribution to estimate the confidence interval for the true proportion of time spent mowing.

The sample proportion of time spent mowing is:

p = 200/400 = 0.5

The standard error of the sample proportion is:

SE = sqrt(p(1-p)/n) = sqrt(0.5 * 0.5 / 400) = 0.025

To find the 95% confidence interval for the true proportion of time spent mowing, we can use the formula:

p ± z*SE

where z* is the critical value from the standard normal distribution corresponding to a 95% confidence level, which is approximately 1.96.

So the 95% confidence interval for the true proportion of time spent mowing is:

0.5 ± 1.96*0.025 = (0.452, 0.548)

Therefore, the manager can be 95% confident that the true proportion of time spent mowing is between 0.452 and 0.548.

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Answer:

Step-by-step explanation:

To solve the equation (T/12) + 5 = (T/3) + (T/4), we need to simplify the right-hand side of the equation by finding a common denominator for T/3 and T/4.

The least common multiple of 3 and 4 is 12, so we can rewrite T/3 and T/4 as (4T/12) and (3T/12), respectively. Substituting these expressions into the equation, we get:

(T/12) + 5 = (4T/12) + (3T/12)

Simplifying the right-hand side, we get:

(T/12) + 5 = (7T/12)

Subtracting (T/12) from both sides, we get:

5 = (6T/12)

Simplifying the right-hand side, we get:

5 = (T/2)

Multiplying both sides by 2, we get:

T = 10

Therefore, the solution to the equation is T = 10.

[tex]\sf\longrightarrow \: \frac{t}{12} + 5 = \frac{t}{3} + \frac{t}{4} \\ [/tex]

[tex]\sf\longrightarrow \: \frac{t + 60}{12} = \frac{t}{3} + \frac{t}{4} \\ [/tex]

[tex]\sf\longrightarrow \: \frac{t + 60}{12} = \frac{4t + 3t}{12} \\ [/tex]

[tex]\sf\longrightarrow \: 12(t + 60) = 12(4t + 3t) \\ [/tex]

[tex]\sf\longrightarrow \: 12t + 720 = 48t + 36t \\ [/tex]

[tex]\sf\longrightarrow \: 12t + 720 = 84t \\ [/tex]

[tex]\sf\longrightarrow \: 720 = 84t - 12t\\ [/tex]

[tex]\sf\longrightarrow \: 720 =72t\\ [/tex]

[tex]\sf\longrightarrow \: 72t = 720\\ [/tex]

[tex]\sf\longrightarrow \: t = \frac{720}{72} \\ [/tex]

[tex]\sf\longrightarrow \: t = 10 \\ [/tex]

[tex]\longrightarrow { \underline{ \overline{ \boxed{ \sf{\: \: \: t = 10 \: \: \: }}}}} \: \: \bigstar\\ [/tex]

The distance from the place Tyler meets his neighbor to his home is 450 meters.

Tyler's speed is 1.2 meters per second.

Part A:

To find the distance from the place Tyler meets his neighbor to his home, we need to find the distance when x=0, since it's the time they meet. Plug x=0 into the equation y=450-72x:

y = 450 - 72(0)

y = 450

The distance from the place Tyler meets his neighbor to his home is 450 meters.

Part B:

To find the speed of Tyler, we need to find the rate at which the distance is decreasing. This can be found by taking the derivative of the distance function with respect to time:

dy/dx = -72

Since the derivative is constant, Tyler's speed is constant, and it is 72 meters per minute. To convert it to meters per second, we need to multiply by the conversion factor (1 minute = 60 seconds):

Speed = 72 m/min * (1 min / 60 s)

Thus, the speed = 1.2 m/s

Tyler's speed is 1.2 meters per second.

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About the equation used to estimate the height E of the tramway t seconds after it left the station, we first need to calculate the rate of elevation gain per second.

Elevation difference: 5632 feet (Range Peak) - 4692 feet (tramway station) = 940 feet
Time: 4 minutes * 60 seconds/minute = 240 seconds

Rate of elevation gain: 940 feet / 240 seconds = 3.9167 feet/second (approximately 3.9 feet/second)

Now, we can write the equation to estimate the height E of the tramway t seconds after it left the station:

E = initial elevation + (rate of elevation gain * t)
E = 4692 + 3.9t

So, the correct equation is: E = 4692 + 3.9t

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Answer:

30 cm²

Step-by-step explanation:

Formula : [tex]Area = \frac{hb}{2}[/tex]

James says the fraction 3/4 has the same value as the expression 4 ÷ 3. I disagree with James' statement.

