Which expression is NOT equivalent to (25 x⁴y)¹/³ ?

a. x ³√25xy

b. 5 x ³√xy

c. ³√25x⁴y

d. ⁶√625 x⁸y²

The equation of the line perpendicular to the tangent to the curve y=x^4+x-1 at the point (1,1) is y = -1x + 2.

To find the equation of the line perpendicular to the tangent, we first need to find the slope of the tangent line. The slope of the tangent line is equal to the derivative of the curve at the given point. Taking the derivative of y=x^4+x-1, we get y'=4x^3+1. Substituting x=1 into the derivative, we get y'=4(1)^3+1=5.

The slope of the tangent line is 5. To find the slope of the perpendicular line, we use the fact that the product of the slopes of perpendicular lines is -1. Therefore, the slope of the perpendicular line is -1/5.

Next, we use the point-slope form of a line to find the equation. Using the point (1,1) and the slope -1/5, we have y-1=(-1/5)(x-1). Simplifying this equation gives us y = -1x + 2. Thus, the equation of the line perpendicular to the tangent to the curve y=x^4+x-1 at the point (1,1) is y = -1x + 2.

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The mean e(t) is 48.393 and the standard deviation sd(t) is approximately 2.850.

To determine the mean e(t) and standard deviation Sd(t) of the relay team's total time in the medley event, we need to calculate the weighted average and standard deviation of each swimmer's time.

The mean e(t) is calculated by adding up the product of each swimmer's time and the corresponding weight (number of strokes) and dividing it by the total weight.

[tex]e(t) = (49.39 * 2 + 55.67 * 1 + 48.76 * 3 + 45.8 * 4) / (2 + 1 + 3 + 4)[/tex]

Calculating this expression, we get:

[tex]e(t) = (98.78 + 55.67 + 146.28 + 183.2) / 10[/tex]

[tex]e(t) = 483.93 / 10[/tex]
[tex]e(t) = 48.393[/tex]

So, the mean e(t) is 48.393.

To calculate the standard deviation sd(t), we need to find the weighted variance and then take its square root.

First, we calculate the weighted variance by summing up the products of each swimmer's squared time difference from the mean and the corresponding weight, and dividing it by the total weight.

variance(t) = [tex][(2 * (49.39 - 48.393)^2) + (1 * (55.67 - 48.393)^2) + (3 * (48.76 - 48.393)^2) + (4 * (45.8 - 48.393)^2)] / (2 + 1 + 3 + 4)[/tex]

Calculating this expression, we get:

variance(t) =[tex][2 * (0.997)^2 + 1 * (7.277)^2 + 3 * (0.367)^2 + 4 * (-2.593)^2] / 10[/tex]
variance(t) =[tex][2 * 0.994 + 1 * 52.94 + 3 * 0.135 + 4 * 6.713] / 10[/tex]
variance(t) =[tex](1.988 + 52.94 + 0.405 + 26.852) / 10[/tex]

variance(t) =[tex]81.185 / 10[/tex]

variance(t) = [tex]8.119[/tex]

Finally, taking the square root of the variance gives us the standard deviation sd(t):

[tex]sd(t) = √8.119[/tex]

[tex]sd(t) ≈ 2.850[/tex]

So, the standard deviation sd(t) is approximately 2.850.

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To determine a rejection region for a test at level α using the fact that the sum of independent Poisson random variables follows a Poisson distribution, we calculate the critical values based on the desired significance level α and compare them with the observed sum of Poisson variables.

To determine a rejection region for a test at level α using the fact that the sum of independent Poisson random variables follows a Poisson distribution, we can follow these steps:

Specify the null and alternative hypotheses: Determine the null hypothesis (H0) and the alternative hypothesis (Ha) for the statistical test. These hypotheses should be stated in terms of the parameters being tested.

Choose the significance level (α): The significance level α represents the maximum probability of rejecting the null hypothesis when it is true. It determines the probability of making a Type I error (rejecting H0 when it is actually true). Common choices for α are 0.05 or 0.01.

Determine the test statistic: Select an appropriate test statistic that follows a Poisson distribution based on the data and hypotheses being tested. The test statistic should be able to capture the effect or difference being examined.

