Two lines intersect. Find the value of b and c. Solution:​

All of the measures of central tendency and spread for these data will need to be changed from the original computations in order to reflect the correct values.

In this case, the initial set of scores was 4, 5, 8, 9, 11, 13, 15, 18, 18, 18, and 20. The teacher initially computed the mean, median, standard deviation, and IQR and then discovered that one of the 18s should have been a 16.The mean is affected by any change in the data, so this measure will need to be changed to reflect the correct values. The new mean is 73.The median is also affected by any change in the data, so this measure must be recalculated to reflect the correct values.

The new median is 15.5.The standard deviation measures the spread of data from the mean, so this measure must also be changed to reflect the correct values. The new standard deviation is 1.4.Finally, the IQR measures the spread of the middle 50% of the data, so this measure must also be changed to reflect the correct values. The new IQR is 6. All of the measures of central tendency will need to be changed.

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Answer: x=211/1296

Step-by-step explanation:

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

The greatest time that the fourth runner can run for the team to reach its goal is 63.23 seconds.

To find the greatest time that the fourth runner can run for the team to reach its goal, we need to subtract the total time of the first three runners from the goal time of 4 minutes (240 seconds).

Total time of the first three runners = 57.38 seconds + 60.92 seconds + 58.47 seconds = 176.77 seconds

Greatest time the fourth runner can run = Goal time - Total time of the first three runners = 240 - 176.77 = 63.23 seconds

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Confidence interval = (y ± t∗s/√n)g1 = y - t*s/√ng2 = y + t*s/√n Substituting the values, g1 = 26.22 - 1.860*(0.11)/√9 = 25.84g2 = 26.22 + 1.860*(0.11)/√9 = 26.59Therefore, the statistics g1 and g2 that will satisfy the required inequality are 25.84 and 26.59 respectively.

The formula for finding the confidence interval is as follows: n − 1, where t is the value of the t-distribution corresponding to the specified confidence level and the sample size minus one.

As per the given exercise 7.11, suppose that in the forest fertilization problem the population standard deviation of basal areas is not known and must be estimated from the sample.

If a random sample of n = 9 basal areas is to be measured, find two statistics g1 and g2 such that p(g1 ≤ (y - u) ≤ g2) = 90

To find the statistics g1 and g2 that will satisfy the required inequality

the following formula can be used: Confidence interval = [tex](y ± t∗s/√n)[/tex]

From the formula, we can see that the confidence interval depends on the values of y, s, t and n.

The value of y is the sample mean

the value of s is the sample standard deviation

And the value of n is the sample size.

The value of t depends on the confidence level desired and the degrees of freedom for the t-distribution. In this case, the confidence level is 90%, which means that we want to find the value of t that will give us a total area of 0.90 under the t-distribution curve with 8 degrees of freedom .Using the t-table, the value of t can be found to be 1.860, where the value for 90% and 8 degrees of freedom is 1.860.t = 1.860Now, we need to calculate the value of s, which is the sample standard deviation.

Since we do not have any information about the population standard deviation, we will use the sample standard deviation as an estimate of the population standard deviations = σ/√nσ = s*√nσ = 0.11*√9σ = 0.33Substituting the values in the confidence interval formula

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The volume of the solid whose base is the region inside the circle with radius 3 if the cross sections taken parallel to one of the diameters are equilateral triangles is 81/2*\sqrt3 by using the slicing method.

To find the volume of the solid whose base is the region inside the circle with radius 3, we need to integrate the area of the cross sections taken parallel to one of the diameters, which are equilateral triangles.

Let's consider a cross section of the solid taken at a distance x from the center of the circle.

Since the cross section is an equilateral triangle, all its sides have the same length.

Let  this length be y. Since the triangle is equilateral, its height can be found using the Pythagorean theorem as follows:

[tex]height = \sqrt{(y^2 - (y/2)^2)} = \sqrt{(3/4y^2)}= \sqrt{3/2y}[/tex]

Therefore, the area of the cross section at a distance x from the center of the circle is:

[tex]A(x) = (1/2)y\sqrt{3/2y} = \sqrt{3/4y^2}[/tex]

Now, we need to integrate this area over the range of x from -3 to 3 (since the circle has radius 3):

[tex]V = \int\ [-3,3]\sqrt{3/4*y^2} dx[/tex]

To find the limits of integration for y, we need to consider the equation of the circle:

[tex]x^2 + y^2= 3^2[/tex]

Solving for y, we get:

[tex]y =\pm\sqrt{(3^2 - x^2)}=\pm\sqrt{(9^2 - x^2)}[/tex]

Since we want the cross sections to be equilateral triangles, we know that y is equal to the height of an equilateral triangle with side length equal to the diameter of the circle, which is 2*3 = 6. Therefore, we can write:

[tex]y = 3*\sqrt{3}[/tex]

Substituting this into the integral, we get:

[tex]V = \int\ [-3,3] \sqrt{3/4*(3\sqrt3)^2} dx[/tex]

[tex]= \int\ [-3,3] 27/4*\sqrt{3} dx[/tex]

Integrating, we get:

[tex]V = [27/4\sqrt{3x}]*[-3,3][/tex]

[tex]= 81/2*\sqrt{3}[/tex]

Therefore, the volume of the solid is [tex]81/2*\sqrt3[/tex]cubic units

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The error bound for the 95% confidence interval is (1.96 x Standard Deviation/√n), which in this case is (1.96 x 11/√50) = 2.56. This means that the true mean weight of newborn elephant calves lies within +/-2.56 pounds of the interval range.

The 95% confidence interval for the population mean weight of newborn elephants can be calculated using the sample mean of 244 pounds and the sample standard deviation of 11 pounds. The confidence interval is calculated using the following formula:
Confidence Interval = (Mean - (1.96 x Standard Deviation/√n)), (Mean + (1.96 x Standard Deviation/√n))
Where n is the sample size.
Therefore, the 95% confidence interval for the population mean weight of newborn elephants is (231.14, 256.86).
This can also be represented in a graph. The graph would have the x-axis representing the confidence interval, with a range from 231.14 to 256.86, and the y-axis representing the probability, which would be 0.95.

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Answer:x=33/4096=0.008

Step-by-step explanation: 1.1     16 = 24

(16)3 = (24)3 = 212

Equation at the end of step

1

:

 ((212 • x) -  1) -  32  = 0

STEP

2

:

Equation at the end of step 2

 4096x - 33  = 0

STEP

3

:

Solving a Single Variable Equation:

3.1      Solve  :    4096x-33 = 0

Add  33  to both sides of the equation :

                     4096x = 33

Divide both sides of the equation by 4096:

                    x = 33/4096 = 0.008

The probability of getting tails in the three coins would be 0.125 or 12.5%.

To calculate the probability of an event happening, first, we need to identify the rate of the desired outcome versus the total possible outcomes. Moreover, to determine the total probability of two or more events happening we need to calculate the probability of each event and then multiply the results.

Probability of getting tails in any of the three coins:

1 / 2 = 0.5

Total probabilityy:

0.5 x 0.5 x 0.5 = 0.125 or 12.5%

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The length, in feet, of BC is the square root of  √242 and, The length, in feet, of CF is the square root of √363

Cube:

In geometry, a cube is a three-dimensional solid object bounded by six faces, facets, or square sides, with three points of intersection at each vertex. Seen from a certain angle, it is a hexagon and its canvas is often represented by a cross.

Cube is the only regular hexahedron and one of the five Platonic solids. It has 6 faces, 12 edges and 8 vertices. Cube is also a cube, an equilateral cuboid, a rhombus and a 3-sonohedron.

Three orientations are square prisms and four orientations are triangular trapezoids. A cube is twice as large as an octahedron. It has cubic or octahedral symmetry. Cube is the only convex polyhedron whose faces are square.

Then the length CB is given by the Pythagorean theorem as:

CB = √(AC² +AB²)

     = √{(2 cm)² + (√2 cm)²}

     = √(4 +2) cm

CB = √6 cm

Now,

The length, in feet, of BC is the square root of  

√242

And,

The length, in feet, of CF is the square root of  

√363

Complete Question:

A cube. The top face has points C, A, B, D and the bottom face has points G, E, F, H. Diagonals are drawn from B to C and from C to F. Side B F is 11 feet.

What are the exact lengths of BC and CF?

The length, in feet, of BC is the square root of

The length, in feet, of CF is the square root of

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Answer:

The length, in feet, of BC is the square root of

✔ 242

The length, in feet, of CF is the square root of

✔ 363

Step-by-step explanation:

[tex]f(x) = -2(x - 0.5)^2 + 6[/tex] is the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8).

What is quadratic function?

f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero, is a quadratic function.

