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Answer:

There are two criterias for. a quadrilateral being a parallelogram. They are:-

1. One pair of opposite sides are equal and parallel

2. Both pairs of oppoosite sides are parallel.

3. Opposite angles are equal.

Answer:

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. If — BC — AD and — BC ≅ — AD , then ABCD is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Step-by-step explanation:

its just the answer

The correct null hypothesis if we want to test for the significance of the slope coefficient is: a. h0: β1 = 0 Therefore, option a. h0: β1 = 0 is correct.

This null hypothesis assumes that there is no linear relationship between the independent and dependent variables, and the slope coefficient is equal to zero.

The alternative hypothesis would be that the slope coefficient is not equal to zero, indicating a significant linear relationship between the variables.

The correct null hypothesis to test for the significance of the slope coefficient is: 1. H0: β1 = 0 In this null hypothesis, H0 represents the null hypothesis, and β1 refers to the slope coefficient.

The hypothesis states that the slope coefficient is not significantly different from zero, implying no significant relationship between the independent and dependent variables.

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The resulting function after these transformations is: y = (1/3)e^(2x) - 2. Starting with the original function y=e^-2x, the vertical compression by a factor of 3 can be achieved by multiplying the function by 1/3: y=(1/3)e^-2x.

Next, reflecting across the y-axis is accomplished by replacing x with -x: y=(1/3)e^2x.

Finally, shifting down 2 units can be achieved by subtracting 2 from the function: y=(1/3)e^2x - 2.

Putting this in the form y=ce^ax+b, we have y=(1/3)e^2x-2. Therefore, c=1/3, a=2, and b=-2.

Given the original function y=e^(-2x), the following transformations occur:

1. Vertically compressed by a factor of 3: y = (1/3)e^(-2x)
2. Reflected across the y-axis: y = (1/3)e^(2x)
3. Shifted down 2 units: y = (1/3)e^(2x) - 2

The resulting function after these transformations is: y = (1/3)e^(2x) - 2

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The change in the amount of water in 12 weeks is [tex]16\frac{1}{2}[/tex] pints.

Let's first find the amount of water that decreases in one week:

[tex]9\frac{5}{8}[/tex] pints / 7 weeks = [tex]1\frac{3}{8}[/tex] pints per week

So the amount of water decreases by [tex]1\frac{3}{8}[/tex] pints per week.

To find the change in the amount of water in 12 weeks

we can simply multiply the amount of decrease per week by the number of weeks:

[tex]1\frac{3}{8}[/tex]  pints per week x 12 weeks

= [tex]16\frac{1}{2}[/tex] pints

Therefore, the change in the amount of water in 12 weeks is [tex]16\frac{1}{2}[/tex] pints.

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If r and d represents the Rafael's age and his dad'age respectively, then the equation which relates the ages of both of Rafael and his dad is equals to r² + 8 = d - 4 .

Let us consider the age of rafael and his dad be equal to 'r' and 'd' respectively. We have to determine a equation which relates the ages of both of Rafael and his dad. Now, Rafael's age squared is equals to r² then plus 8 in resultant, i.e., r² + 8. This situation of rafael'a age is equivalent to 4 less the age of rafael's dad. So, we can write as r² + 8 = d - 4

Simplify the expression,

=> r² - d + 8 + 4 = 0

=> r² - d + 12= 0

Which is a trinomial ( contains three terms). Hence, required relation is r² - d + 12= 0.

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Complete question:

Rafael's age squared plus 8 is equivalent to 4 less the age of rafael's dad. Enter the equation related to Rafael's age, r and rafael's dad age, d.

The amount of Iodine-131 remaining after six days is 0.6023682337x grams.

Let's suppose the initial amount of Iodine-131 present in the sample is x grams. After one day, the amount of Iodine-131 remaining in the sample will be 91.7% of the original amount. We can represent this mathematically as:

Amount after one day = x - (8.3/100) * x

Amount after one day = x * (1 - 8.3/100)

Amount after one day = 0.917x

Similarly, after two days, the amount of Iodine-131 remaining in the sample will be:

Amount after two days = 0.917x - (8.3/100) * 0.917x

Amount after two days = 0.917x * (1 - 8.3/100)

Amount after two days = 0.841489x

We can use a unitary method to find out how much Iodine-131 will remain after six days. We know that the amount of Iodine-131 decreases by 8.3% every day, so the amount of Iodine-131 remaining after two days is 84.15% of the initial amount.

