How many different strings of length 12 containing exactly five a's can be chosen over the following alphabets? (a) The alphabet {a,b) (b) The alphabet {a,b,c}

For the first question, the equation for the line is y = -x + 11. This comes from the fact that the slope for a line perpendicular to the line x = 5 is -1. From there, we can use the point (5,6) to calculate the y-intercept, which is 11.

For the second question, the equation for the line is y = 7x. This comes from the fact that the slope for a line parallel to the line 7x - y = -7 is 7. Since the point (0,0) is already on the line, the equation is already solved.

For the third question, the equation for the line is x = 8. This comes from the fact that the slope for a line with an undefined slope is 0. Since the point (8,2) is already on the line, the equation is already solved.

Answer:

The unit circle has many different angles that each have a corresponding point on the circle. The coordinates of each point give us a way to find the tangent of each angle. The tangent of an angle is equal to the y-coordinate divided by the x-coordinate.

As a result, the probability that the average battery life exceeds 553.9 minutes is 0.0262 (or 2.62%). The answer, rounded to four decimal places, is 0.0262.

Probability serves as an indicator of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing an unlikely event and 1 representing an unavoidable event. Switching a fair coin and coin flips has a probability of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probability theory is a branch of mathematics that studies happenings rather than their properties. It is applied in many fields, including statistics, fund, science, and engineering.

The central limit theorem can be used to approximate the sample mean distribution as a normal distribution with a mean of the population mean and a standard deviation of the population standard deviation divided by the square root of the sample size.

The standard error of the mean (SE) is calculated as follows:

SE = σ/√n

Where n is the sample size and is the population standard deviation.

SE = 83/√160 = 6.575

Z = (X - μ) / SE

Where X represents the sample mean, is the population mean, and SE represents the standard error of the mean.

Z = (553.9 - 541) / 6.575 = 1.94

As a result, the probability that the average battery life exceeds 553.9 minutes is 0.0262 (or 2.62%). The answer, rounded to four decimal places, is 0.0262.

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Answer:

--------------------------

Find the prime factors of 1352:

We need another factor of 2 as a minimum added in order to have even number of same factors:

Hence we need to multiply the given number by 2.

The value of standard deviation of the stock returns is 10.246%.

The variance or dispersion of a group of data points is measured by standard deviation. Finding the square root of the variance, which is the sum of the squared deviations between each data point and the mean, is how it is determined. In statistics, the term "standard deviation" is used to characterise the distribution of a data collection and to estimate the probability of certain outcomes or events.

The standard deviation is determined using the formula:

√(V).

The mean of the given data is:

(−12 + 10 + 5 − 7 + 3) / 5 = −0.2%

Now, the variance is:

Variance = [ (−12 − (−0.2))² + (10 − (−0.2))² + (5 − (−0.2))² + (−7 − (−0.2))² + (3 − (−0.2))² ] / 5

Variance = 104.96

Now, for standard deviation:

Standard deviation = √(104.96) = 10.246%

Hence, the value of standard deviation of the stock returns is 10.246%.

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The area of the enlarged logo will be 9 times larger than the original logo.

When an object is enlarged or scaled up how does it area change ?

When an object is enlarged or scaled up by a factor of [tex]k[/tex], both its length and width are multiplied by [tex]k[/tex]. Therefore, the new length is [tex]k[/tex] times the original length, and the new width is [tex]k[/tex] times the original width.

The area of the new object is the product of the new length and width, which is ([tex]k[/tex] times the original length) multiplied by ([tex]k[/tex] times the original width), or [tex]k^2[/tex] times the original area.

Therefore, the area of an object increases by a factor of [tex]k^2[/tex] when the object is enlarged or scaled up by a factor of [tex]k[/tex].

Calculating how many times larger the area of the enlarged logo will be :

The original logo has dimensions of 3 inches by 5 inches, so its area is 3 x 5 = 15 square inches.

Mr. James wants to enlarge the logo so that the dimensions are 3 times larger than the original. This means the new dimensions will be 9 inches by 15 inches.

To determine how many times larger the area of the enlarged logo will be, we need to compare the areas of the original logo and the enlarged logo. The area of the enlarged logo is 9 x 15 = 135 square inches.