Calculate the critical region: The critical region is the set of values of the test statistic for which the null hypothesis will be rejected. To determine the critical region, we need to find the values of the test statistic that correspond to the rejection region based on the significance level α.

Use the Poisson distribution: Since the sum of independent Poisson random variables follows a Poisson distribution, we can utilize the Poisson distribution to determine the probabilities associated with different values of the test statistic. We can calculate the probabilities for the test statistic under the null hypothesis.

Compare the probabilities: Compare the probabilities calculated under the null hypothesis with the significance level α. If the calculated probability is less than or equal to α, it falls in the rejection region, and we reject the null hypothesis. Otherwise, if the probability is greater than α, it falls in the acceptance region, and we fail to reject the null hypothesis.

It is important to note that the specific details of determining the rejection region and performing hypothesis testing depend on the specific test being conducted, the data at hand, and the nature of the hypotheses being tested. Different tests and scenarios may require different approaches and considerations.

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The required answer is the volume of the sphere is approximately 65 cubic inches.

To find the volume of the sphere inscribed in a cube with a volume of 125 cubic inches,  the formula for the volume of a sphere.

The volume of a sphere is given by the formula V = (4/3) * π * r^3, where r is the radius of the sphere.

In this case, since the sphere is inscribed in the cube, the diameter of the sphere is equal to the side length of the cube. the side length of the cube as s.

Since the volume of the cube is 125 cubic inches, we have s^3 = 125.

Taking the cube root of both sides gives us s = 5.

Therefore, the diameter of the sphere is 5 inches, and the radius is half of the diameter, which is 2.5 inches.

Plugging the value of the radius into the volume formula, we get V = (4/3) * π * (2.5)^3.

Evaluating this expression gives us V ≈ 65.4 cubic inches.

Rounding this answer to the nearest whole number, the volume of the sphere is approximately 65 cubic inches.

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The value of the missing angle x is approximately 26.015 degrees.

Real functions called trigonometric functions link the angle of a right-angled triangle to the ratios of its two side lengths. The sine, cosine, tangent, cotangent, secant, and cosecant are the six trigonometric functions. These formulas reflect the right triangle side ratios.

To determine the value of the missing angle, we can use the fact that the sine function and cosine function are related in a right triangle.

Given that sin(26) = 0.4384, we can find the value of the missing angle by using the inverse sine function (also known as arcsine). Let's denote the missing angle as x.

sin(x) = 0.4384

Taking the inverse sine of both sides:

x = arcsin(0.4384)

Using a calculator, we can find the approximate value of arcsin(0.4384) to be approximately 26.015 degrees.

Therefore, the angle x that is lacking has a value of about 26.015 degrees.

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The directional derivative of f(x, y) = ln(6 + x² + y²) at the point P(-2, 1) in the direction of the vector (3, 2) is -8 / (9 √(13)).

To compute the directional derivative of the function f(x, y) = ln(6 + x² + y²) at the point P(-2, 1) in the direction of the given vector (3, 2), we need to calculate the dot product of the gradient of f at P and the unit vector in the direction of (3, 2).

First, let's find the gradient of f(x, y):

∇f(x, y) = (∂f/∂x, ∂f/∂y)

Taking partial derivatives:

∂f/∂x = 2x / (6 + x² + y²)

∂f/∂y = 2y / (6 + x² + y²)

Now, let's evaluate the gradient at the point P(-2, 1):

∇f(-2, 1) = (2(-2) / (6 + (-2)² + 1²), 2(1) / (6 + (-2)² + 1²))

          = (-4 / 9, 2 / 9)

Next, we need to calculate the unit vector in the direction of (3, 2):

Magnitude of (3, 2) = sqrt(3² + 2²) = √(13)

Unit vector = (3 / √(13), 2 / √(13))

Finally, we take the dot product of the gradient and the unit vector to find the directional derivative:

Directional derivative = ∇f(-2, 1) · (3 / sqrt(13), 2 / sqrt(13))

                     = (-4 / 9)(3 / √(13)) + (2 / 9)(2 / √(13))

                     = (-12 / (9 √(13))) + (4 / (9 √(13)))

                     = -8 / (9 √(13))

Therefore, the directional derivative of f(x, y) = ln(6 + x² + y²) at the point P(-2, 1) in the direction of the vector (3, 2) is -8 / (9 √(13)).