Let's represent the amount of Iodine-131 remaining after six days as y. We can use the unitary method to find y as follows:

Amount after 2 days = 0.841489x

Amount after 3 days = 0.841489x - (8.3/100) * 0.841489x

Amount after 3 days = 0.841489x * (1 - 8.3/100)

Amount after 3 days = 0.7738631721x

Amount after 4 days = 0.7738631721x - (8.3/100) * 0.7738631721x

Amount after 4 days = 0.7738631721x * (1 - 8.3/100)

Amount after 4 days = 0.7117127535x

Amount after 5 days = 0.7117127535x - (8.3/100) * 0.7117127535x

Amount after 5 days = 0.7117127535x * (1 - 8.3/100)

Amount after 5 days = 0.6544992961x

Amount after 6 days = 0.6544992961x - (8.3/100) * 0.6544992961x

Amount after 6 days = 0.6544992961x * (1 - 8.3/100)

Amount after 6 days = 0.6023682337x

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Rounded to three decimal places, the approximate change in f is Δf ≈ -0.173. To approximate the change in f(x), we need to use the formula for differentials:

df ≈ f'(x)Δx

where f'(x) is the derivative of f(x) and Δx is the change in x.

First, we find f'(x) by taking the derivative of f(x):

f'(x) = -12 cos(x) sin(x)

Then, we plug in the values of x:

f'(π/6) = -12 cos(π/6) sin(π/6) = -6

Next, we calculate Δx:

Δx = π/6 + 0.1 - π/6 = 0.1

Finally, we substitute these values into the formula for differentials:

df ≈ f'(π/6)Δx = -6(0.1) = -0.6

Rounding to three decimal places, the approximate change in f(x) is -0.600.
To compute the differential df for f(x) = 6 cos^2(x), we need to find its derivative with respect to x. Using the chain rule, we have:

df/dx = 12 cos(x)(-sin(x))

Now, we can approximate the change in f when x changes from x = π/6 to x = π/6 + 0.1. Using the formula:

Δf ≈ (df/dx)(Δx)

We can plug in the values for x = π/6 and Δx = 0.1:

Δf ≈ 12 cos(π/6)(-sin(π/6))(0.1)

Δf ≈ 12 * (√3/2) * (-1/2) * 0.1

Δf ≈ -√3/10

Rounded to three decimal places, the approximate change in f is Δf ≈ -0.173.

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The measure of AC is given as follows:

AC = 26.25 units.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

For this problem, we have that segment AC represents the hypotenuse of a right triangle of a right triangle of sides 25 and 8, as AB is tangent to the circle, hence:

(AC)² = 25² + 8²

AC = sqrt(25² + 8²)

AC = 26.25 units.

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The probability that 0 < Z < 2.03 is approximately 0.4788. This means that about 47.88\% of the values in a standard normal distribution are between 0 and 2.03.

To find the probability using a z-score, you need to use a formula that involves subtracting the mean and dividing by the standard deviation of the normal distribution. Then, you can look up the corresponding probability in a z-table, which shows the probability of a value being less than, greater than, or between certain z-scores.¹²

To answer your question, you need to use the formula and the z-table.

The formula for finding a z-score is:

z = \frac{x - \mu}{\sigma}

where x is the value, \mu is the mean, and \sigma is the standard deviation of the normal distribution.

Since you are given that Z follows a standard normal distribution, you can assume that \mu = 0 and \sigma = 1. Therefore, the formula simplifies to:

z = x

To find the probability that 0 < Z < 2.03, you need to find the area under the curve between these two values. You can do this by using the z-table.

First, look up the value 0 in the z-table. You will find that the probability that Z < 0 is 0.5. This means that half of the area under the curve is to the left of 0.

Next, look up the value 2.03 in the z-table. You will find that the probability that Z < 2.03 is 0.9788. This means that most of the area under the curve is to the left of 2.03.

To find the probability that 0 < Z < 2.03, you need to subtract these two probabilities:

P(0 < Z < 2.03) = P(Z < 2.03) - P(Z < 0)

P(0 < Z < 2.03) = 0.9788 - 0.5

P(0 < Z < 2.03) = 0.4788

Therefore, the probability that 0 < Z < 2.03 is approximately 0.4788. This means that about 47.88\% of the values in a standard normal distribution are between 0 and 2.03.

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the complete question is:

Find The Specified Probability. Round Your Answer To Four Decimal Places, If Necessary. P(0<Z&Lt;2.03)

The correct label is ∠CBA.