To find the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8), we can use the vertex form of the quadratic function, which is:

[tex]f(x) = a(x - h)^2 + k[/tex]

[tex]f(1) = a(1 - h)^2 + k\\\\6 = a(1 - h)^2 + k[/tex]

We can use a second point to find a relationship between h and k. Let's use the point (0, 8):

[tex]f(0) = a(0 - h)^2 + k\\\\8 = a(-h)^2 + k\\\\6 - 8 = a(1 - h)^2 + k - (a(-h)^2 + k)\\\\-2 = a(1 - h)^2 - a(h)^2\\\\-2 = a(1 - 2h + h^2) - a(h^2)\\\\-2 = a - 2ah + ah^2 - ah^2\\\\-2 = a - 2ah\\\\a = -2/(2h - 1)[/tex]

Let's use the second equation:

[tex]8 = a(-h)^2 + k\\\\8 = (-2/(2h - 1))(h^2) + k\\\\8(2h - 1) = -2h^2 + k(2h - 1)\\\\16h - 8 = -2h^2 + k(2h - 1)\\\\-2h^2 + 16h - 8 = k(2h - 1)\\\\k = (-2h^2 + 16h - 8)/(2h - 1)[/tex]

Now we can substitute this value of h into our expressions for a and k to get:

[tex]a = -2/(2(0.5) - 1) = -2\\\\k = (-2(0.5)^2 + 16(0.5) - 8)/(2(0.5) - 1) = 6[/tex]

So the equation of the quadratic function is:

[tex]f(x) = -2(x - 0.5)^2 + 6[/tex]

Therefore, [tex]f(x) = -2(x - 0.5)^2 + 6[/tex] is the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8).

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Answer:

Pretty sure its C

Step-by-step explanation:

To cut through y axis, x axis is always 0, So it would be (0, a) where a is any real number, not (a,0) as is given in reason.

The derivative of the implicit function x · cos y = 1 at point (2, π / 3) is equal to y' = √3 / 6.

In this problem we find the case of a implicit function of the form f(x, y), whose derivative must be found. This can be done by implicite differentiation, whose procedure is shown:

Step 1 - Derive the expression by derivative rules:

cos y - x · sin y · y' = 0

Step 2 - Clear y' within the expression:

y' = cos y / (x · sin y)

Step 3 - Clear x and y in the resulting expression:

y' = cos (π / 3) / [2 · sin (π / 3)]

y' = 1 / [2 · tan (π / 3)]

y' = √3 / 6

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Answer:

Since 1 000 paperclips = 1 kilogram

5,600 paperclips x 0.001 kilograms per paperclip = 5.6 kilograms

Answer: 5.6 kilograms

Step-by-step explanation:

5,600 grams = 5.6 kilograms

In the given scenario, the average current produced within the loop is approximately 2.13 A.

We can begin by computing the change in magnetic flux across the loop as it expands to determine the average current generated within the loop.

The following equation provides the magnetic flux across a loop:

Φ = B * A * cos(θ)

ΔΦ = B * ΔA

ΔA = A₂ - A₁ = π * (1.00 m)² - π * (0.40 m)² = π * (1.00² - 0.40²) = π * (1.00 + 0.40)(1.00 - 0.40) = π * (1.40)(0.60) = 0.84π m²

So,

ΔΦ = B * ΔA = (24 mT) * (0.84π m²) = 20.25π m²·T

emf = ΔΦ / Δt = (20.25π m²·T) / (1.0 s) = 20.25π V

As:

emf = I * R

So, again

I = emf / R = (20.25π V) / (30 Ω) ≈ 2.13 A

Thus, the average current produced within the loop is approximately 2.13 A.

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They can present in 96 different orders. Given, there are five student groups to present in class and one group cannot go first because they need additional set-up time.

Permutation is to select an object then arrange it and it cares about the orders while Combination is about only selecting an object without caring the orders.

We have 5 positions to fill here.

First position: 4 ways (one of the rest 4 groups will present first)

Second position: 4 ways (one of the rest 3 groups and the group which could not present first, will present second)

Third position: 3 ways (one of the rest 3 groups will present third)

Fourth position: 2 ways (one of the rest 2 groups will present fourth)

Fifth position: 1 way (rest group will present last)

Total ways in which they can present = 4*4*3*2*1 = 96

Hence, the answer is 96.

