y=x^2+7x-3
complete the square to re-write the quadratic function in vertex form.
pls help

This procedure is repeated for each interval until the overall frequency of 35 is reached.

A data summary called a frequency distribution displays the frequency, or amount of occurrences, of each value or range of values in a data set. It is frequently displayed as a table or graph, with the values enumerated along one axis and the frequencies of those values listed along the other. The pattern or shape of a data collection can be described using frequency distributions, which can also be used to spot outliers or other unusual values. They can also shed light on the data's central trend and variability.

given

To complete a cumulative frequency table:

Class interval Frequency Cumulative frequency

0-10               5                                   5

10-20       8                                  13

20-30       12                                 25

30-40        7                                 32

40-50         3                                35

Total        35                               35

The number of the first class interval is 5.

For the second interval, we multiply the total frequency of 5 plus the frequency of 8 to get 13.

This procedure is repeated for each interval until the overall frequency of 35 is reached.

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The missing value in the cumulative table is 36.

The missing value in the cumulative table can be determined by examining the frequencies given in the table. Let's analyze the frequencies step by step:

1. The frequency for scores less than 145 is given as 16.
2. The frequency for scores less than 150 is given as 26.
3. The frequency for scores less than 155 is given as 36.
4. The frequency for scores less than 160 is not explicitly given in the table, but we can determine it by subtracting the frequency for scores less than 155 (36) from the frequency for scores less than 150 (26).

This gives us a value of [tex]26 - 36 = -10.[/tex]

Since a frequency cannot be negative, we can conclude that there is an error in the given table. The cumulative frequency for scores less than 160 should be 36 instead of -10.

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then the area would be: [tex]Area=\frac{(a+b)}{2*h}[/tex] = (16 ft + 10 ft)/2 x 15 ft = 150 ft²

Area is a mathematical term that refers to the measurement of the size or extent of a two-dimensional region or surface. It is typically expressed in square units, such as square meters (m²), square centimeters (cm²), square feet (ft²), or square inches (in²). The area of a shape is determined by multiplying the length and width of the shape in the case of a rectangle or square, or by using more complex formulas for irregular shapes such as circles, triangles, or polygons. The concept of area is important in various fields such as mathematics, geometry, physics, engineering, and architecture, among others.

by the question.

. If we assume that these are the dimensions of a rectangle, then the area would be:

Area = length x width = 20 ft x 12 ft = 240 ft²

However, if we assume that the area is a trapezoid with a height of 15 ft, and the parallel sides of length 16 ft and 10 ft.

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Option C. 6,160 yards. To convert miles to yards, we multiply the number of miles by 1,760 (which is the number of yards in one mile). So, 3.5 miles x 1,760 yards/mile = 6,160 yards.

To convert miles to yards, we need to know that there are 1,760 yards in one mile. Therefore, to find out how many yards are in three and one-half miles, we need to multiply 1,760 by 3.5:

1,760 x 3.5 = 6,160

Therefore, there are 6,160 yards in three and one-half miles.

The other options provided are:

4,400 yards: This is equivalent to 2.5 miles, since 1 mile is 1,760 yards. Therefore, this is not the correct answer.

5,280 yards: This is equivalent to 3 miles, since 1 mile is 1,760 yards. Therefore, this is also not the correct answer.

7,040 yards: This is equivalent to 4 miles, since 1 mile is 1,760 yards. Therefore, this is not the correct answer.

So, the correct answer is 6,160 yards, which is the result of multiplying 1,760 yards by 3.5 miles.

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Numerical integration involves approximating the definite integral of a function using numerical methods.

There are various techniques for numerical integration, including the Trapezoidal rule, Simpson's rule, and Gaussian quadrature.In general, these methods divide the integration interval into smaller sub-intervals and approximate the area under the curve within each sub-interval using mathematical formulas.

The approximations are then summed up to estimate the integral over the entire interval.The accuracy of the numerical integration depends on the number of sub-intervals used and the specific method used. Generally, increasing the number of sub-intervals leads to a more accurate approximation.

Overall, numerical integration is a useful tool for evaluating integrals that cannot be solved analytically and is commonly used in various fields such as physics, engineering, and finance.

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Answer:

The submarine travels 100 ft / min.

Step-by-step explanation:

To determine how long it will take the submarine to reach the lab from the surface of the ocean, we need to know how far it is from the surface of the ocean to the lab. Since this is not given, let us present the distance as D

The submarine travels at 100ft/min

He also travels a distance of D ft

Then  the ratio below is correct:1/100 = x/Dx = D/100

Where the x is the time we want to find. If you have omitted the distance D by mistake, all you need to do is divide it by 100 to get the time you are looking for.

It will take 29.2 minutes in order for the submarine to reach the lab from the surface of the​ ocean.

Solution

To determine how long it will take the submarine to reach the lab from the surface of the ocean, we need to know how far it is from the surface of the ocean to the lab. Since this is not given, let us present the distance as D.

The submarine travels at 100ft/min

He also travels a distance of Dft

Then the ratio below is correct:

[tex]1/100 = x/D[/tex]

[tex]x = D/100[/tex]

Where x is the time we want to find.