The correct label for the angle formed by rays BC and BA is ∠CBA. When  Angles labeling , it is important to consider the vertex of the angle, which is the point where the two rays meet. The vertex is usually labeled with a capital letter, and the angle itself is labeled with three letters, with the vertex letter in the middle. In this case, the vertex is at point B, and the two rays are BC and BA. Therefore, the angle is labeled as ∠CBA. It is important to use the correct labeling when communicating about angles in mathematics, as it ensures clarity and accuracy in solving problems and expressing ideas.

The correct label for the angle formed by rays BC and BA is ∠CBA.

Therefore, in this case, the correct label for the angle formed by rays BC and BA is ∠CBA.

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Answer:

The answer is <CBA

Step-by-step explanation:

Bc if the vertex is on "b" its going to be {bc} and {ba}. and sinces the vertex is in the middle in the image so should the letter. So its <CBA.

hope this helps

The probability that the person selected is older than 20 years old and watching the drama movie is 0.765.

We have,

The total number of people who watch drama.

= 12 + 20 + 19

= 51

The total number of people who is older than 20 years who watch drama.

= 20 + 19

= 39

Now,

The probability that the person selected is older than 20 years old and watching the drama movie.

= 39/51

= 0.765

Thus,

The probability that the person selected is older than 20 years old and watching the drama movie is 0.765.

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The percentage of sites with either purple loosestrife or common reed present is 67%.

Write down the formula to calculate the probability of the union (or) of two events:

P(A or B) = P(A) + P(B) - P(A and B)

This formula says that to find the probability of A or B occurring, you need to add the probability of A occurring, the probability of B occurring, and then subtract the probability of both A and B occurring at the same time.

This is because if you simply add the probabilities of A and B, you would be double-counting the cases where A and B both occur.

Identify the probabilities given in the problem statement:

P(Purple loosestrife) = 0.35

P(Common reed) = 0.55

P(Purple loosestrife and Common reed) = 0.23

Substitute the probabilities into the formula for P(A or B):

P(Purple loosestrife or Common reed) = P(Purple loosestrife) + P(Common reed) - P(Purple loosestrife and Common reed)

P(Purple loosestrife or Common reed) = 0.35 + 0.55 - 0.23

Simplify the expression:

P(Purple loosestrife or Common reed) = 0.67

Convert the probability to a percentage by multiplying by 100:

P(Purple loosestrife or Common reed) = 67%

Therefore, the percentage of sites with either purple loosestrife or common reed present is 67%.

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The equation of the line with a gradient of 2/3 that passes through point C in the form ax + by + c = 0 is 2x - 3y + 24 = 0.

Let's begin by finding the coordinates of point C, which is where the line y = 1/2x + 6 intersects the x-axis. Since the x-axis has a y-coordinate of 0, we can substitute y = 0 into the equation of the line and solve for x:

0 = 1/2x + 6

-6 = 1/2x

-12 = x

So point C is (-12, 0). Now we need to find the equation of a line with a slope of 2/3 that passes through point C. We can use the point-slope form of a linear equation:

y - y₁ = m(x - x₁)

where m is the slope and (x₁, y₁) is a point on the line. We substitute m = 2/3 and (x₁, y₁) = (-12, 0) to get:

y - 0 = 2/3(x - (-12))

y = 2/3x + 8

This is the equation of the line we were asked to find, but it's not in the form ax + by + c = 0. To convert it to that form, we can rearrange the terms:

2/3x - y + 8 = 0

Multiplying both sides by 3 to get rid of the fraction, we get:

2x - 3y + 24 = 0

So the final answer is a = 2, b = -3, and c = 24.

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a) The probability of the sample proportion of voters against the mayor being less than 50% is 0.8461 or about 84.61%.

b) The maximum proportion of voters in favor of the reelection that would result in the lowest 30% of groups of 50 voters being against the reelection is 0.4097 or about 40.97%.

Using the normal approximation to the binomial distribution, we can find the probability of the sample proportion of voters against the mayor being less than 50% as follows:

First, we need to calculate the mean and standard deviation of the sampling distribution:

Mean (μ) = p = 0.4

Standard deviation (σ) = =√(p(1-p)/n) = √(0.4*0.6/40) = 0.09798

Next, we need to standardize the sample proportion using the formula z = (x - μ)/σ, where x is the sample proportion. We want to find the probability that z is less than (0.5 - 0.4)/0.09798 = 1.02. Using a standard normal distribution table or calculator, we find that the probability is approximately 0.8461.

for b), We want to find the maximum proportion of voters in favor of the reelection that would result in the lowest 30% of groups of 50 voters being against the reelection.

