Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?

Interval notation is a way to represent an interval of real numbers on the number line. Set notation is a way to represent a set of elements.

Interval notation is a way to represent an interval of real numbers on the number line.

The notation uses parentheses, brackets, and infinity symbols to indicate whether the endpoints of the interval are included or excluded from the set of numbers.

For example, [3, 8) represents the interval of real numbers from 3 (included) to 8 (excluded), while (-∞, 4) represents the interval of real numbers less than 4 (excluding 4), and extending to negative infinity.

Set notation is a way to represent a set of elements. It uses curly braces to enclose the elements of the set and can include various symbols to indicate properties of the set.

For example, {2, 3, 5, 7, 11} represents the set of prime numbers less than 12, while {x | x is an even number} represents the set of even numbers.

The vertical bar | is used to separate the variable (x in this case) from the condition that must be met for elements to be included in the set (x is an even number).

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Therefore, the exponential function that passes through the points (2,45) and (5,3072) is: y = (45/16) ([tex]4^{x}[/tex]).

In mathematics, a function is a rule that assigns to each element in a set called the domain, a unique element in another set called the range. In other words, a function is a mathematical object that takes an input and produces a specific output, according to a specific set of rules or operations.

by the question.

The general form of the exponential function is y = [tex]ab^{x}[/tex] where a and b are constants. To find the values of a and b that satisfy the given conditions, we can use the two points (2,45) and (5,3072) and solve for a and b.

First, we can write two equations based on the two points:

45 = a[tex]b^{2}[/tex] (1)

3072 = ab^5 (2)

We can then divide (2) by (1) to eliminate a:

3072/45 = (a[tex]b^{5}[/tex])/ (a[tex]b^{2}[/tex])

68 = [tex]b^{3}[/tex]

Solving for b, we get:

b = 4

Substituting this value of b into either (1) or (2) to solve for a, we get:

45 = a ([tex]4^{2}[/tex])

a = 45/16

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There are no values in the wrong partition after the first pass across the array using quicksort when the pivot is [28].

The given problem states that using quicksort, if the pivot is [28], which values are found to be in the wrong partition after the first pass across the array? {28, 17, 45, 51, 35, 14, 88, 24}.Quicksort is a sorting algorithm that uses the divide and conquer approach. Here’s how to find out which values are found to be in the wrong partition after the first pass across the array .Step-by-step The first thing to do is to pick the pivot value. It can be any element in the array. The pivot value is 28.Then, the partition function is performed to place the pivot element in its final position in the sorted array.

All the values smaller than the pivot will be on the left side, and all the values greater than the pivot will be on the right side of the pivot. Here’s the array after the partition: 17 24 14 [28] 35 45 51 88Now, the elements to the left of the pivot are 17 24 14, and the elements to the right of the pivot are 35 45 51 88. Since the pivot is placed in its final position, there are no values in the wrong partition after the first pass across the array. Therefore, there are no values in the wrong partition after the first pass across the array using quicksort when the pivot is [28].

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Multiplication and division of regular numbers are the same as fractions and we will learn them with the multiplication and division of rational numbers worksheet.

To do the multiplication, first find the greatest common divisor of the numerator and the denominator. The factors are then grouped together so that the score equals one. Then multiply by the remaining factors. To divide, first rewrite the division as multiplication by the inverse of the denominator.

Cuemath's interactive math worksheets contain visual simulations to help your child visualize the concepts being taught, "see things as they really are, and reinforce learning from them". The Rational Numbers Multiplication and Division Worksheet follows a step-by-step learning process that helps students better understand concepts, identify errors, and eventually develop strategies for solving future problems.

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We can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles

To answer this question, we need to use the following formula for a confidence interval for the mean: CI = (μ - z*(σ/√n), μ + z*(σ/√n)), Where μ is the population mean, z is the z-score for the given confidence level, σ is the population standard deviation, and n is the sample size. Using the given information, we can calculate the confidence interval for the mean:CI = (23500 - 2.575*(3900/√100), 23500 + 2.575*(3900/√100)), CI = (21342.6, 24637.4)

To summarize, we used the formula for a confidence interval for the mean and the given information to calculate the confidence interval for the average number of miles an automobile is driven annually in Alabama. This confidence interval is (21342.6, 24637.4), which means we can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles.

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The Verde River's discharge and sediment load determine how much eroded material it can move beyond a given site in 6 hours, or 1/4 of a day. We cannot accurately determine the amount of eroded material carried by the river without knowing these numbers.

The amount of water that flows past a specific place in a river per unit of time is known as its discharge. The amount of sediment (including eroded material) that a river carries is referred to as its sediment load.

