PLEASE !! HELP i’ve been on this question for a while

The result (recurrent value), A = sum j=1 to 89 ln(j), is true for every n. This is the desired result.

As in T(n) = T(n/2) + n, T(0) = T(1) = 1, a recurrence or recurrence relation specifies an infinite sequence by explaining how to calculate the nth element of the sequence given the values of smaller members.

We can start by proving the base case in order to demonstrate the first portion through recurrence. Let n = 1. Next, we have:

Being true, ln(a1) = ln(a1). If n = k, let's suppose the formula is accurate:

Sum j=1 to k ln = ln(prod j=1 to k aj) (aj)

Prod j=1 to k aj * ak+1 = ln(prod j=1 to k+1 aj)

(Using the logarithmic scale) = ln(prod j=1 to k aj) + ln(ak+1)

Using the inductive hypothesis, the property ln(ab) = ln(a) + ln(b)) = sum j=1 to k ln(aj) + ln(ak+1) = sum j=1 to k+1 ln (aj)

(b), we can use the just-proven formula:

A = ln(1, 2,...) + ln + ln (89)

= ln(j=1 to 89) prod

sum j=1 to 89 ln = ln(prod j=1 to 89 j) (j).

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There will be a population of 2,097,152 rabbits after 5 years.

A form of growth known as exponential growth occurs when a quantity's rate of expansion is proportionate to its present value. In other words, a quantity expands more quickly the greater it is. A prime example of exponential expansion is the rabbit population, which doubles in size every three months.

Given that, population of rabbits is doubling every 3 months.

That is,

5 years = 5 x 12 = 60 months

Number of doublings = 60 / 3 = 20

For every doubling, the population will be twice as large.

Thus,

P = 2 x 2²⁰ = 2 x 1,048,576 = 2,097,152 rabbits

Therefore, there will be approximately 2,097,152 rabbits after 5 years.

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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 new position of Rhombus ABCD after the translation can be described as follows: point A is now at (-1,-8), point B is at (2,-3), point C is at (7,0), and point D is at (4,-5).

To translate Rhombus ABCD using the rule (x, y) = (x+2, y-6), we add 2 to the x-coordinate and subtract 6 from the y-coordinate for each vertex.

Thus, the new vertices for the translated rhombus are:

A' = (-3+2, -2-6) = (-1, -8)

B' = (0+2, 3-6) = (2, -3)

C' = (5+2, 6-6) = (7, 0)

D' = (2+2, 1-6) = (4, -5)

Therefore, the coordinates for the translated Rhombus ABCD are A'(-1,-8), B'(2,-3), C'(7,0), and D'(4,-5).

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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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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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[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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The line x + y = 9 is y = -x + 6 is keeps through the point (5,1).

To find the equation of the line that passes through the point (5,1) and is parallel to the line x + y = 9, we need to first find the slope of the line x + y = 9.

Rearranging the equation in slope-intercept form, we get y = -x + 9

The slope of this line is -1, since the coefficient of x is -1.

Since the line we want to find is parallel to this line, it will have the same slope of -1.

Using the point-slope form of a line, the equation of the line passing through the point (5,1) and with a slope of -1 is: y - 1 = -1(x - 5)

Simplifying and rearranging the equation, we get:

y - 1 = -x + 5

y = -x + 6

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Regardless of the appearance of the normal distribution or the size of the standard deviation, approximately less than 2.98% of observations consistently fall within two deviations types (one high and one low).

The normal distribution, also known as the Gaussian distribution, is a probability distribution symmetrical about the mean, indicating that data near the mean occurs more frequently than data far from the mean. In graphical form, the normal distribution is represented by a "bell curve".

The normal distribution is important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distribution is unknown. Their importance is partly due to the central limit theorem. It states that in some cases the mean of many samples (observations) of a random variable with finite mean and variance is itself a random variable - whose distribution converges to a normal distribution as the size of the l sample increases. Therefore, physical quantities assumed to be the sum of many independent processes, such as measurement errors, tend to have a near-normal distribution.

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The wave is polarized parallel to the y-axis, and the magnetic field lines are parallel to the x-axis. Here option D is the correct answer.

The oscillating electric dipole along the z-axis creates an electromagnetic wave with electric and magnetic fields perpendicular to each other and to the direction of wave propagation. At a position on the y-axis far from the origin, the electric field will be parallel to the y-axis.

The polarization of the wave refers to the orientation of the electric field vector. Since the electric field is parallel to the y-axis, the wave is polarized parallel to the y-axis.

According to the right-hand rule, the direction of the magnetic field lines will be perpendicular to both the electric field and the direction of wave propagation, which is along the z-axis. Therefore, the magnetic field lines will be parallel to the x-axis.