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1. Calculate the gradient of the function at point p. The gradient is a vector that points in the direction of the steepest increase of the function at that point.

2. Normalize the given direction vector to obtain a unit vector. To normalize a vector, divide each of its components by its magnitude.

3. Compute the dot product between the normalized direction vector and the gradient vector. The dot product measures the projection of one vector onto another. This gives us the magnitude of the directional derivative.

4. To find the actual directional derivative, multiply the magnitude obtained in step 3 by the magnitude of the gradient vector. This accounts for the rate of change of the function in the direction of the given vector.

5. The directional derivative represents the rate of change of the function at point p in the direction of the given vector. It indicates how fast the function is changing in that particular direction.

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To calculate z-scores, use the formula z1 = (x1 - mean) / standard deviation and z2 = (x2 - mean) / standard deviation. Use a standard normal table or calculator to find the proportion of measurements between z1 and z2.Using a standard normal table or a calculator, we can find the proportion of measurements between -0.769 and 0.769.

To find the proportion of measurements in a given interval, we can use the properties of the normal distribution. Since the distribution is mound-shaped, we can assume that it follows the normal distribution.

First, we need to determine the z-scores for the lower and upper bounds of the given interval. The z-score formula is given by: z = (x - mean) / standard deviation.

Let's say the lower bound of the interval is x1 and the upper bound is x2. To find the proportion of measurements between x1 and x2, we need to find the area under the normal curve between the corresponding z-scores.

To calculate the z-scores, we use the formula:
z1 = (x1 - mean) / standard deviation
z2 = (x2 - mean) / standard deviation

Once we have the z-scores, we can use a standard normal table or a calculator to find the proportion of measurements between z1 and z2.

For example, if x1 = 50 and x2 = 70, the z-scores would be:
z1 = (50 - 60) / 13 = -0.769
z2 = (70 - 60) / 13 = 0.769

Using a standard normal table or a calculator, we can find the proportion of measurements between -0.769 and 0.769.

Note: Since the question does not specify the specific interval, I have provided a general approach to finding the proportion of measurements in a given interval based on the mean and standard deviation. Please provide the specific interval for a more accurate answer.

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In this particular scenario, if the height of the cone is doubled while the radius remains the same, the volume of the cone will be doubled as well.

The volume of a cone can be calculated using the formula V = (1/3)πr²h, where V represents the volume, r is the radius, and h is the height of the cone.

In the given scenario, the cone has a radius of 4 centimeters and a height of 9 centimeters. If we consider the initial volume of the cone as V₁, we can calculate it using the formula: V₁ = (1/3)π(4²)(9) = (1/3)π(16)(9) = 48π cm³.

Now, let's consider the situation where the height is doubled. In this case, the new height would be 2 times the original height, which is 2(9) = 18 centimeters. Let's denote the new volume of the cone as V₂. Using the formula, we can calculate it as follows: V₂ = (1/3)π(4²)(18) = (1/3)π(16)(18) = 96π cm³.

Comparing the two volumes, we have V₂ = 96π cm³ and V₁ = 48π cm³. The ratio of V₂ to V₁ is 96π/48π = 2. This indicates that the volume of the cone is indeed doubled when the height is doubled.

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The expression is [tex]∑(i=1 to n) (a * x_i + b * y_i + c) = a * ∑(i=1 to n) x_i + b * ∑(i=1 to n) y_i + c * n[/tex]

To show that the given expression is true, we will use the properties of summation notation. Let's break it down step-by-step:

1. Start by expanding the left side of the equation using the properties of summation:
[tex]a * x_1 + b * y_1 + c + a * x_2 + b * y_2 + c + ... + a * x_n + b * y_n + c[/tex]

2. Now, group the terms together based on their constants (a, b, and c):
[tex](a * x_1 + a * x_2 + ... + a * x_n) + (b * y_1 + b * y_2 + ... + b * y_n) + (c + c + ... + c)[/tex]