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As per the combination method, the number of tickets that we put in that are marked "outer" is 3 (option d).

In this case, we want to choose the number of tickets marked "outer." Let's call this number k. We know that we already put one ticket marked "inner" in the box, so the total number of tickets in the box is 2. Therefore, n = 2.

Now we need to determine k. We want to know how many tickets we need to put in that are marked "outer." We can represent this as a. So we have:

ᵃC₁ = a! / ((1!)(a-1)!) = a

We want to find the value of a that satisfies the condition that the probability of choosing an "inner" ticket is 1/3 and the probability of choosing an "outer" ticket is 3/2.

Since we already put in 1 ticket marked "inner," the probability of choosing an "inner" ticket is 1/2, which means the probability of choosing an "outer" ticket is also 1/2.

We know that the probability of choosing an "outer" ticket is 3/2, so we can set up the following equation:

ᵃC₁ / 2 = 3/2

Solving for a, we get:

a = 3

In conclusion, the answer is (d) 3.

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Answer:

Step-by-step explanation:

To solve this problem, we can use the formula for calculating simple interest:

I = P * r * t

where:

I = interest earned

P = principal (initial amount of money)

r = rate of interest

t = time (in years)

We can rearrange the formula to solve for the principal:

P = I / (r * t)

In this case, we know that Gill earned $2306 in interest after one year at a rate of 6%. So:

I = $2306

r = 0.06

t = 1 year

Substituting these values into the formula, we get:

P = $2306 / (0.06 * 1)

P = $38,433.33

Therefore, the initial amount of money that Gill deposited into her account was $38,433.33.

Answer:

To solve the integral, we can use partial fractions and then integrate each term separately. The integrand can be written as:

[tex]\dfrac{x^4(1-x)^4}{1+x^2} = \dfrac{x^4(1-x)^4}{(x+i)(x-i)}[/tex]

Using partial fractions, we can write:

[tex]\dfrac{x^4(1-x)^4}{(x+i)(x-i)} = \dfrac{Ax+B}{x+i} + \dfrac{Cx+D}{x-i}[/tex]

Multiplying both sides by (x+i)(x-i), we get:

[tex]x^4(1-x)^4 = (Ax+B)(x-i) + (Cx+D)(x+i)[/tex]

Substituting x=i, we get:

[tex]i^4(1-i)^4 = (Ai+B)(i-i) + (Ci+D)(i+i)[/tex]

Simplifying, we get:

[tex]16 = 2Ci + 2B[/tex]

Substituting x=-i, we get:

tex^4(1+i)^4 = (Ci+D)(-i-i) + (Ai+B)(-i+i)[/tex]

Simplifying, we get:

[tex]16 = 2Ai + 2D[/tex]

Substituting x=0, we get:

[tex]0 = Bi + Di[/tex]

Substituting x=1, we get:

[tex]0 = A+B+C+D[/tex]

Solving these equations simultaneously, we get:

A = -22/7 + π

B = 0

C = 22/7 - π

D = -2/5

Therefore, the integral can be written as:

[tex]\int_0^1 \dfrac{x^4(1-x)^4}{1+x^2}dx = \int_0^1 \left[\dfrac{-22/7+\pi}{x+i} + \dfrac{22/7-\pi}{x-i} - \dfrac{2/5}{1+x^2}\right]dx[/tex]

Integrating each term separately, we get:

[tex]\int_0^1 \dfrac{-22/7+\pi}{x+i}dx = [-22/7+\pi]\ln(x+i) \bigg|_0^1 = [\pi-22/7]\ln\left(\dfrac{1+i}{i}\right)[/tex]

[tex]\int_0^1 \dfrac{22/7-\pi}{x-i}dx = [22/7-\pi]\ln(x-i) \bigg|_0^1 = [22/7-\pi]\ln\left(\dfrac{1-i}{-i}\right)[/tex]

[tex]\int_0^1 \dfrac{-2/5}{1+x^2}dx = -\frac{2}{5}\tan^{-1}(x)\bigg|_0^1 = -\frac{2}{5}\tan^{-1}(1) + \frac{2}{5}\tan^{-1}(0) = -\frac{2}{5}\tan^{-1}(1)[/tex]

Therefore, the correct options are:

A) [tex]\pi-\frac{22}{7}[/tex]

B) [tex]\frac{2}{105}[/tex]

C) 0

D) [tex]\frac{71}{15}-\frac{3\pi}{2}[/tex]

We need to use mathematical logic to prove that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

     Let's take an example of two sets A and B.