Both of these numbers can fluctuate considerably based on a number of variables, including the time of year, regional geology, and weather patterns.

In order to precisely determine the amount of eroded debris carried by the Verde River past a specific point in 1/4, thus we need further information.

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The system of equation are

The profit function is = 20x + 6y

Let's define the following variables:

x: the number of boards of mahogany sold

y: the number of boards of black walnut sold

Based on the problem statement, we can write the following system of inequalities to represent the constraints:.

x ≤ 260 (there are only 260 boards of mahogany available)

y ≤ 320 (there are only 320 boards of black walnut available)

x + y ≤ 380 (the total number of boards sold can't exceed 380)

The profit function is:

P(x, y) = 20x + 6y

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a. The answer to this question is the solution to the equation:

84000(1 + 0.04)^a = 132000

Where "a" represents the number of years since the beginning of 2020, and 0.04 is the decimal equivalent of 4%.

We use the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we assume that the population growth rate is compounded annually, so n = 1.

b. Solving the equation in part (a), we get:

(1 + 0.04)^a = 132000/84000

1.04^a = 1.5714

a = log(1.5714)/log(1.04)

a ≈ 9.9 years

Therefore, it will take approximately 9.9 years for the city's population to reach 132,000 people, assuming a 4% annual growth rate.

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Answer:

A is correct

Step-by-step explanation:

Whatever value you put in for "y" will keep both of the equations equal

There are 4060 different groups of 3 pictures that can be displayed on the frame.

A combination is a method of choosing a subset of items from a bigger set when the order of the items is irrelevant. For instance, the two-letter possibilities for a set of three letters A, B, and C are AB, AC, and BC. Combinations are employed in many different mathematical and practical situations, including combinatorial optimization, statistics, and probability theory. Choosing a password with a specific amount of characters, picking a combination of things from a menu, and assembling a team from a group of players are all examples of how they are utilised in daily life.

Given that, from 30 pictures we need to form groups of 3.

The combination of selecting groups is given as:

nCk = n! / (k! * (n - k)!)

Substituting the values we have:

30C3 = 30! / (3! * (30 - 3)!)

= 30! / (3! * 27!)

= (30 * 29 * 28) / (3 * 2 * 1)

= 4060

Hence, there are 4060 different groups of 3 pictures that can be displayed on the frame.

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Answer:

Angle CAB= 28°

Angle ABC= 45°

Angle DCA= 73°

Angle DCE=107°

Step-by-step explanation:

Step-by-step explanation:

xy=6

x=6/y (1)

now,

XY=6

6/y-y=6

The system of equations x + y = 15 and 2x + 3y = 24

What is an equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.

To substitute for the variable x in the equation x + y = 15, we can rearrange the equation to solve for x in terms of y:

x = 15 - y

Then we can substitute this expression for x in any other equation that involves x.

2(15 - y) + 3y = 24

Simplifying this equation gives:

30 - 2y + 3y = 24

y = -6

Then we can substitute this value back into our expression for x to get:

x = 15 - (-6) = 21

So the solution to the system of equations x + y = 15 and 2x + 3y = 24 is (x, y) = (21, -6), but the question only asked us to rewrite the equation to substitute for x.

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a) Graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380

b) The maximum profit of $5200 achieved.

Selling profit is the profit that a business makes on the sale of its products or services. It is the difference between the selling price and the cost of product.

a) Let x be the number of boards of mahogany sold and y be the number of boards of black walnut sold. Then, the constraints of the problem can be represented by the following system of inequalities:

x ≥ 0 (non-negative constraint)

y ≥ 0 (non-negative constraint)

x ≤ 260 (maximum number of mahogany boards available)

y ≤ 320 (maximum number of black walnut boards available)

x + y ≤ 380 (maximum number of boards that can be sold)

To graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380 on a coordinate plane and shade the feasible region that satisfies all of the constraints. The feasible region is the area that is bounded by these lines and includes the origin (0, 0).

b) The profit function P(x, y) can be defined as follows:

P(x, y) = 20x + 6y

To maximize the profit, we need to find the values of x and y that satisfy all of the constraints and maximize the profit function P(x, y).

One way to do this is to use the corner-point method. We can evaluate the profit function at each of the corners of the feasible region and find the corner that gives the maximum profit.

The corners of the feasible region are (0, 0), (0, 320), (260, 0), and (120, 260).

P(0, 0) = 0

P(0, 320) = 6(320) = 1920

P(260, 0) = 20(260) = 5200

P(120, 260) = 20(120) + 6(260) = 4720

Therefore, the maximum profit of $5200 can be achieved by selling 260 boards of mahogany and 0 boards of black walnut.