We can use the binomial distribution to find the probability that in a group of 50 voters, the number of voters against the reelection is greater than or equal to 25 (50% of the sample). We can then find the maximum proportion of voters in favor of reelection such that this probability is less than or equal to 0.3.

Using a binomial distribution calculator or formula, we find that the probability of 25 or more voters being against the reelection in a group of 50 voters is approximately 0.0747. We want this probability to be less than or equal to 0.3, so we need to find the maximum value of p such that P(X >= 25) <= 0.3.

Using a binomial distribution table or calculator, we can find that the maximum value of p is approximately 0.4097.

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Complete Question:

Suppose that 40% of voters in a city are in favor of re-election of the current mayor. a) What is the probability that the sample proportion of voters against the mayor is less than 50%, in a sample of 40 voters? b) What is the maximum proportion of voters in favor of re-election that could be observed in the lowest 30% of groups of 50 voters towards re-election?

Yes, it is possible for a connected graph with 7 vertices and 10 edges to be drawn so that no edges cross and create 4 faces.

A planar graph is a graph that can be drawn in a plane without any edges crossing. According to Euler's formula for planar graphs, V - E + F = 2, where V represents vertices, E represents edges, and F represents faces. In this case, we have V = 7 and E = 10, and we want to find out if there can be a graph with F = 4.

Substituting the values into Euler's formula, we get:

7 - 10 + F = 2

Solving for F, we find:

F = 2 - 7 + 10
F = 5 - 7
F = 4

Since the formula holds true, it is possible to draw a connected graph with 7 vertices, 10 edges, and 4 faces without any edges crossing.

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Dot plots are better suited for maintaining the identity of individual observations, especially in smaller data sets, while histograms are useful for visualizing the distribution of larger data sets, even though they lose the identity of each specific data point.

Regarding dot plots and histograms, the true statement is that dot plots do not lose the identity of individual observations. Dot plots display each data point as a dot on a number line or axis, preserving information about individual data points. This is especially useful when dealing with small to moderate-sized data sets, as it allows for easy identification of patterns, clusters, or outliers.

On the other hand, histograms are a graphical representation that organizes data into intervals or bins, which can provide an overview of the distribution of a larger data set. While histograms are often easier to construct and can help visualize patterns and trends for large data sets, they lose the identity of individual observations, as the data points are grouped together in bins.

In summary, dot plots are better suited for maintaining the identity of individual observations, especially in smaller data sets, while histograms are useful for visualizing the distribution of larger data sets, even though they lose the identity of each specific data point.

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An equation that could be used as a line of best fit to approximate the data in the scatterplot is y = 0.601x + 21.757.

In order to determine an equation for the line of best fit that models the data points contained in the graph (scatter plot), we would have to use a graphing calculator (Microsoft Excel).

Based on the scatter plot (see attachment) which models the relationship between the x-values and y-values, an equation for the line of best fit is given by:

y = 0.601x + 21.757

In conclusion, we can reasonably infer and logically deduce that the scatter plot most likely indicates a linear relationship between the x-values and y-values.

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The boundary value problem for the temperature u(x,t) of the rod is:

∂u/∂t = [tex]\alpha^2[/tex]∂[tex]^2u[/tex]/∂[tex]x^2[/tex] + f(x,t).

To set up the boundary value problem for the temperature u(x,t) of the rod, we need to consider the heat equation, which is given by:

ρc∂u/∂t = ∂/∂x (k∂u/∂x) + Q

where ρ is the density, c is the specific heat, k is the thermal conductivity, Q is the heat source or sink, and u(x,t) is the temperature at position x and time t.

Assuming that the rod is homogeneous and has constant density and specific heat, we can simplify the heat equation to:

∂u/∂t = [tex]\alpha^2[/tex]∂[tex]^2u[/tex]/∂[tex]x^2[/tex] + f(x,t)

where [tex]\alpha^2[/tex] = k/ρc is the thermal diffusivity and f(x,t) = Q/ρc is the heat source or sink per unit volume.