To find out how many times larger the area of the enlarged logo is compared to the original logo, we divide the area of the enlarged logo by the area of the original logo:

135 square inches ÷ 15 square inches = 9

Therefore, the area of the enlarged logo will be 9 times larger than the area of the original logo.

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Answer:

Step-by-step explanation:

D

Answer: The poster should have width 17.50 CM

Step-by-step explanation:

Given that boasted I have a total area 245 cm square area of a poster is 200 and 45 20 m square. And it is in the rectangle format. So we know that the poster is always in the rectangle format and the area of rectangular area is equal to L N T W. So 245 will be equal to Ln tW. From this. We need to find L. So L is equal to 245 divided by W. Now consider the diplomatic representation of the poster. So it has mentioned that there is a margin around the edges of six cm at the top. This is 6cm and four cm at the sites. All the four sides 3 sides are four cm. So from this we need to find land and the doctor posted area that is printed area. The first wine printed with www. Z. Quilter. Now let this be total birth will be W. And this will be four and this will be four. Therefore posted with will be we need to find this part alone. So W -4 -4 will give the this part with. So W -4 -4. So post printed with PW will be equal to W -8. Similarly printed lunch will be equal to The total length is already we have found 245 by W. And we need to find this part length. So we have to subtract six and four from the total length so that that will give them this part length, So -6 -4. So printed length will be equal there 245 Divided by W -10. And we know that formula for area of a rectangle. Urz D is equal to L W. Now substitute the printed with and printed length in the area formula. We have to find the printed area. So Printed area Zeke Walter W -8 into 245 Divided by W -10. To simplify this, we get 325 minus 10. W -100 and 30,960 W. to the power of -1. Now fine D A by D. T. Not D D. This is D. W. So this is equal to differentiation of constant alma zero, this is minus 10 plus 900 and 60 W. to the power of -2 and equate this to be equal to zero. We out to find the maximum width so D A by D W is equal to zero, therefore minus 10 1960 W. To the power of -2 will be equal to zero. To simplify this, we get them maximum with value the W is equal term I wrote off 196. Therefore the value of W. Z quilter plus or minus 14. We will neglect the negative values since we cannot be negative. So one we assume the positive values. So what will be equal to 14 and length will be equal to 245 divided by 14, So which will be equal to 17.50 centimeters. And they have concluded that At W is equal to 14 cm and lunch will be equal to 17.50 cm needed to print the largest area. I hope you found the answer to school. Thank you.

The inequality is X > 0 and a word sentence represent the graph is  X  the graph of a number line with an open circle on zero and an arrow pointing to the right.

The inequality X > 0 represents the graph of a number line with an open circle on zero to left and an arrow pointing to the right. This means that any value of X that is greater than zero is a valid solution for the inequality.

In other words, X can be any positive number, such as 1, 2, 3, and so on. However, X cannot be zero or any negative number, as those values do not satisfy the inequality. Therefore, the word sentence that represents this inequality is "X is greater than zero."

This means that X must be a positive number, and it can be any value that is greater than zero.

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Answer:

1. x = 11

Step-by-step explanation:

1. x + 7 = 18

move 7 to right then change the sign
        x = 18 - 7
        x = 11
2. x - 13 = 15

move -13 to right then change the sign
        x = 15 + 13
        x = 28
3. 8x = 64
   8      8
divided by 8 both side
        x = 8
4. 5x-13-12=0 it this the correct given?
add same variable
   5x = 13 + 12
   5x = 25
   5      5

divided by 5 both side
         x = 5

In response to the stated question, we may respond that Given that she is an M, the probability that the chosen female is both a M and a C is 0.6.

Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

a) The girl has a 0.65 chance of being a Clothes Pony (C).

b) The girl has a 50% chance of being a Makeup Missie (M).

c) The girl has a 0.30 chance of being both a Clothing Pony and a Makeup Missie.

d) Using the conditional probability formula, we can calculate the likelihood that the chosen female is both a M and a C provided that she is a M:

P(C|M) = P(C and M) divided by P (M)

Part (c) tells us that P(C and M) = 0.30, while Part (b) tells us that P(M) = 0.50. Therefore:

P(C|M) = 0.30 / 0.50 = 0.6

Given that she is an M, the likelihood that the chosen female is both a M and a C is 0.6.