The fraction 3/4 is not the same as the expression 4 ÷ 3. A fraction can be interpreted as division of the numerator (top number) by the denominator (bottom number). In this case, 3/4 represents the division of 3 by 4, whereas 4 ÷ 3 represents the division of 4 by 3. These two expressions have different values and are not equal.

Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English.

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If p : q = 2/3 : 2 and p : r = 3/4 : 1/2 , the ratio of p : q : r in its simplest form is 32 : 27 : 24.

To calculate the ratio p : q : r, we need to first find the values of p, q, and r. We can use the given proportions to set up a system of equations and solve for the variables.

From the first proportion, we know that:

p/q = 2/3 : 2

We can simplify this by cross-multiplying:

p = (2/3) * 2q

p = (4/3)q

From the second proportion, we know that:

p/r = 3/4 : 1/2

Again, we can cross-multiply and simplify:

p = (3/4) * r/(1/2)

p = (3/2)r

Now we have two equations for p in terms of q and r. We can substitute these into each other and solve for q and r:

(4/3)q = (3/2)r

r/q = (8/9)

q/r = (9/8)

Now we have the ratios of r to q and q to r. We can use these to find the ratio of p, q, and r:

p : q : r = p : q * (9/8) : r * (8/9)

Substituting the values we found for p in terms of q and r:

p : q : r = (4/3)q : q * (9/8) : r * (8/9)

Simplifying:

p : q : r = 32 : 27 : 24

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The rotated vector is (-2,0). To see why, imagine the original vector (0,2) plotted on the coordinate plane. To rotate it 270° clockwise about the origin, we can first rotate it 90° clockwise to get (2,0), then rotate that 180° clockwise to get (-2,0).

To understand this geometrically, think of the vector (0,2) as pointing straight up on the y-axis. Rotating it 90° clockwise means it now points to the right on the x-axis. Then, rotating it another 180° clockwise means it points straight down on the negative y-axis, which corresponds to the vector (-2,0).

In general, to rotate a vector (x,y) by an angle θ about the origin, we can use the following formulas: x' = x cos θ - y sin θ. y' = x sin θ + y cos θ In this case, θ = 270°, so cos θ = 0 and sin θ = -1.

Plugging in x=0, y=2, we get: x' = 0 - 2(-1) = 2 y' = 0(270) + 2(0) = 0. So the rotated vector is (2,0), which corresponds to (-2,0) because we rotated it clockwise instead of counterclockwise.

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Answer:

A

Step-by-step explanation:

2.17 > 2.017

because

2.017 = 2 + 17/1000

while

2.17 = 2 + 17/100 = 2 + 170/1000

170/1000 is larger than 17/1000.

for that reason D is wrong, of course.

2.17 is NOT equal to 2.017. 17/1000 is NOT equal to 170/1000.

2.018 = 2 + 18/1000

2.17 = 2 + 17/100 = 2 + 170/1000

also 18/1000 is NOT larger than 170/1000.

2.16 = 2 + 16/100 = 2 + 160/1000

2.017 = 2 + 17/1000

17/1000 are NOT larger than 160/1000.

length of AC in the triangle is 32 units.

The triangle proportionality theorem, also known as the side-splitter theorem, states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides proportionally.  

In mathematical terms, let ABC be a triangle with a line parallel to one side, say line DE || AB, where D lies on BC and E lies on AC. Then, the theorem states that:

BD/DC = AE/EC

In the given triangle ABC;

GH and AC are parallel

AG=BG

BH=HC

Using proportional rule

BG/AB=GH/AC

BG/2BG=16/AC

1/2=16/AC

AC=32 units

Hence, length of AC in the triangle is 32units.

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Answer:

1. 32 units

Step-by-step explanation:

sorry abt the other person but the answer is 32... i just took it

The probability that the mean of the sample would differ from the population mean by less than 43 points is approximately 0.7597 or 0.7600 (rounded to four decimal places).

Given that the standard deviation of the scores on a skill evaluation test is 320 points with a mean of 1434 points. And we have a sample of size n = 338.

We need to find the probability that the mean of the sample would differ from the population mean by less than 43 points.

The standard error of the mean is given by:

SE = σ/√n

where σ is the population standard deviation and n is the sample size.

Substituting the given values, we get:

SE = 320/√338

SE ≈ 17.398

To find the probability, we need to standardize the sample mean using the standard error as follows:

Z = (X - μ) / SE

where X is the sample mean, μ is the population mean, and SE is the standard error of the mean.

Substituting the given values, we get:

Z = (1434 - 1434) / 17.398

Z = 0

Since the mean difference is 0, we can find the probability of a difference less than 43 points by finding the probability that Z lies between -43/17.398 and 43/17.398.