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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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Three numbers have a product of 115 and a sum of 3(√115), which is the smallest possible sum.

In mathematics, a positive number is any number that is greater than zero. This includes all numbers that are written without a minus sign or are explicitly denoted as positive, such as 1, 2, 3, 4, 5, and so on

To find three positive numbers whose product is 115 and whose sum is as small as possible, we can use the AM-GM inequality. In other words, if we have three positive numbers x, y, and z, then:

(x + y + z)/3 ≥ (xyz)^(1/3)

If we rearrange this inequality, we get:

x + y + z ≥ 3(√(xyz))

Now, let's apply this inequality to the given problem. We want to find three positive numbers x, y, and z whose product is 115 and whose sum is as small as possible. Therefore, we want to minimize x + y + z while still satisfying the condition xyz = 115.

Using the AM-GM inequality, we have:

x + y + z ≥ 3(√(xyz)) = 3(√115) ≈ 16.75

Therefore, the sum of the three numbers is at least 16.75. To find three numbers that achieve this minimum sum, we can use trial and error or solve the system of equations:

xyz = 115

x + y + z = 3(√115)

One solution to this system is:

x = √(115/3)

y = √(115/3)

z = 3(√(115/3)) / 5

These three numbers have a product of 115 and a sum of 3(√115), which is the smallest possible sum.

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The complete question is Find three positive numbers whose product is 115.

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.

Answer:

X+4

Step-by-step explanation:

Area = l *b

x^2 + 13x + 36 = (X+9) * b

x^2 + 9x + 4x + 36 = (X+9) * b

X(X+9) + 4(X+9) = (X+9) * b

(X+4) (X+9) = (X+9) * b

b = (X+4)

Answer:

When it was purchased (year 0) the coin was worth $7

Step-by-step explanation:

we have

[tex]f(t) = 7(2)^t[/tex]

This is a exponential function of the form

[tex]y=a(b)^x[/tex]

where

a is the initial value

b is the base

In this problem we have

[tex]a=\$7[/tex]

[tex]b=2[/tex]

[tex]b=1+r[/tex]

so

[tex]2=1+r[/tex]

[tex]r=1[/tex]

[tex]r=100\%[/tex]

The y-intercept is the value of the function when the value of x is equal to zero

In this problem

The y-intercept is the value of a rare coin when the year t is equal to zero

[tex]f(0)=7(2)^0[/tex]

[tex]f(0)=\$7[/tex]

therefore

The meaning of y-intercept is

When it was purchased (year 0) the coin was worth $7

Answer:

Value of the coin when it was first released

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

The y-intercept is the value of f(0).

Substitute t = 0 and find the y-intercept:

This is representing the value of the coin when it was released.

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:

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.

Answer:

3x, 4x | 5, 3

Step-by-step explanation:

The decoration's volume, to the closest tenth of an inch cubic, is: 308.9 cubic inches make up V.

The quantity of space that an object or substance occupies is measured by its volume. Usually, it is expressed in cubic measures like cubic metres, cubic feet, or cubic inches. By multiplying an object's length, width, and height, or by applying a formula unique to the shape of the object, one can determine the volume of the object.

given

The cylinder's radius is equal to half of its diameter, or 14/2, or 7 inches. The new water level is 9 + 2 = 11 inches because the initial water level was 9 inches and the decoration raised the water level by 2 inches.

The decoration's volume is equivalent to the volume of water it removed from the area.

We can determine the volume of the ornamentation by using the following formula: V = r2h.

V = (72/2), which equals 98 cubic inches.

The decoration's volume, to the closest tenth of an inch cubic, is: 308.9 cubic inches make up V.

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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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Answer: x=211/1296

Step-by-step explanation:

Answer:

The answer is Nitrogen

Hope this helps :)

I can give you with a general explanation of the functions of muscles, bones, and sensitive organs, as well as how they work together.

Muscles are responsible for movement and give the force needed to move bones. They're attached to bones via tendons and work in dyads or groups to produce coordinated movement. Muscles are also responsible for maintaining posture and generating heat.

Bones give a rigid frame for the body, cover internal organs, and serve as attachment points for muscles. They also store minerals similar as calcium and produce blood cells in the bone gist.

sensitive organs, similar as the eyes, cognizance, nose, and skin, descry and respond to stimulants in the terrain. They transmit information to the brain, which processes the information and generates an applicable response.

All three body corridor work together in the musculoskeletal system to produce movement, maintain posture, and respond to external stimulants. Muscles attach to bones and work together to produce coordinated movement. sensitive organs descry stimulants in the terrain and transmit information to the brain, which coordinates muscle movement and generates a response. Bones give the rigid frame and attachment points for muscles, as well as cover internal organs.