If you have omitted the distance D by mistake, all you need to do is divide it by 100 to get the time you are looking for.

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

The formula for calculating a confidence interval, on the other hand, varies based on the population parameter being investigated. The standard error of the mean is used to estimate the precision of a sample mean in the case of population means.

The larger the confidence, the lesser the chance that it will capture the population mean. In statistical analysis, confidence intervals are used to estimate the value of a population mean or a proportion, as well as to provide a degree of precision for these estimates.

The higher the confidence level, the larger the interval, which implies that the less exact the estimation, but the more assured we are that it captures the true population parameter. However, if the confidence interval is set too wide, there is a greater possibility of missing the actual population mean.

A confidence interval (CI) is a range of values that is expected to include the true value of a population parameter (e.g., mean, proportion, variance) with a specified degree of confidence.

It is a range of values that encloses the population parameter of interest, indicating that we have a degree of certainty in our estimates.Confidence intervals may be computed for means, proportions, or differences between means or proportions using the same procedure.

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The null hypothesis would be that the lives of these brands of batteries are the same and the alternative hypothesis would be that they are different. With an alpha level of 0.05, you can determine if the difference is significant.

(a)  Use a = 0.05.
Yes, the lives of these brands of batteries are different. To analyze whether this difference is significant, you can use an ANOVA (Analysis of Variance) to determine whether the variances between the groups are statistically different.

(b) Analyze the residuals from this experiment.

The residuals can be calculated by subtracting the mean life of each brand from the observed lifetimes of each battery. For Brand 1, the residuals would be (100-84), (96-84), (92-84), (75-84), and (96-84). The residuals for Brand 2 and Brand 3 can be calculated in the same way.

(c) Construct a 95 percent interval estimate on the mean life of battery brand 2.

The 95 percent interval estimate on the mean life of battery brand 2 is 81.7 weeks to 97.3 weeks.

(d) Construct a 99 percent interval estimate on the mean difference between the lives of battery brands 2 and 3.

The 99 percent interval estimate on the mean difference between the lives of battery brands 2 and 3 is -1.8 weeks to 12.2 weeks.

(e)  Based on the results, the most reliable brand of battery is Brand 3.

(f) If the manufacturer will replace without charging any battery that fails in less than 85 weeks, what percentage would the company expect to replace?

Using the data, the company can expect to replace 3/5, or 60%, of the batteries that fail in less than 85 weeks.

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Answer:

13) F(-5, 0), so F' will be at (1, 2)

G(-1, -1), so G' will be at (5, 1)

H(-2, 3), soH' will be at (4, 5)

14) G(-2, 1), so G' will be at (-1, 3)

H(-1, 3), so H' will be at (0, 5)

I(3, 1), so I' will be at (4, 3)

J(2, -1), so J' will be at (3, 1)

Step-by-step explanation:

13) Add 6 to each original x-coordinate and add 1 to each original y-coordinate.

14) Add 1 to each original x-coordinate and add 2 to each original y-coordinate.

You may plot the new points to confirm my answer.

The Stevia is a low-calorie alternative to sugar and is a good option for people who are trying to reduce their sugar intake.

Sweet 'n Low - Modified SugarSweet 'n Low is an artificial sweetener that is made from modified sugar. Sweet 'n Low is not as sweet as some other artificial sweeteners like Splenda and Truvia. However, Sweet 'n Low is still used in many products like gum, candy, and other sweet treats. Sweet 'n Low has been around since the 1950s and is still used today as a low-calorie alternative to sugar.Splenda - 600 times sweeter than sugarSplenda is a popular artificial sweetener that is around 600 times sweeter than sugar. Splenda is often used in diet drinks, desserts, and other sweet products. Splenda is made from sugar but is modified to be much sweeter. Splenda is a low-calorie alternative to sugar and can be used by people who are trying to reduce their sugar intake.Stevia - Made from the Stevia PlantStevia is an artificial sweetener that is made from the Stevia plant. Stevia is a natural sweetener and is often used in tea and other drinks. Stevia is not as sweet as some other artificial sweeteners, but it is still a popular alternative to sugar. Stevia is also used in some foods and desserts. Stevia is a low-calorie alternative to sugar and is a good option for people who are trying to reduce their sugar intake.

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x and y are both differentiable functions of t and find the required values of  [tex]dy/dt[/tex]  and Given that [tex]x2+y2=100[/tex] and that [tex]dx/dt = 4[/tex]  and [tex]dy/dt = -2[/tex], So the required values of [tex]dy/dt[/tex] and [tex]dx/dt.[/tex] are y = 8 and x = 8.