Using a standard normal distribution table or calculator, we find that this probability is approximately 0.7597.

Therefore, the probability that the mean of the sample would differ from the population mean by less than 43 points is approximately 0.7597 or 0.7600 (rounded to four decimal places).

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The variable Q, representing the number of questions Violet answers correctly before she misses one in a computer-adaptive test, is a Geometric variable.

This is because a geometric distribution models the number of trials needed for the first success in a series of Bernoulli trials with a constant probability of success.

Where as all aspects of a logarithmic articulation that is isolated by a short or in addition to sign is known as the term of the algebraic expression and an algebraicexpression with two non-zero terms is called a binomial.

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The number of adult tickets purchased that day was 36 if the charge is $3.25 for each adult and $1.50 for each student and 400 people paid $1237

Let the number of adults be x

the number of students be y

Total people = 400

x + y = 400

Total receipts = $1,237

Cost of an adult ticket = $3.25

Cost of a student ticket = $1.50

Cost of x adults tickets = 3.25x

Cost of y student tickets = 1.50y

3.25x + 1.50y = 1237

Multiply the first equation by 1.50

1.50x + 1.50y = 600

Subtract the second and above equation

1.75x = 637

x = 364

364 + y = 400

y = 36

Thus, the number of adult tickets purchased is 36.

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Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

The rule of 5% slack states that the cable should have 5% of the cable's length in slack, so for a 130 feet cable, there would need to be 6.5 feet of slack. Since Chantal's design only has 10 feet of vertical change, that would not leave enough slack to meet the rule.

Also, the 6-8 feet of vertical change per 100 feet of horizontal change rule states that the vertical change should be between 6-8 feet for every 100 feet of horizontal change. In this case, the horizontal change is 130 feet, so the vertical change should be between 7.8-10.4 feet. Since Chantal's design has only 10 feet of vertical change, it does not meet this rule either.

Therefore, Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

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The recursive formula for this sequence is f(n) = f(n-3) + 6n - 18.

We can use the method of finite differences to find a possible recursive formula for this sequence.

First, let's compute the first few differences:

f(4) - f(1) = 6

f(7) - f(4) = 6

Since the second differences are zero, we can assume that the sequence is a quadratic sequence. Let's write it in the form f(n) = an^2 + bn + c. We can solve for the coefficients using the given values:

f(1) = a(1)^2 + b(1) + c = 6

f(4) = a(4)^2 + b(4) + c = 12

f(7) = a(7)^2 + b(7) + c = 18

Solving for a, b, and c, we get:

a = 1

b = 5

c = 0

Therefore, the recursive formula for this sequence is f(n) = f(n-3) + 6n - 18.

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It would take approximately one minute and a half (or 1.6 minutes) to reach 100 jumping jacks at this rate.

What was the activity?

The dependent variable is the number of jumping jacks completed in a specific time period. In this case, the number of jumping jacks completed in one minute is the dependent variable.

The independent variable is the time in minutes. This means that the number of jumping jacks completed is influenced by the time spent doing the activity.

The relationship between the dependent variable (number of jumping jacks completed) and the independent variable (time in minutes) is directly proportional, with a ratio of approximately 63 jumping jacks per minute. This means that for every one minute spent doing jumping jacks, approximately 63 jumping jacks can be completed.

To calculate how much of the activity would be completed if this rate was maintained for 12 minutes, we can multiply the rate (63 jumping jacks per minute) by the time (12 minutes), which gives us 756 jumping jacks.

To find out how long it would take to reach 100 jumping jacks, we can set up a ratio:

63 jumping jacks / 1 minute = 100 jumping jacks / x minutes

We can solve for x by cross-multiplying:

63x = 100

x = 100 / 63

x ≈ 1.6

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The mean absolute deviation (MAD) of the data in the pictograph is equals to the one basketball.

We have a data in the pictograph. In mathematics, a pictograph is a pictorial representation of data using images, icons. It is also known as a pictogram. We have a pictograph, in which the vertical axis is labeled Baskets made and the horizontal axis is labeled Student. Here, one basketball picture equals two baskets. Mean absolute deviation (MAD) is a statistical measure of the average absolute distance between each data value and the mean of a data set. It is a parameter or statistic that measures the spread, or variation, in data.

Mean is defined as the sum of data values divided by number of values.

Sum of data values = 4 + 4×2 + 3×2 + 5×2

= 28

So, mean = 28/4 = 7

Now, | 4 - 7| + |8 - 7| + |6 -7 | + | 10 - 7|

= 3 + 1 + 1 + 3 = 8

So,, mean absolute deviation (MAD) of the data = 8/7 = 1.1 ~ 1. Hence, required value is 1.