There are 1124 ways to choose a committee of 7 members with at least one member from each group and at least 3 artists.

Here, we have to solve this problem, we can use the concept of combinations, which involves counting the ways to choose a specific number of items from a larger set without regard to the order of selection.

Given the conditions that at least one member must be chosen from each group (artists, singers, writers) and there must be at least 3 artists, we can break down the problem into cases.

Case 1: Choosing 1 artist, 1 singer, and 5 members from the remaining groups (writers).

Case 2: Choosing 2 artists, 1 singer, and 4 members from the remaining groups (writers).

Case 3: Choosing 3 artists, 1 singer, and 3 members from the remaining groups (writers).

For each case, we will calculate the number of ways to choose members and then sum up the results from all three cases to get the total number of ways.

Let's calculate the number of ways for each case:

Case 1:

Number of ways to choose 1 artist: 6C1 (6 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 5 writers: 5C5 (1 way)

Total ways for case 1: 6C1 * 4C1 * 5C5 = 6 * 4 * 1 = 24

Case 2:

Number of ways to choose 2 artists: 6C2 (15 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 4 writers: 5C4 (5 ways)

Total ways for case 2: 6C2 * 4C1 * 5C4 = 15 * 4 * 5 = 300

Case 3:

Number of ways to choose 3 artists: 6C3 (20 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 3 writers: 5C3 (10 ways)

Total ways for case 3: 6C3 * 4C1 * 5C3 = 20 * 4 * 10 = 800

Now, add up the total ways from all three cases:

Total ways = 24 + 300 + 800 = 1124

So, there are 1124 ways to choose a committee of 7 members with at least one member from each group and at least 3 artists.

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The greater number is [tex]$n^3+8$[/tex]

Let x be the greater number and y be the smaller number. We know that x-y=8.

We are also given that the smaller number is n³.

So we can set up the equation:

x = y + 8

x = n³ + 8

Therefore, the greater number is [tex]$n^3+8$[/tex].

The greater number is given as n³ + 8. If the smaller number we get is represented by the n³,  then by adding 8 to that value gives the greater number. The difference between the two numbers is always going to be 8, regardless of the value of n.

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The volume of cylindrical tin is 1584 [tex]cm^3[/tex]. The depth of the oil in the tin is 9cm.

[tex]V_1 =[/tex] VOLUME OF CYLINDRICAL TIN

   [tex]= \pi r^2 h[/tex]

  [tex]=\frac{22}{7}[/tex] x 6 x 6 x 14

  = 44 x 36

  = 1584 [tex]cm^3[/tex]

[tex]V_2 =[/tex] VOLUME OF RECTANGULAR TIN

    = lbh = 1584

    = (16)(11)(h) = 1584

    = 176h =1584

    = h = 1584 / 176

    = h = 9 cm

A cylinder is a three-dimensional shape that consists of a circular base and a curved surface that extends upward to meet at a point known as the apex. The volume of a cylinder is the amount of space occupied by the shape and is given by the formula V = πr²h, Once we have calculated the area of the circular base, we can multiply it by the height of the cylinder to get the volume.

To calculate the volume of a cylinder, we need to know its dimensions, which are the radius and height. The radius is the distance from the center of the circular base to the edge, while the height is the distance between the two circular bases.

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The mean number of burps after drinking root beer is between 0.66 and 4.24 burps fewer than after drinking cola.

Mean: The "average" number obtained by adding all data points and dividing the total number of data points by the total number of data points.

Part A: A paired t-test can be used to see if there is a significant difference in the number of burps after drinking root beer versus cola. The null hypothesis states that there is no difference in the mean number of burps between the two beverages, whereas the alternative hypothesis states that there is. Using a two-tailed test with a significance level of = 0.05, we find that the t-value is -3.365 and the p-value is 0.003. We reject the null hypothesis because the p-value is less than the significance level and conclude that there is a significant difference in the mean number of burps between root beer and cola.

Part B: We can use the paired t-test formula to generate a 95% confidence interval for the difference in the mean number of burps between root beer and cola:

(xd - d) / (sd / n) t

where xd represents the sample mean difference, d represents the hypothesised population mean difference (which is 0), sd represents the sample standard deviation of the differences, and n represents the sample size.

We calculate the sample mean difference to be -2.45 and the sample standard deviation of the differences to be 2.69 using the data in the table. We get a t-value of -3.365 with 19 degrees of freedom after plugging in these values. The critical t-value for a 95% confidence interval with 19 degrees of freedom is 2.093, according to a t-distribution table.

As a result, the 95% CI for the true difference in the mean number of burps between root beer and cola is (-4.24, -0.66). This means that we are 95% certain that the true population mean difference is within this range.

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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 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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