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Answer: Quadratic polynomial.

Step-by-step explanation: Polynomials with 2 as the degree.

The 95% confidence interval of .61 to 1.19 cm provides strong evidence that there is a difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not. Furthermore, the small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.

The 95% confidence interval for the difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not is .61 to 1.19 cm. This implies that the true mean difference is likely to be between .61 and 1.19 cm. The p-value is a measure of how likely it is that the difference in means is zero, and can be used to assess the statistical significance of the difference in means.

Therefore, the p-value for the hypothesis that the mean difference is zero is very small and can be considered statistically significant.

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The small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.

A 95% confidence interval (CI) is a range of values that is expected to include the true population mean with a probability of 0.95. The CI for the difference in mean head circumference between non-users and users of marijuana during pregnancy is 0.61 to 1.19 cm.

This means that the sample mean difference (nonuse minus use) falls within this interval.

If the true population mean difference were zero, the sample mean difference would fall within the range of random sampling error, and the null hypothesis that there is no difference between the groups would be supported.

However, since the 95% CI does not include zero, we can conclude that the difference is statistically significant at the 0.05 level.

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A particular solution to the differential equation is [tex]y = Ae^(iwt) =(1/4)e^({i\sqrt{3t} )}.[/tex]

A particular solution to the differential equation,

[tex]\frac{d^4y}{dt^4} +5\frac{d^2y}{dy^2} +3y=2[/tex],

we assume that y has the form [tex]y = Ae^(iwt)[/tex], where A is a complex constant and w is a real constant to be determined. Substituting this form of y into the differential equation, we get:

[tex](-w^4 + 5w^2 + 3)Ae^(iwt) = 2.[/tex]

we solve the quadratic equation [tex]-w^4 + 5w^2 + 3 = 0[/tex], which can be factored as:

[tex](w^2 - 3)(-w^2 + 1) = 0.[/tex]

The roots are w = ±√3 and w = ±i. Since w must be real, we choose w = √3. Substituting this value of w and solving for A, we get:

[tex]A= \frac{2}{\sqrt{(-\sqrt{3} )^4 + 5(\sqrt{3})^2 +3 } }[/tex] [tex]=\frac{2}{5+3} = \frac{1}{4}[/tex]

Therefore, the particular solution is:

[tex]y = Ae^(iwt) =(1/4)e^({i\sqrt{3t} )}.[/tex]

A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).

Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.

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The area of the quadrilateral is 161.24 cm².

We can see that we can divide the quadrilateral into two triangles: ABD and CBD. We know that the height of ABD is 6 cm and the height of CBD is 8 cm. We also know that BD is 15 cm. To find the area of each triangle, we need to find the base of each triangle. We can do this using the Pythagorean theorem.

For triangle ABD:

AB² = AD² + BD²

AB² = (6 cm)² + (15 cm)²

AB² = 261 cm²

AB = [tex]\sqrt(261) cm[/tex]

For triangle CBD:

BC² = CD² + BD²

BC² = (8 cm)² + (15 cm)²

BC² = 289 cm²

BC = 17 cm

Now we can find the areas of the triangles:

Area of ABD =[tex]\frac{1}{2}[/tex] * AB * 6 cm

Area of ABD = [tex]\frac{1}{2}[/tex] * [tex]\sqrt(261) cm[/tex] * 6 cm

Area of ABD = 93.24 cm^2

Area of CBD = [tex]\frac{1}{2}[/tex] * BC * 8 cm

Area of CBD = [tex]\frac{1}{2}[/tex] * 17 cm * 8 cm

Area of CBD = 68 cm²

Finally, we can find the area of the quadrilateral by adding the areas of the triangles:

Area of ABCD = Area of ABD + Area of CBD

Area of ABCD = 93.24 cm² + 68 cm²

Area of ABCD = 161.24 cm²

Therefore, the area of the quadrilateral is 161.24 cm².

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The best statistical measure to represent the entire spread of the data set when a data set is symmetrical is the standard deviation.

The standard deviation is a measure of how much the individual data points vary from the mean of the data set.

It is calculated by taking the square root of the variance.

It is the average of the squared differences of each data point from the mean.

In a symmetrical data set, the mean, median, and mode are all equal and located at the center of the data set.

This represents that the data points are evenly distributed around the center.

The standard deviation can provide a good measure of the spread of the data points from this center.

Other measures of spread representing the range or interquartile range, may not be as useful in a symmetrical data set .

As they are more affected by outliers and skewness in the data.

Therefore, the symmetrical data set in best way represented by statistical measure known as standard deviation.