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Therefore , the solution of the given problem of function comes out to be  the zeros of the function y = x² + x - 2 are x = -2 and x = 1, and the polynomial in factored form is y = (x + 2)(x - 1).

All of the subjects, including actual and fictitious locales and arithmetic variable design, will be covered in the midterm exam questions. a schematic illustrating the connections between various components that work together to produce the same outcome. A service is made up of many unique parts that work together to produce unique outcomes for each input. Every postbox has a specific location that could serve as a refuge.

Here,

We can draw points for different values of x and y to graph the function => y = x² + x- 2:

|x| |y=x²+x-2|

|---|-----------|

|-3 | 4 |

|-2 | 0 |

|-1 | -2 |

|0 | -2 |

|1 | 0 |

|2 | 6 |

|3 | 12 |

We can use the elements in the table above to plot them on a graph and link them to create a parabolic shape.

We can search for the values of x where the function y = 0 to determine the function's zeros.

Using the zeros we discovered, we can recast the function as the polynomial in factored form is:

=> y = (x + 2)(x - 1)

Therefore, the zeros of the function y = x² + x - 2 are x = -2 and x = 1, and the polynomial in factored form is y = (x + 2)(x - 1).

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Answer:

Step-by-step explanation:

there is an 85% chance that the two cards drawn at random from an ordinary deck of 52 cards will not be sixes.

The probability of drawing a card from an ordinary deck without replacement can be determined using the concept of conditional probability. Conditional probability is the probability of an event occurring, assuming that another event has already occurred.

In order to calculate the probability that the two cards drawn are not sixes, we can use the formula:

P(A and B) = P(A) x P(B|A)

Where A and B represent two independent events, P(A) is the probability of event A occurring, and P(B|A) is the conditional probability of event B occurring given that event A has already occurred.

The probability of drawing the first card that is not a six is:

P(A) = 48/52 = 0.9231

The probability of drawing the second card that is not a six, given that the first card drawn was not a six, is:

P(B|A) = 47/51 = 0.9216

Therefore, the probability of drawing two cards at random from an ordinary deck of 52 cards and having neither of them be a six is:

P(A and B) = P(A) x P(B|A) = 0.9231 x 0.9216 = 0.8503 or approximately 85%.

This means that there is an 85% chance that the two cards drawn at random from an ordinary deck of 52 cards will not be sixes.

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If we total up the dots plot for 3, 4, and 5 pets, we find that 3 people have 2 pets at home, 10 individuals have at least 3 pets at home.

The fact that dogs are the most common pet in the world shouldn't be shocking. There is a reason why there are tens of millions of dogs living in the United States alone, which is why some people say that dogs are a man's greatest friend. Around the world, at least one dog is kept in one-third of all households.

A fully domesticated animal kept constitutes a "household pet." a pet kept by you for personal company, like a dog, cat, reptile, bird, or mouse. Any kind of horse, cow, pig, sheep, goat, chicken, turkey, other captive fur-bearing animal is not considered a household pet, nor is any animal that is typically kept for food or profit.

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The total number of people with pets at home is 11, which is the sum of the heights of the columns.

What is equation?

A math equation is a method that links two claims and represents equivalence using the equals sign (=). An equation is a mathematical statement that establishes the equivalence of two mathematical expressions in algebra.

Based on the given dot plot, we can answer the following questions:

How many people have 2 pets at home?

Answer: Two people have 2 pets at home, as indicated by the two dots in the second column.

How many people have at least 3 pets at home?

Answer: Six people have at least 3 pets at home, as indicated by the dots in the third column and beyond.

How many more people have 2 pets than 5 pets?

Answer: There are no dots in the last column, which represents 5 pets. Therefore, the difference between the number of people with 2 pets and those with 5 pets is 2 - 0 = 2.

How many people have less than 3 pets at home?

Answer: Three people have less than 3 pets at home, as indicated by the dots in the first two columns.

Therefore, the total number of people with pets at home is 11, which is the sum of the heights of the columns.

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Let X and Y be independent random variables, uniformly distributed in the interval [0, 1 ]. The CDF of X - Y is FZ(z) = (1/2)(1+z)^2 for -1 ≤ z ≤ 0, 1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1, 0 for z < -1 or z > 1. The PDF of X - Y is fZ(z) = z + 1 for -1 < z < 0, 1 - z for 0 < z < 1, 0 otherwise.