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Complete question :

Find the mean absolute deviation (MAD) of the data in the pictograph below. Baskets

The key says one basketball picture equals two baskets. The key says one basketball picture equals two baskets. A picture graph labeled Baskets each student made. The vertical axis is labeled Baskets made. The horizontal axis is labeled Student. The names from left to right on the horizontal axis are Reynaldo, Marcelle, Allie, and Fernando. There are two basketball pictures above Reynaldo. There are four basketball pictures above Marcelle. There are three basketball pictures above Allie. There are five basketball pictures above Fernando

To determine how many biscuit packages, pemmican packages, and butter and cocoa packages are needed per day for one person.

Calculating an individual's daily Caloric food requirements based on calorie intake and percentage of calories from each food group requires several pieces of information. The first is the total daily calorie requirement, which varies based on factors such as age, gender, height, weight, and physical activity level.

The second is the percentage of daily calories that should come from each food group, which is determined by dietary guidelines and varies based on factors such as age and gender. Finally, the calorie content of each food item must be known to determine how much of each food is needed to meet daily calorie and nutrient requirements.

Once these factors are known, it is possible to calculate how many biscuit packages, pemmican packages, and butter and cocoa packages are needed per day for one person. However, without knowing the specific calorie content and nutritional value of each food item, it is impossible to provide a specific answer.

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The surface area of the pyramid is 310.00 square centimeters.

To find the surface area of the pyramid, we need to find the area of each of its faces and add them up.

First, let's find the area of the base of the pyramid;

Area of base = (side length)² = 7² = 49 cm²

Now, we find the area of each triangular face. To do this, we need to find the length of the slant height (l), which is the hypotenuse of a right triangle with one leg equal to half the base and the other leg equal to the height of  pyramid.

Half of the base = 7/2 = 3.5 cm

Using Pythagorean theorem, we find the slant height;

l² = (3.5)² + (15)²

l² = 122.25 + 225

l² = 347.25

l = √(347.25) ≈ 18.64 cm

Now we find the area of each triangular face;

Area of face = (1/2) x base x height = (1/2) x 7 x 18.64 ≈ 65.24 cm²

Finally, we can find the total surface area by adding up the areas of all four faces;

Total surface area = 4 x (area of face) + (area of base)

Total surface area = 4 x 65.24 + 49

Total surface area = 261.00 + 49

Total surface area = 310.00 cm²

Therefore, the surface area of the pyramid is 310.00 cm².

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If you paid $25,000 for your car and you sell it after owning the car for 3 years, then the worth of the car is t₃=$7,910. 15 (option c).

To find the value of your car after owning it for 3 years, we need to evaluate the function at t=3. This means we need to substitute t=3 into the function and simplify the expression.

f(3) = 0.75(3) = 2.25

The output of the function when t=3 is 2.25. But what does this number mean? It represents the fraction of the original value of the car that remains after owning it for 3 years.

To find the actual value of the car, we need to multiply this fraction by the original value of the car, which is given as $25,000.

Value of car after 3 years = 2.25 x $25,000 = $56,250

Therefore, the value of the car after owning it for 3 years is $7,910.15. This is the option (C) in the given choices.

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The proportion of students from a random sample of 500 fail to reject the null hypothesis and can not support Mr. Smith's claim as per the data.

Percent of students claim they have at least two cell phones = 20%

Sample size = 500

Significance level α = 0. 05

This is a hypothesis testing problem with the following hypotheses,

Null hypothesis (H₀),

The proportion of students who have at least two cell phones is 0.20.

Alternative hypothesis (Hₐ),

The proportion of students who have at least two cell phones is greater than 0.20.

Use a one-tailed z-test for proportions to test the null hypothesis at a significance level of α = 0.05.

The test statistic is calculated as,

z = (p₁ - p₀) / √(p₀(1-p₀)/n)

where p₁ is the sample proportion,

p₀ is the null hypothesis proportion,

and n is the sample size.

Using the values in the problem, we get,

p₁ = 88/500

   = 0.176

p₀ = 0.20

n = 500

z = (0.176 - 0.20) / √(0.20(1-0.20)/500)

  = -1.34

Using a standard normal distribution table,

the p-value for z = -1.34 is approximately 0.0901.

Since the p-value (0.0901) is slightly greater than the significance level (0.05),

Fail to reject the null hypothesis.

Do not have sufficient evidence to conclude that the proportion of students who have at least two cell phones is greater than 0.20.

Therefore, cannot support Mr. Smith's claim based on the given data as fail to reject the null hypothesis.