3. Observe that each sum within the parentheses represents the summation of the sequences x_i, y_i, and a sequence of c's respectively:
[tex]a * ∑(i=1 to n) x_i + b * ∑(i=1 to n) y_i + c * n[/tex]

4. This matches the right side of the equation, which proves that the given expression is true.

Therefore, we have shown that:
[tex]∑(i=1 to n) (a * x_i + b * y_i + c) = a * ∑(i=1 to n) x_i + b * ∑(i=1 to n) y_i + c * n.[/tex]

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The dataset representing total points scored during recent football games is as follows: 12, 14, 15, 17, 17, 21, 24, 25, 27, 31, 33 so  the quartiles of the given dataset are Q1 = 15, Q2 = 21, and Q3 = 27.

To determine the quartiles of this dataset, we need to find the values that divide the dataset into four equal parts. The first quartile (Q1) represents the 25th percentile, the second quartile (Q2) represents the 50th percentile (also known as the median), and the third quartile (Q3) represents the 75th percentile.

To find the quartiles, we first need to arrange the dataset in ascending order: 12, 14, 15, 17, 17, 21, 24, 25, 27, 31, 33.

There are a total of 11 data points in the dataset. To find the median (Q2), we take the middle value. Since there are 11 data points, the middle value is the 6th value, which is 21. Therefore, Q2 (the median) is 21.

To find Q1, we need to locate the 25th percentile. This means that 25% of the data points in the dataset should be below Q1. Since 25% of 11 is 2.75, we round it up to 3. The third value in the dataset is 15, so Q1 is 15.

To find Q3, we locate the 75th percentile, which means that 75% of the data points should be below Q3. 75% of 11 is 8.25, which we round up to 9. The ninth value in the dataset is 27, so Q3 is 27.

Therefore, the quartiles of the given dataset are Q1 = 15, Q2 = 21, and Q3 = 27.

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ANCOVA allows for the comparison of mean differences while controlling for the influence of covariates. This analysis will help Erin assess the impact of the after-school program on the outcomes while accounting for potential differences in the schools attended.

To examine the outcomes of an after-school program and account for the potential impact of the school the youth attend, Erin can use a statistical analysis called Analysis of Covariance (ANCOVA). ANCOVA is suitable when there is a need to control for the effect of a covariate, in this case, the school attended.

Erin can conduct a pre-assessment in September to gather baseline data and then a post-assessment in May to measure the program's effectiveness. Along with these assessments, Erin should also collect information about the school attended by each student. By including the school as a covariate in the analysis, Erin can determine whether any observed differences in the program outcomes are due to the after-school program itself or other factors related to the school.

ANCOVA allows for the comparison of mean differences while controlling for the influence of covariates. This analysis will help Erin assess the impact of the after-school program on the outcomes while accounting for potential differences in the schools attended.

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The probability that a randomly chosen young adult has at least a high school education can be found using the rule of probability called the "complement rule".

To find the answer, we need to subtract the probability that a randomly chosen young adult does not have at least a high school education from 1. In other words:

Probability of having at least a high school education = 1 - Probability of not having at least a high school education.

By using this rule, we can calculate the probability.

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We can see that when the radius and slant height are doubled, the surface area of the cone is quadrupled.

If both the radius and the slant height of a cone are doubled, the surface area of the cone will be affected as follows:

The surface area of a cone can be calculated using the formula:

[tex]\[A = \pi r (r + l)\][/tex]

where [tex]\(A\)[/tex] represents the surface area, [tex]\(r\)[/tex] is the radius, and [tex]\(l\)[/tex] is the slant height.

When the radius and slant height are doubled, the new values become [tex]\(2r\)[/tex] and [tex]\(2l\)[/tex] respectively.

Substituting these new values into the surface area formula, we have:

[tex]\[A' = \pi (2r) \left(2r + 2l\right)\][/tex]

Simplifying further:

[tex]\[A' = \pi (2r) \left(2(r + l)\right)\][/tex]

[tex]\[A' = 4 \pi r (r + l)\][/tex]

Comparing this new surface area [tex]\(A'\)[/tex] to the original surface area [tex]\(A\),[/tex] we can see that when the radius and slant height are doubled, the surface area of the cone is quadrupled.