To find the CDF of X - Y, we first note that the range of X - Y is [0, 1]. Let Z = X - Y, then:

FZ(z) = P(Z ≤ z) = P(X - Y ≤ z)

We can write this as an integral over the joint distribution of X and Y:

FZ(z) = ∫∫[X - Y ≤ z] fXY(x, y) dx dy

Since X and Y are independent, the joint distribution is simply the product of their marginal distributions:

fXY(x, y) = fX(x) fY(y) = 1 * 1 = 1

for 0 ≤ x, y ≤ 1.

Thus, we have:

FZ(z) = ∫∫[X - Y ≤ z] dx dy

= ∫∫[Y ≤ X - z] dx dy

= ∫0^1 ∫0^(x-z) 1 dy dx + ∫0^1 ∫(x-z)^1 1 dy dx

= ∫0^(1+z) (1-z) dx

= (1/2)(1+z)^2 for -1 ≤ z ≤ 0

= 1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1

Therefore, the CDF of X - Y is:

FZ(z) =

(1/2)(1+z)^2 for -1 ≤ z ≤ 0

1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1

0 for z < -1 or z > 1

To find the PDF of X - Y, we differentiate the CDF:

fZ(z) = dFZ(z)/dz =

z + 1 for -1 < z < 0

1 - z for 0 < z < 1

0 otherwise

Therefore, the PDF of X - Y is:

fZ(z) =

z + 1 for -1 < z < 0

1 - z for 0 < z < 1

0 otherwise

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Answer:

x = 2

y = -3/4

Step-by-step explanation:

1. Substitute y=1/8x -1 in −5x+4y=−13

-5x+4(1/8x -1) = -13

2. Solve for x

-5x + 4/8x - 4 = -13

-9/2x - 4 = -13

-9/2x = -9

x = 2

3. Now that you know x = 2, plug it into y=1/8x - 1 to find what y is.

y= 1/8(2) - 1

y= 2/8 - 1

y= -3/4

The discount rate that should be applied to a new project's cash flows is the Weighted Average Cost of Capital (WACC). To calculate WACC, you need to first calculate the cost of debt. This is done by taking the face value of the bonds ($1000) multiplied by the coupon rate (8%) multiplied by (1 - the tax rate (26%)), which equals 5.92%. The cost of debt is then calculated by taking the market value of the debt (6,380 x $1,000 x 99.1%) and dividing this by the total market value of the debt plus the market value of the equity (6,380 x $1,000 x 99.1% + 76,650 x $50 + 14,250 x $80), which equals 5.22%.

Next, you need to calculate the cost of equity using the Capital Asset Pricing Model (CAPM). This is done by taking the risk-free rate (3.21%) plus the market risk premium (9.79%) multiplied by the firm's beta (1.47), which equals 17.18%.

The WACC is then calculated by taking the cost of equity multiplied by the proportion of equity (76,650 x $50 + 14,250 x $80 divided by the total market value of the debt plus the market value of the equity) plus the cost of debt multiplied by the proportion of debt (6,380 x $1,000

The problem asks for the time, we know that the blue whale will reach a depth of 190.65 meters below sea level after 26.7 minutes.

If a function is linear or not, its graph can be examined to determine. A straight line results from the graphing of a linear function. A nonlinear function has some sort of curve in contrast to a linear function.

The slope and y-intercept must first be determined in order to build a linear function. The slope can be determined using the two provided locations (4, -33.7) and (13, -99.4):

slope is the product of (y-change) and (change in x)

slope = (-99.4 - (-33.7)) / (13 - 4)

slope = -65.7 / 9

slope = -7.3

We can now use the point-slope form of a linear equation to obtain the equation of the line since we know its slope:

y - y1 = m(x - x1)

y - (-33.7) = -7.3(x - 4)

y + 33.7 = -7.3x + 29.2

y = -7.3x - 4.5

The following is the linear function that plots the blue whale's depth below sea level against time (x):

y = -7.3x - 4.5

We can plug in this number for y and solve for x to get how long it will take the blue whale to descend 190.65 metres below sea level:

190.65 = -7.3x - 4.5

195.15 = -7.3x \sx = -26.7

We know that the blue whale will descend to a depth of 190.65 metres below sea level in 26.7 minutes because the problem specifically asks for the time.