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Answer:

{51

Step-by-step explanation:

10^2 - 7^2= 51

You can’t square 51 because no two same whole numbers multiplied creates 51

B= 51

Check answer

7^2 + 51= 100

{100 = 10^2

Answer:

3(x - 2)(2x + 3)

Step-by-step explanation:

6x² - 3x - 18 ← factor out GCF of 3 from each term

= 3(2x² - x - 6) ← factor the quadratic

consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term

product = 2 × - 6 = - 12 and sum = - 1

the factors are - 4 and + 3

use these factors to split the x- term

2x² - 4x + 3x - 6 ( factor the first/second and third/fourth terms )

= 2x(x - 2) + 3(x - 2) ← factor out (x - 2) from each term

= (x - 2)(2x + 3) ← in factored form

then

6x² - 3x - 18 = 3(x - 2)(2x + 3)

There is no significant difference in the proportion of college enrollees between females and males who have completed high school within the past 12 months.

To determine if the difference in proportions is statistically significant or if it could be due to chance.

We will conduct a hypothesis test. Our null hypothesis (H₀) is that there is no difference in the proportion of college enrollees between females and males. Our alternative hypothesis (H₁) is that there is a difference in the proportion of college enrollees between females and males.

We can use a two-sample z-test to test this hypothesis. The formula for the test statistic is:

z = (p₁ - p₂) / √(p'* (1 - p') * ((1 / n₁) + (1 / n₂)))

where p₁ and p₂ are the sample proportions, p' is the pooled proportion, n₁ and n₂ are the sample sizes.

Given, p₁ = 0.72, p₂ = 0.65, n₁ = 175, n₂ = 160

p' = (x₁ + x₂) / (n₁ + n₂)

x₁ = 126 (0.72 * 175) and x₂ = 104 (0.65 * 160).

p' = (126 + 104) / (175 + 160) = 0.684

By applying the above values we get,

z = (0.72 - 0.65) / √(0.684 * (1 - 0.684) * ((1 / 175) + (1 / 160))) ≈ 2.11

The critical value for a two-tailed test with alpha = 0.01 is approximately ±2.58. Since our calculated z-value (2.11) is less than the critical value, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that there is a significant difference in the proportion of college enrollees between females and males.

Therefore, there is no significant difference in the proportion of college enrollees between females and males who have completed high school within the past 12 months.

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The possible rectangles that Wes can make with his 20 feet of fencing are:

L = 3, W = 7

L = 4, W = 6

L = 5, W = 5

L = 6, W = 4

L = 7, W = 3

Let L be the length of the rectangular garden and W be the width of the garden. Since the garden is enclosed by four sides, Wes will need 2L+2W feet of fencing to enclose it. We know that he has 20 feet of fencing, so we have the equation:

2L + 2W = 20

We also know that the smallest side of the garden should be 3 feet or longer, so:

L >= 3

W >= 3

To find the possible rectangles Wes can make, we can solve the equation for one variable in terms of the other:

2L + 2W = 20

2L = 20 - 2W

L = 10 - W

Now we can substitute this expression for L into the inequality L >= 3 to get:

10 - W >= 3

W <= 7

Similarly, we can substitute L = 10 - W into the inequality W >= 3 to get:

10 - L >= 3

L <= 7

Therefore, the possible values for L and W are:

3 <= L <= 7

3 <= W <= 7

We can also use the equation 2L + 2W = 20 to find the combinations of L and W that add up to 10, since the total length of fencing is 20 feet:

L = 3, W = 7

L = 4, W = 6

L = 5, W = 5

L = 6, W = 4

L = 7, W = 3

These are the possible rectangles that Wes can make with his 20 feet of fencing, where the smallest side is 3 feet or longer.

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z = 4(x - 1) - ln (y - 4)

Therefore, the equation of the tangent plane to the surface z = y ln x at the point (1, 4, 0) is z = 4(x - 1) - ln (y - 4).

To determine the equation of the tangent plane to the surface z = y ln x at the point (1, 4, 0), we first need to find the partial derivatives of the surface with respect to x and y.

∂z/∂x = y/x

∂z/∂y = ln x

Then, we can use these partial derivatives along with the point (1, 4, 0) to find the equation of the tangent plane using the formula:

z - z0 = ∂z/∂x(x0, y0)(x - x0) + ∂z/∂y(x0, y0)(y - y0)

where (x0, y0, z0) is the given point.

Plugging in the values, we have:

z - 0 = (4/1)(x - 1) + ln 1(y - 4)

Simplifying:

z = 4(x - 1) - ln (y - 4)

Therefore, the equation of the tangent plane to the surface z = y ln x at the point (1, 4, 0) is z = 4(x - 1) - ln (y - 4).

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