      Let's say, A={a, b} and B={1, 2, 3}

      To find the cardinality of the cross-product of A and B, we need to make all possible ordered pairs with A and B.

             {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

 Counting the number of ordered pairs, we get six.

  So, the cardinality of the cross-product of two sets A and B equals the cardinality of each set multiplied by the other. Here,

           |A| = 2 and |B| = 3,

          so |A x B| = 2 x 3

                           = 6.

          Therefore, it is proved that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

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Answer:

Step-by-step explanation:

You want the measures of AO and DO in a trapezium in which AB║CD, the diagonals intersect at O, and AB = 3 cm, CD = 6 cm, CO = 4 cm, OB = 3 cm.

Diagonal AC is a transversal to parallel lines AB and CD, so alternate interior angles BAO and DCO are congruent. Vertical angles AOB and COD are also congruent, so ∆ABO ~ ∆CDO by the AA similarity postulate.

This means the side lengths are proportional, so ...

  AB/CD = AO/CO = BO/DO

  3/6 = AO/4 = 3/DO   ⇒   AO = 2, DO = 6

The measures of AO and DO are 2 cm and 6 cm, respectively.

__

Additional comment

It can help to draw a diagram.

The sample mean is 1450g/cm², the standard deviation is 44.3 g/cm² and  the probability that, for a random sample of n = 10 test pieces of paper, x is greater than 1400 g/cm2 is 0.8708

a) The sampling distribution of the sample mean of n = 10 test pieces of paper is approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size:

mean of sample mean = mean of population = 1450 g/cm²

standard deviation of sample mean = standard deviation of population / square root of sample size

= 140 g/cm2 / √(10)

= 44.3 g/cm²

Therefore, the sampling distribution of the sample mean is approximately normal with mean 1450 g/cm2 and standard deviation 44.3 g/cm2.

b) To find the probability that, for a random sample of n = 10 test pieces of paper, x is greater than 1400 g/cm2, we need to standardize the sample mean using the sampling distribution calculated in part (a):

z = (x - mean of sample mean) / standard deviation of sample mean

= (1400 - 1450) / 44.3

= -1.13

Using a standard normal distribution table or calculator, we can find the probability that z is less than -1.13 and subtract that probability from 1 to find the probability that z is greater than -1.13:

P(z > -1.13) = 1 - P(z < -1.13)

= 1 - 0.1292

= 0.8708

Therefore, the approximate probability that, for a random sample of n = 10 test pieces of paper, x is greater than 1400 g/cm² is 0.8708.

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Answer:

45 yo

Step-by-step explanation:

Let's start by defining some variables to represent the ages of Ted and Rosie:

- Let's call Ted's current age "T"

- Let's call Rosie's current age "R"

From the problem statement, we know that:

- Ted is five times as old as Rosie was when Ted was Rosie's age. Written as an equation, this becomes:

T = 5(R - (T - R))

Simplifying this equation, we get:

T = 5(R - T + R)

T = 10R - 5T

- When Rosie reaches Ted's current age, the sum of their ages will be 72. Written as an equation, this becomes:

R + T = 72 - T

We now have two equations with two variables. We can use substitution to solve for T.

Substitute the second equation into the first equation to eliminate R:

T = 10R - 5T

T = 10(72 - T) - 5T

T = 720 - 15T

16T = 720

T = 45

Therefore, Ted's current age is 45.

When solving arithmetic expressions, Oracle 12c resolves addition and subtraction operations first from left to right in the expression. This statement is true.

By following these rules, Oracle 12c is able to resolve arithmetic expressions accurately and efficiently. This is particularly important when dealing with large amounts of data, where the correct order of operations is critical to ensure that the correct result is obtained.

Oracle 12c is a database management system that is used to handle large amounts of data. It has a variety of features that enable it to manage and maintain data effectively.

Oracle 12c has a SQL engine that allows it to execute SQL statements and expressions to retrieve and manipulate data.

When performing arithmetic operations in Oracle 12c, the order of operations is critical. The order of operations is the set of rules that dictate the sequence in which arithmetic operations should be performed in a mathematical expression.

These rules are also known as the rules of precedence.In Oracle 12c, the rules of precedence dictate that arithmetic operations should be resolved from left to right in an expression. This means that addition and subtraction operations should be resolved before multiplication and division operations.