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Using pascal's triangle, the fifth term of the binomial expansion of [tex](x-y)^5[/tex] is [tex]-5yx^4[/tex].

Below is the image attached of pascal's triangle. Pascal's triangle is a triangular array of the binomial coefficients arising in probability theory, combinatorics, and algebra.

To find the expansion of [tex](x-y)^5[/tex], we need the 5th row of the pascal's triangle.

The expansion becomes,

[tex](1)(-y^5)(x^0)+(5)(-y^4)(x^1)+(10)(-y^3)(x^2)+(10)(-y^2)(x^3) +(5)(-y)(x^4)+(x^5)[/tex]

The fifth term becomes, [tex]-5yx^4[/tex].

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You can substitute the value of x into the formula to calculate the probability for any specific number of no-change audits.

To determine the probability that the sample has a specific number of no-change audits, we can use the binomial probability formula.

The binomial probability formula is given by:

[tex]P(X = k) = C(n, k) * p^k * (1 - p)^{(n - k)}[/tex]

Where:

P(X = k) is the probability of having exactly k successes (in this case, no-change audits),

n is the sample size,

k is the number of successes,

p is the probability of success in a single trial (in this case, the probability of a no-change audit), and

C(n, k) is the binomial coefficient, also known as "n choose k," which represents the number of ways to choose k successes from n trials.

In this scenario, n = 200 (sample size) and p = 0.9 (probability of no-change audit). We want to calculate the probability of having a specific number of no-change audits. Let's say we want to find the probability of having x no-change audits.

[tex]P(X = x) = C(200, x) * 0.9^x * (1 - 0.9)^{(200 - x)}[/tex]

Now, let's calculate the probability of having a specific number of no-change audits for different values of x. For example, if we want to find the probability of having exactly 180 no-change audits:

[tex]P(X = 180) = C(200, 180) * 0.9^{180} * (1 - 0.9)^{(200 - 180)}[/tex]

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John wanted to bring attention to the fact that litter was getting out of hand at his neighborhood park. He created a poster where giant pieces of trash came to life and stomped on the park. The type of humor that he used in the poster is exaggeration.

What is exaggeration?

Exaggeration is the action of describing or representing something as being larger, better, or worse than it genuinely is. It is a representation of something that is far greater than reality or what the person is used to.

In this case, John used an exaggerated approach to convey the message that litter was getting out of hand in the park.

Incongruity: This is a type of humor that involves something that doesn't match the situation.

Parody: This is a type of humor that involves making fun of something by imitating it in a humorous way.

Reversal: This is a type of humor that involves changing the expected outcome or situation.

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Answer:

The type of satire that John used in his poster is exaggeration.Exaggeration is a technique used in satirical writing, art, or speech that highlights the importance of a certain issue by making it seem bigger than it actually is. It is used to make people aware of a problem or issue by amplifying it to the point of absurdity.In the case of John's poster, he exaggerated the issue of litter by making it appear as if giant pieces of trash were coming to life and stomping on the park, which highlights the importance of keeping the park clean.

To construct an equilateral triangle inscribed in a circle, Zahid would need to draw three specific segments.

First, Zahid would need to draw the radius of the circle, which is a line segment connecting the center of the circle to any point on its circumference. This segment serves as the base of the equilateral triangle.

Next, Zahid would draw two more line segments from the endpoints of the base (radius) to another point on the circumference of the circle. These segments should be of equal length and form angles of 60 degrees with the base. These segments complete the equilateral triangle by connecting the remaining two vertices. Zahid needs to draw the radius of the circle (base of the equilateral triangle) and two additional line segments connecting the endpoints of the radius to other points on the circle's circumference. These line segments should be equal in length and form angles of 60 degrees with the base.

It is important to note that an equilateral triangle is a special case where all sides are equal in length and all angles are 60 degrees. In the context of a circle, an equilateral triangle is inscribed when all three vertices lie on the circumference of the circle.

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The expression for Cam's amount would be: bbbb dollars - 7777 dollars.

To find the number of dollars Cam has, we need to subtract 7777 from Ben's amount.

Let's represent Ben's amount as "bbbb dollars."
The expression for Cam's amount would be: bbbb dollars - 7777 dollars.