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Answer:

44%

Step-by-step explanation:

the answer is 44% because 55/125 can be simplifed by dividing both sides by 5 to get 11/25

Suppose that the Quick-Fix Car Service franchise finds that the income generated by its stores can be modeled by assuming that the total income is a continuous stream with a monthly rate of flow at time t given by f(t) = 10,000e^0.02t (dollars per month).

The given function is f(t) = 10,000e^0.02t (dollars per month)So, the monthly rate of flow of income is given by f(t) = 10,000e^0.02t (dollars per month)The total income generated in the first 2 years of operation can be found using integration.

So, the integral of f(t) with respect to t from 0 to 24 months will give us the total income generated in the first 2 years of operation.∫₀²₄f(t)dt = ∫₀²₄10,000e^0.02tdt= [500,000e^0.02t] from 0 to 24= 500,000(e^0.02*24 - e^0)≈ $223,443.95Thus, the total income generated from a Quick-Fix store for the first 2 years of operation is approximately $223,443.95.

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Answer:

Option B

Step-by-step explanation:

Using Proportionality, we get

Let x be the quantity of water in bathtub. Y be time it be removed.

By formula, x2/y2= k = x1/y1

x/y = 5/1(x1=5, y1= 1)[Given]

5x2= 35.5

35.5 = 5y2

=> y = 35.5/5 = 7.1 minutes

97

Step-by-step explanation:

a prime number is a number that can only divide by itslef and by 1 and the result is a whole number (no fractions or dicemals)

91 can be divided by 13 and 7

which makes it not a prime number

95 can be divided by 5 and 19 so it's not a prime number either

The correct option is B. The value of the quartile Q3 is 65.3 inches

Quartile is a type of statistical value that represents one-fourth of a dataset. It divides a dataset into four equal parts. Q1, Q2, and Q3 are the first, second, and third quartiles, respectively.

The formula for the third quartile is as follows:

Q3 = μ + (0.67 × σ), where μ is the mean and σ is the standard deviation.

Given:

Mean = μ = 63.6 inches

Standard deviation = σ = 2.5 inches

The value of the quartile Q3 can be found by substituting the given values in the formula.

Q3 = μ + (0.67 × σ)

Q3 = 63.6 + (0.67 × 2.5)

Q3 = 63.6 + 1.675

Q3 = 65.275 ≈ 65.3

Therefore, the value of the quartile Q3 is 65.3 inches. Therefore, option B is the correct answer.

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Length of the median of the triangle is b. 18

How to find the length of the median of a triangle?
Given that the centroid of a triangle is located 12 units from one of the vertices of a triangle. We need to find the length of the median of the triangle drawn from that same vertex.

Let ABC be a triangle.

Let AD be median of the triangle.

Let E be centroid of ΔABC

Length of the median of triangle = AD

AD = Length of AE + Length of ED

But,

AE = 12

ED = x

So,

AD = 12 + x

Since, centroid divides median in 2:1 ratio.

Then,

[tex]12 : x = 2 : 1\\12/x = 2/1\\\\12= 2x\\[/tex]

Divide by 2 each side

[tex]x = 6[/tex]

So,  AD = 12 + x = 12 + 6 = 18

Thus, Length of the median of triangle is 18.

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16h^(10/2) *remove the root sign

16h^5 *simplify the exponent

Answer: 16h^5

Step-by-step explanation: im correct

Thus, from the attached graph the, solution of the equation in set notation is found as: Domain = (-∞, +∞) and Range: [6, ∞).

The visual method actually only requires three easy actions to complete:

Identify the system's solution.

The given equation is-

5x^2+2x-6=0

Since it is a quadratic equation it will have 1 value on y axis for every two values for x.

The graph will be quadratic with 2 degree.

Using the graphing tool, draw the graph.

Thus, from the attached graph the, solution of the equation in set notation is found as:

Domain = (-∞, +∞)

Range: [6, ∞)

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By using addition calculation, we determine that Angela will end up making 42 cards in total.

Angela made 23 cards for her friends, but she wants to make even more to share with others. To determine how many cards she will make in total, we need to add the number of cards she has already made with the number of cards she plans to make.

So, we add 23 (the number of cards she has made) and 19 (the number of cards she plans to make) so in total, she will make:

23 + 19 = 42

Therefore, Angela will make 42 cards in all.

By using simple arithmetic calculation of basic addition, we find that  Angela will make 42 cards in all.

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Answer:

-2x + 10

Step-by-step explanation:

-2(x - 5)

-2(x) -2(-5)

-2x + 10

Helping in the name of Jesus.

Answer:

One solution

Step-by-step explanation:

I added a photo of my solution

Answer:

The system has one solution: (0, 4).