This is because addition and subtraction operations have a lower precedence than multiplication and division operations. As a result, Oracle 12c always resolves addition and subtraction operations first from left to right in an expression .This order of operations ensures that the correct result is obtained when performing arithmetic operations in Oracle 12c.

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Axes were labeled, then triangle ABC was drawn followed by a rotation of 90 degrees clockwise, and then it was translated three units down.

What are triangles?

Three vertices make up a triangle, a three-sided polygon. The angles of the triangle are formed by the connection of the three sides end to end at a single point. The sum of all three angles of the triangle is equal to 180 degrees.

How rotation of a point by 90 degrees clockwise will take place?

When points will be rotated 90 degrees clockwise, the x will become y while y will become -x.

After rotation:

A (-3, 0) will become A' (0,3)

B (-2, 4) will become B'(4,2)

C (1, -1). will become C'(-1,-1)

After rotation of 90 degrees clockwise is performed, translation of three units down will be performed as follows:

A' (0,3) will become A'' (0,0)

B'(4,2) will become B''(4,-1)

C'(-1,-1) will become (-1,-4)

Thus, Axes were labeled, then triangle ABC was drawn followed by a rotation of 90 degrees clockwise, and then it was translated three units down.

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An inequality representing the amount that the island of Martinique can received for hurricane relief efforts, where x is the number of days is y ≤ 32,000 + 4,500x.

Inequality is an algebraic statement that two or more mathematical expressions are unequal.

Inequalities can be represented as:

The total amount received by the island = $32,000

The daily receipt of donations = $4,500

Let the number of days = x

Let the funds raised for aid = y

y ≤ 32,000 + 4,500x

Thus, the inequality for the funds that the island can fundraise for hurricane relief aid by the end of the month is y ≤ 32,000 + 4,500x.

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The probability that the first ball is yellow if the second ball is black is 1/14. The correct option is D.

The given question is a classic example of dependent events in probability. As the balls are drawn without replacement, the second event's outcome will depend on the outcome of the first event.

Probability = Number of favorable events/ Total number of events

The probability of the first ball being yellow is [tex](5/15)[/tex], while the probability of the second ball being black is [tex](3/14)[/tex].

Mathematically represented as P(Yellow ball on the first draw) = P(Yellow ball) = [tex]5/15[/tex]

P(Blackball on second draw given Yellow ball on the first draw) = P(Blackball | Yellow ball) = [tex]3/14[/tex]

As both the events are dependent, we need to find the joint probability of both the events, which can be calculated as P(Yellow ball on the first draw and Blackball on the second draw) = P(Yellow ball) × P(Blackball | Yellow ball)

P (Yellow ball on the first draw and blackball on second draw) = [tex](5/15) × (3/14) = 3/42 = 1/14.[/tex]

Therefore, the correct option is D.

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Answer:

12

Step-by-step explanation:

6=a/4+2

L.C.M=4

4(6)=a+6(2)

24=a+12

24-12=a

a=12

The centre οf the circle is (-4, -11).

We knοw that the general equatiοn fοr a circle is (x - h)² + (y - k)² = r² with (h, k) representing the centre and r representing the radius. Sο multiply bοth sides by 21 tο get the cοnstant term οn the right side οf the equatiοn. Then, fοr the y terms, cοmplete the square.

Tο write a circle equatiοn in standard fοrm, we must cοmplete the square fοr bοth x and y.

Tο begin, cοnsider the fοllοwing equatiοn: x²+ y² + 8x + 22y + 37 = 0.

Let's separate the terms with x frοm the terms with y:

[tex](x^2 + 8x) + (y^2 + 22y) + 37 = 0[/tex]

We add (8/2)² = 16 tο bοth sides tο cοmplete the square fοr x: (x²+ 8x + 16) + (y² + 22y) + 37 = 16

Simplifying the left side οf the equatiοn and cοmbining cοnstants οn the right:

[tex](x + 4)^2 + (y^2 + 22y + 121) = 16 - 37 - 121\s(x + 4)^2 + (y + 11)^2 = 50[/tex]

The equatiοn can nοw be written in standard fοrm:

[tex](x + 4)^2/50 + (y + 11)^2/50 = 1[/tex]

The circle's centre is (-4, -11).

As a result, the standard fοrm οf the circle's equatiοn is (x + 4)²/50 + (y + 11)²/50 = 1, and the circle's centre is (-4, -11).

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