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A line chart is a special type of scatter plot in which the data points in the scatter plot are connected with a line. A line chart is a graphical representation of data that is used to display information that changes over time. The line chart is also known as a line graph or a time-series graph. The data points are plotted on a grid where the x-axis represents time and the y-axis represents the value of the data.

The data points in the scatter plot are connected with a line to show the trend or pattern in the data. Line charts are commonly used to visualize data in business, economics, science, and engineering.Line charts are useful for displaying information that changes over time. They are particularly useful for tracking trends and changes in data. Line charts are often used to visualize stock prices,

sales figures, weather patterns, and other types of data that change over time. Line charts are also used to compare two or more sets of data. By plotting multiple lines on the same graph, you can easily compare the trends and patterns in the data.Overall, line charts are a useful tool for visualizing data and communicating information to others. They are easy to read, understand, and interpret, and can be used to display a wide range of data sets.

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The seasonal total prior to the recent storm was 76.42 inches.

To calculate the seasonal total prior to the recent storm, we need to subtract the rainfall from the recent storm (50 inches) from the updated seasonal total (26.42 inches).

Let's assume that the seasonal total prior to the recent storm is represented by "x" inches.

So, we can set up the equation:

x - 50 = 26.42

To solve for x, we can add 50 to both sides of the equation:

x - 50 + 50 = 26.42 + 50

This simplifies to:

x = 76.42

Therefore, the seasonal total prior to the recent storm was 76.42 inches.

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The percent of increase in the price of the basketball is approximately 10.4%.

When a sporting goods store raises the price of a basketball from $16.75 to $18.50,

the percent of increase in the price can be calculated using the percent increase formula which is given as:\[\% \text{ increase} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\]

Substituting the given values in the above formula,

we get:\[\% \text{ increase} = \frac{18.50 - 16.75}{16.75} \times 100\]\[\% \text{ increase} = \frac{1.75}{16.75} \times 100\]\[\% \text{ increase} = 10.4478...\]

To round this answer to the nearest tenth, we look at the second decimal place which is 4.

Since 4 is less than 5, we round down the first decimal place which gives us:\[\% \text{ increase} \approx 10.4\]

Therefore, the percent of increase in the price of the basketball is approximately 10.4%.

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The number of square tiles required to completely cover the play area is equal to the product of the length and width of the play area.

To determine the number of square tiles required to cover the play area, we need to know the dimensions of the play area. Specifically, we need to know the length and width of the play area.

Let's assume the length of the play area is L meters and the width is W meters.

The area of the play area can be calculated by multiplying the length and width:

Area = L * W

Since each square tile has sides of 1 meter, the area of each tile is 1 * 1 = 1 square meter.

To find the number of tiles required, we can divide the area of the play area by the area of each tile:

Number of tiles = Area / Area of each tile

Number of tiles = Area / 1

Therefore, the number of square tiles required to completely cover the play area is equal to the area of the play area.

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The student should subtract the correct value, -2, instead of -3  is the answer.

The student made an error while performing synthetic division. To correctly divide x³-x²-2x by x+1 using synthetic division, we start by writing the coefficients of the polynomial in descending order, which in this case are 1, -1, and -2. Next, we write the opposite of the divisor, which is -1, on the left side.

We then bring down the first coefficient, 1, and multiply it by -1, which gives us -1. Adding this result to the second coefficient, -1, we get -2. We then multiply -2 by -1, which gives us 2, and add it to the last coefficient, -2. The result is 0.

The correct division would be x²-2. So, the student's error was in the second step of synthetic division, where they incorrectly added -1 and -2 to get -3 instead of the correct result, which is -2. To correct the error, the student should subtract the correct value, -2, instead of -3.

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The total cost of the flowers, we need to multiply the cost of each flower by the total number of flowers. The given expression is D. 48x.

Let's assume that the cost of each flower is represented by the variable "x". Since all the flowers are identical, the cost of each flower is the same.

To find the total cost, we multiply the cost of each flower (x) by the total number of flowers (48):

Total cost = x * 48

So, the expression 48x represents the total cost of the flowers.

The correct answer is D).