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The returned value would be 70 which is the missing value in the data set. Hence, option D is correct. We have some X values; we called these numeric inputs and some Y value that we are trying to predict.

This set of data would yield a result of 70 if a median imputer were run on it. In regression, we have some X values that are referred to as independent variables and some Y values that are referred to as dependent variables (this is the variable we are trying to predict). Several Y values are possible, but they are uncommon.

Learning a function that can predict Y given X is the fundamental concept behind a regression. Depending on the data, the function may be linear or non-linear.

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Complete question is:

Imagine X in the below is a missing value. If I were to run a median imputer on this set of data. What would the returned value be? 50 , 60 , 70 , 80 , 100 , 60 , 5000 , x (It's okay to have to look up how to do this!)

50

An error

80

70

100

The basic idea of a regression is very simple. We have some X values, we called these ______ and some Y value (this is the variable we are trying to _______.

We could have multiple Y values, but that is not but that is not re-ordered ordinals intercepts features numeric inputs.

Answer:

B) Group 2

Step-by-step explanation:

Group 1 wrote it correctly because there are 3 groups of 4x and 3 groups of 3

Group 2 wrote it incorrectly because that would mean 9 groups of 4x, which is not the case here

Group 3 wrote it correctly because 3(4x) + 3(3) = 3(4x+3) due to the Distributive Property

Group 4 wrote it correctly because it is an expansion of Group 3's expression

The appropriate inference from the data is (B) Since [tex]20%[/tex] of this group read below grade level, we cannot draw any generalizations about the population. Thus, option B is correct.

A representative sample is a subset of data, often drawn from a wider population, that can show qualities that are similar.

Because the data produced contains more manageable, smaller representations of the larger group, representative sampling aids in the analysis of bigger groups.

Although the sample may be representative of seventh-grade American students, it is not necessarily representative of all seventh-graders or all American children. Hence, without additional data or research, we cannot extrapolate the sample's results to the overall population.

Therefore, 20%20, percent of all American students in seventh grade were reading below grade level.

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The equation of the line that is parallel to the given line and passes through the point (-3, 2) is 3x - 4y = -17.

If the line's equation is axe + by + c = 0 and the coordinates are (x1, y1).

To find the equation of a line parallel to a given line, we must first understand that parallel lines have the same slope. As a result, we must first determine the slope of the given line.

3x - 4y = -17 is the given line. To determine its slope, solve for y and write the equation in slope-intercept form:

-3x - 4y = -3x - 17 y = (3/4)x + (17/4)

This line has a 3/4 slope.

Now we want to find the equation of a parallel line that passes through the point (-3, 2). Because the new line is parallel to the given line, it has a slope of 3/4.

We can write the equation of the new line using the point-slope form of the equation of a line as:

y - y1 = m(x - x1)

where m represents the slope and (x1, y1) represents the given point (-3, 2).

When m = 3/4, x1 = -3, and y1 = 2, we get:

y - 2 = (3/4)(x - (-3))

y - 2 = (3/4)(x + 3)

Divide both sides by 4 to get rid of the fraction.

4y - 8 = 3x + 9

3x - 4y = -17

As a result, the equation of the parallel line that passes through the point (-3, 2) is 3x - 4y = -17.

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Answer:

23.8 cm

Step-by-step explanation:

17 * 140% = 17 * 1.4 = 23.8 cm

The image of the house would be 23.8 cm tall on the screen.

To calculate the height of the image of the house on the screen, we can use the given zoom factor of 140%.

The zoom factor of 140% means that the images on the screen are 140% as long and 140% as wide compared to when they are printed on a sheet of paper.

To calculate the height of the image on the screen, we need to multiply the printed height by the zoom factor (140% or 1.4).

Height on the screen = Printed height * Zoom factor

Height on the screen = 17 cm * 1.4

Height on the screen = 23.8 cm

Therefore, the image of the house would be 23.8 cm tall on the screen.

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(a) The interval (-∞, ∞).

(b) The interval (-∞, ∞).

(c) The interval (-∞, ∞).

(d) The interval (-π/2, π/2) \ {0}.