The boundary conditions for the rod are:

u(0,t) = 0 (left end held at temp zero)

∂u(L,t)/∂x = 0 (right end insulated)

The initial condition for the rod is:

u(x,0) = f(x) (initial temp is f(x) throughout)

Therefore, the boundary value problem for the temperature u(x,t) of the rod is:

∂u/∂t = [tex]\alpha^2[/tex]∂[tex]^2u[/tex]/∂[tex]x^2[/tex] + f(x,t)

subject to the boundary conditions:

u(0,t) = 0

∂u(L,t)/∂x = 0

and the initial condition:

u(x,0) = f(x)

This is a well-posed boundary value problem that can be solved using appropriate analytical or numerical techniques.

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The approximate cost per square inch of the cookie cake is $0.25 per square inch. Then the correct option is D.

Given that:

Diameter, d = 8 inches

Let d be the diameter of the circle. Then the area of the circle will be

A = πd²/4 square units

The area of the cake is calculated as,

A = 3.14 x 8 x 8 / 4

A = 50.24 square inches

The approximate cost per square inch of the cookie cake is calculated as,

Cost = $12.56 / 50.24

Cost = $0.25 per square inch

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Answer: The new price of the rice in March 2023 was $180.

Step-by-step explanation:

> 20% of $150 is 0.20 ($150) = $30

> The price of the rice increased by $30, so the new price is:

> $150 + $30 = $180

Therefore, the new price of the rice in March 2023 was $180.

The maximum value of P subject to the given constraints is 9.

To solve this problem, we can use the method of linear programming. We need to maximize the objective function P = 2x1 + 3x2 + x3 subject to the constraints:

x1 + x2 + x3 ≤ 4

2x1 + x2 - x3 ≤ 0

x1, x2, x3 ≤ 10

x1, x2, x3 ≥ 0

We can start by graphing the feasible region defined by the constraints:

  x3

  |

10 |\

  | \

  |  \   x1 + x2 + x3 <= 4

  |   \

4 |    \ 2x1 + x2 - x3 <= 0

  |     \

  |      \

  |       \

  |________\

  0  10  20 x1,x2

The feasible region is a polygon with vertices at (0,0,4), (0,2,2), (1,1,2), (2,0,0), and (0,0,0). We can then evaluate the objective function P = 2x1 + 3x2 + x3 at each vertex:

P(0,0,4) = 4

P(0,2,2) = 8

P(1,1,2) = 9

P(2,0,0) = 4

P(0,0,0) = 0

We can see that the maximum value of P is 9, which occurs at the vertex (1,1,2). Therefore, the maximum value of P subject to the given constraints is 9.

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The range of (uºv)(x) can be represented in interval notation as: [tex]\mathbf{(-\infty,3)}[/tex]

In a function, the range is the set of all valid values of y and this can be better determined from the graphical representation of the given function.

Here, we are given:

u(x) = -2x² +3

v(x) = 1/x

To find (uºv)(x) which can be written as u(v(x)), we need to input all the values of v(x) into where we find the variable x in u(x), by doing so, we have:

u(1/x) = -2(1/x)² +3

u(1/x) = -2/x² +3

Now, the range of u(1/x) which is the set of all valid values of y can be represented in interval notation as: [tex]\mathbf{(-\infty,3)}[/tex]

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It is possible that blood pressure levels could change after listening to soothing music, but it is unclear from the data given whether this is the case or not.

A sample of 15 people is a relatively small sample size, so it may not be representative of the population as a whole. In addition, it is not clear what method was used to select the sample, so there may be issues with sampling bias.

Furthermore, the presence of an outlier in the data could indicate that there are other factors influencing the change in blood pressure, such as an underlying medical condition or a reaction to a specific type of music. This outlier could also significantly affect the overall results of the study, making it difficult to draw reliable conclusions.

Therefore, it is not possible to determine from the information given whether or not blood pressure levels change after listening to soothing music. A larger and more carefully selected sample would be needed to provide more reliable results.

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Answer:

square root of (a^2 + b^2)

Step-by-step explanation:

According to the pythagorean theorem, the hypotenuse of a triangle is equal to its two shortest sides squared and added together. Then you get the square root to get rid of the squaring done in the equation.

We do not have enough evidence to conclude that the population mean height of bass singers is greater than the population mean height of tenor singers

We can conduct a two-sample t-test to determine if the population mean height of bass singers is greater than the population mean height of tenor singers.

The null hypothesis is that there is no difference between the population means, while the alternative hypothesis is that the population mean height of bass singers is greater than the population mean height of tenor singers.