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--The given question is incomplete, the complete question is given below " Susie purchased 48 identical flowers . Which expression represents the total cost of the flowers

A. 48+x B. 48 - x C. 48÷x D. 48x"--

Answer:

A

Step-by-step explanation:

Given quadratic function:

[tex]y=3x^2 - 9, \qquad x \leq 0[/tex]

The domain of the given function is restricted to values of x less than or equal to zero. Therefore:

As 3x² ≥ 0, then range of the given function is restricted to values of y greater than or equal to -9.

[tex]\hrulefill[/tex]

To find the inverse of the given function, first interchange the x and y variables:

[tex]x = 3y^2 - 9[/tex]

Now, solve the equation for y:

[tex]\begin{aligned}x& = 3y^2 - 9\\\\x+9&=3y^2\\\\\dfrac{x+9}{3}&=y^2\\y&=\pm \sqrt{\dfrac{x+9}{3}}\end{aligned}[/tex]

The range of the inverse function is the domain of the original function.

As the domain of the original function is restricted to x ≤ 0, then the range of the inverse function is restricted to y ≤ 0.

Therefore, the inverse function is the negative square root:

[tex]f^{-1}(x)=-\sqrt{\dfrac{x+9}{3}}[/tex]

The  domain of the inverse function is the range of the original function.

As the range of the original function is restricted to y ≥ -9, then the domain of the inverse function is restricted to x ≥ -9.

[tex]\boxed{f^{-1}(x)=-\sqrt{\dfrac{x+9}{3}}\qquad x \geq -9}[/tex]

So the correct statement is:

Approximately 0.12% of high school athletes will go on to play professional sports. What we are given is that about 9% of high school athletes proceed to play sports in college. And of these college athletes, only 1.3% will play professional sports. Now we have to calculate the probability of a high school athlete going on to play professional sports.

It is important to remember that only college athletes can go pro, so the probability we are looking for is the probability that a high school athlete will go on to play in college and then become a professional athlete. We can solve this by multiplying the two probabilities:

Probability of a high school athlete playing in college = 9% = 0.09Probability of a college athlete playing professionally = 1.3% = 0.013Probability of a high school athlete playing college and then professionally = (0.09) (0.013) = 0.00117 or 0.12% (rounded off to two decimal places)Therefore, the probability that a high school athlete will go on to play professional sports is approximately 0.12%.

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Answer:

Step-by-step explanation: The negative sign needs to be enclosed in parentheses if you want the result to be 9

If you write (-3)^2 the result is 9

and -3^2 = -9 is right

The ordered pair that can be added to the given set and still have the set represent the same linear function is (3, 1).

To determine which ordered pair can be added to the given set and still have the set represent the same linear function, we need to identify the pattern or rule governing the set. We can do this by examining the x and y values of the ordered pairs.

Looking at the x-values, we can see that they increase by 1 from -2 to 2. This suggests that the x-values follow a constant increment pattern.

Next, let's examine the y-values. We can see that they also increase by 2 from -9 to -1. This indicates that the y-values follow a constant increment pattern as well.

Based on these observations, we can conclude that the linear function rule is y = 2x - 5.

Now, let's check if the ordered pair (3, 1) follows this rule. Plugging in x = 3 into the linear function equation, we get y = 2(3) - 5 = 1. Since the y-value matches, we can add (3, 1) to the given set and still have the set represent the same linear function.

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a) The square root of 196 is 14.

b) The square root of 441 is 21.

To find the square root of a number using the prime factorization method, we need to express the number as a product of its prime factors and then take the square root of each prime factor.

a) Let's find the square root of 196:

First, we find the prime factorization of 196:

196 = 2 * 2 * 7 * 7

Now, we group the prime factors into pairs:

196 = (2 * 2) * (7 * 7)

Taking the square root of each pair:

√(2 * 2) * √(7 * 7) = 2 * 7

Therefore, the square root of 196 is 14.

b) Let's find the square root of 441:

First, we find the prime factorization of 441:

441 = 3 * 3 * 7 * 7

Now, we group the prime factors into pairs:

441 = (3 * 3) * (7 * 7)

Taking the square root of each pair:

√(3 * 3) * √(7 * 7) = 3 * 7

Therefore, the square root of 441 is 21.

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