(a) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient function, 3t, is continuous and bounded. Since 3t is a continuous and bounded function for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(b) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, t(t - 4), 3t, and 4, are continuous and bounded. Since t(t - 4), 3t, and 4 are continuous and bounded functions for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(c) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, cost and In|t|, are continuous and bounded. Since cost and In|t| are continuous and bounded functions for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(d) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, x - 2, 1, and (x - 2)tanx, are continuous and bounded. Since x - 2, 1, and (x - 2)tanx are continuous and bounded functions for all x in the interval (-π/2, π/2) \ {0} , the given initial value problem is certain to have a unique twice-differentiable solution for all x in (-π/2, π/2) \ {0}.

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The given question is incomplete, the complete question is:

determine the longest interval in which the given initial value problem is certain to have a unique twice- differentiable solution. Do not attempt to find the solution. (a) ty" + 3y = 1, y(1) = 1, y'(1) = 2   (b)   t(t – 4)y" + 3ty' + 4y = 2, y(3) = 0, y'(3) = -1 (c)   y" + (cost)y' + 3( In |t|) y = 0, y(2) = 3, y'(2) = 1 (d) (x - 2)y"+y' +(x - 2)(tan x) y = 0, y(3) = 1, y'(3) = 2

Answer:

56

90

56

Step-by-step explanation:

easy easy lol.

aaaaa

Answer:

E.  x² + (y - 3)² = 16

Step-by-step explanation:

The equation of a circle in the standard (x, y) coordinate plane with center (h, k) and radius r is given by:

[tex]\boxed{(x - h)^2 + (y - k)^2 = r^2}[/tex]

To find the equation of the circle with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0, we need to first write the equation of the second circle in the standard form.

We can complete the square for y to rewrite this equation in standard form. To do this move the constant to the right side of the equation:

[tex]\implies x^2 + y^2 - 6y + 4 = 0[/tex]

[tex]\implies x^2 + y^2 - 6y = -4[/tex]

Add the square of half the coefficient of the term in y to both sides of the equation:

[tex]\implies x^2 + y^2 - 6y +\left(\dfrac{-6}{2}\right)^2= -4+\left(\dfrac{-6}{2}\right)^2[/tex]

[tex]\implies x^2 + y^2 - 6y +9= -4+9[/tex]

[tex]\implies x^2 + y^2 - 6y +9=5[/tex]

Factor the perfect square trinomial in y:

[tex]\implies x^2+(y-3)^2=5[/tex]

[tex]\implies (x-0)^2 + (y-3)^2=5[/tex]

So the center of this circle is (0, 3) and its radius is √5 units.

Since the new circle has the same center, its center is also (0, 3).

We know its radius is 4 units, so we can write the equation of the new circle as:

[tex]\implies (x - 0)^2 + (y - 3)^2 = 4^2[/tex]

[tex]\implies x^2 + (y - 3)^2 = 16[/tex]

Therefore, the equation of the circle in the standard (x, y) coordinate plane with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0 is x² + (y - 3)² = 16.

To find:-

Answer:-

The given equation of the circle is ,

[tex]\implies x^2+y^2-6y + 4 = 0 \\[/tex]

Firstly complete the square for y in LHS of the equation as ,

[tex]\implies x^2 + y^2 -2(3)y + 4 = 0 \\[/tex]

Add and subtract 3² ,

[tex]\implies x^2 +\{ y^2 - 2(3)(y) + 3^2 \} -3^2 + 4 = 0 \\[/tex]

The term inside the curly brackets is in the form of a²-2ab+b² , which is the whole square of "a-b" . So we may rewrite it as ,

[tex]\implies x^2 + (y-3)^2 -9 + 4 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 - 5 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 = 5\\[/tex]

can be further rewritten as,

[tex]\implies (x-0)^2 + (y-3)^2 = \sqrt5^2\\[/tex]

now recall the standard equation of circle which is ,

[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]

where,

So on comparing to the standard form, we have;

[tex]\implies \rm{Centre} = (0,3)\\[/tex]

Now we are given that the radius of second circle is 4units . On substituting the respective values, again in the standard equation of circle, we get;

[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]

[tex]\implies (x-0)^2 + (y-3)^2 = 4^2 \\[/tex]

[tex]\implies \underline{\underline{\red{ x^2 + (y-3)^2 = 16}}}\\[/tex]

and we are done!