Let's calculate the t-statistic:

t = (xb - xt) / sqrt(s^2/nb + s^2/nt)

where xb and xt are the sample means, sb and st are the sample standard deviations, and nb and nt are the sample sizes.

Plugging in the given values, we get:

t = (70.99 - 69.41) / sqrt((2.52)^2/64 + (2.79)^2/42) = 2.18

Using a two-tailed t-distribution table with degrees of freedom of 64+42-2=104 and a significance level of 0.01, we find the critical t-value to be 2.364.

Since our calculated t-value of 2.18 is less than the critical t-value of 2.364, we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that the population mean height of bass singers is greater than the population mean height of tenor singers at a 1% level of significance.

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To maximize the area of the rectangular garden, we need to find its dimensions using the given 120 feet of fencing.

Let's assume the length of the garden to be x and the width to be y.

Since there are three sides that need fencing, we can write the equation:

2x + y = 120
Solving for y, we get y = 120 - 2x.
Now, we can write the area of the rectangle as A = xy.
Substituting y in terms of x, we get A = x(120-2x) = 120x - 2x^2.

To maximize the area, we need to find the value of x that gives the highest value of A. To do this, we can take the derivative of A with respect to x and set it equal to zero.
dA/dx = 120 - 4x = 0
Solving for x, we get x = 30.

Therefore, the dimensions of the rectangular garden that maximize its area are 30 feet by 60 feet.

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Answer:

(e = 2u - 220) I need to write at least 20 characters to post this only read what is in the parenthesis

Equations that can be used to solve for y, the length of the room is [tex]y^{2}[/tex] - 5y = 750, y(y - 5) = 750 and 750 - y(y - 5) = 0.

Let's assume that the length of the room is y, then the width of the room will be y - 5 (as per the given information).

The area of the rectangular room can be calculated as the product of its length and width, i.e., y(y - 5) = 750.

Now we can simplify this equation to a quadratic equation by bringing all the terms to one side:

[tex]y^{2}[/tex] - 5y - 750 = 0

So, the equations that can be used to solve for y, the length of the room are:

y^2 - 5y - 750 = 0 (This is the simplified quadratic equation)

y(y - 5) = 750 (This is the original equation obtained from the area formula)

750 - y(y - 5) = 0 (This is the same as the equation in option 2, but with terms rearranged)

Therefore, the correct options are:

[tex]y^{2}[/tex] - 5y = 750

y(y - 5) = 750

750 - y(y - 5) = 0

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To determine the correct two-column table for the given data of Tamara's reading pages and time taken, we need to compare the given data with the values in each row of tables. The table with "Pages Time: 25 36, 63 48, 74.5 52" is the correct one. So, the correct answer is C).

Identify the data given, Tamara read 25 pages in 36 minutes, 48 pages in 63 minutes, and 52 pages in 74.5 minutes.

Based on the given data, create a two-column table that has one column for the number of pages Tamara read and another column for the time it took her to read them.

Compare the values in each row of the table to the given data to make sure they match.

The first table, "Pages Time: 25 36, 63 48, 74.5 52" matches the given data and has two columns for the number of pages and the time taken to read them, so it is the correct answer.

The other tables do not match the given data or do not have two columns for the number of pages and the time taken to read them.

Therefore, the table "Pages Time: 25 36, 63 48, 74.5 52" is the correct two-column table for the given data. So, the correct option is C).

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We can prove that S(n+1,k)=S(n,k−1)+kS(n,k).

To prove that S(n+1,k)=S(n,k−1)+kS(n,k), we will use combinatorial argument.

Consider the set {1,2,...,n+1}. We want to partition this set into k unlabelled nonempty parts. There are two cases to consider:

Case 1: The element n+1 belongs to a part of size 1.

In this case, we have n elements to partition into k-1 parts. The number of ways to do this is S(n,k-1) since we are partitioning n elements into k-1 parts.

Case 2: The element n+1 belongs to a part of size m>1.

In this case, we have n elements to partition into k parts, with one part having size m-1. There are k ways to choose the part of size m-1, and m-1 ways to choose the element of that part that will be n+1. The remaining n-m+1 elements are partitioned into k-1 parts. The number of ways to do this is k(m-1)S(n-m+1,k-1).

Therefore, the total number of partitions of {1,2,...,n+1} into k unlabelled nonempty parts is S(n,k-1)+kS(n,k), which proves the desired formula S(n+1,k)=S(n,k−1)+kS(n,k).

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