We can explain that x and y are both differentiable functions of t is,

a) To find [tex]dy/dt[/tex]when x=6 and y=8, we first solve the equation for y:

         [tex]y2 = 100-x2[/tex]

           [tex]y2 = 100-62[/tex]

            [tex]y2 = 100-36[/tex]

              y = √(100-36)

              y = √64

              y = 8

           Therefore, [tex]dy/dt = 4.[/tex]

Now, for,

b) To find [tex]dx/dt[/tex] when x=8 and y=6, we again solve the equation for x:

           [tex]x2 = 100-y2[/tex]

         [tex]x2 = 100-62[/tex]

        [tex]x2 = 100-36[/tex]

          x = √(100-36)

           x = √64

           x = 8

      Therefore, [tex]dx/dt=-2.[/tex]

the required values of [tex]dy/dt[/tex] and [tex]dx/dt[/tex]. with given conditions are y = 8 and x = 8.

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1. 48,000 oz = 1.5 ton, 2. 4 and 1/5 T = 8,400 lb, 3. 7 gal = 56 pt, 4. 125 fl oz = 3.90625 qt 5.  the perimeter of the rectangle is 132.8 m.

Measurement units are standard quantities used to express physical quantities, such as length, mass, time, volume, and temperature. They provide a standard framework for expressing and comparing quantities, allowing for precise and accurate communication and measurement.

There are two types of measurement units: fundamental units and derived units. Fundamental units are the basic units used to express physical quantities, such as meter (for length), kilogram (for mass), and second (for time). Derived units are combinations of fundamental units used to express other physical quantities, such as square meters (for area) or meters per second (for speed).

1. To convert 48,000 oz to T (tons):

1 ton = 32,000 oz (since there are 2,000 pounds in a ton, and 1 pound = 16 ounces)

48,000 oz ÷ 32,000 oz/ton = 1.5 T

Therefore, 48,000 oz = 1.5 T

2. To convert 4 and 1/5 T to lb (pounds):

1 ton = 2,000 lb

4 T = 8,000 lb

1/5 T = 1/5 x 2,000 lb = 400 lb

4 and 1/5 T = 8,000 lb + 400 lb = 8,400 lb

Therefore, 4 and 1/5 T = 8,400 lb

3. To convert 7 gal to pt (pints):

1 gal = 8 pt

7 gal x 8 pt/gal = 56 pt

Therefore, 7 gal = 56 pt

4. To convert 125 fl oz to qt (quarts):

1 qt = 32 fl oz

125 fl oz ÷ 32 fl oz/qt = 3.90625 qt

Therefore, 125 fl oz = 3.90625 qt

5. The perimeter of a rectangle with sides of 17.1 m and 49.3 m can be found by adding up all four sides:

Perimeter = 2(17.1 m) + 2(49.3 m) = 34.2 m + 98.6 m = 132.8 m

Therefore, the perimeter of the rectangle is 132.8 m.

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Answer:

ex9wjsxswhciwebwec938129389038092

Step-by-step explanation:

gte betterv

Answer:

x = -4

y = -3

Step-by-step explanation:

-5y + 3x = 3

-8y + 9x = -12​

Time the first equation by -3

15y - 9x = -9

-8y + 9x = -12

7y = -21

y = -3

Now put -3 in for y and solve for x

-5(-3) + 3x = 3

15 + 3x = 3

3x = -12

x = -4

Let's Check

-5(-3) + 3(-4) = 3

15 - 12 = 3

3 = 3

So, x = -4 and y = -3 is the correct answer.

The tangential component of the acceleration vector at point t = 1 is aT(1) = 233/3 and The normal component of the acceleration vector at point t = 1 is aN(1) = (1/3)√10459

The acceleration vector can be found from the following formula:

[tex]a(t) = r''(t) = (-3,-10t,-8t3).[/tex]

To find the tangential component of the acceleration vector, we first need the velocity vector v(t).

[tex]v(t) = r'(t) = (-3,-10t,-8t3) .[/tex]

Next, we need to normalize the velocity vector using the following formula:

[tex]T(t) = v(t) / ||v(t)||,[/tex]

Where ||v(t)|| is the magnitude of the velocity vector.

[tex](1) = (-3,-10,-8) / \sqrt{(3^2 + 10^2 + 8^2)} = (-3/3, -10/3, -8/3) = (-1 , -10/3, -8/3) .[/tex]

Then, the tangential component of a(1) is:

[tex]aT(1) = a(1) T(1) = (-3, -10, -8) (-1, -10/3, -8/3) = 3 + 100/3 + 64/3 = 233/3.[/tex]

To find the normal component of a(1), we simply need to find the magnitude of the tangential component and subtract it from the magnitude of the acceleration vector.

[tex]aN(1) = \sqrt{ (a^2 - aT(1)^2)} = \sqrt{(3^2 + (10)^2 + (8)^2 - (233/3)^ 2)}  = \sqrt{(9 + 100 + 64 - 54289/9)} = \sqrt{(10459/9)} = (1/3)\sqrt{10459}[/tex]

Therefore, the tangential and normal components of the acceleration vector at the point t = 1 are:

[tex]aT(1) = 233/3[/tex] and [tex]aN(1) = (1/3)\sqrt{10459}[/tex]

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Answer:

To determine the total amount of interest the student paid, we can break down the problem into several steps:

Step 1: Calculate the interest that accrues during the six-month grace period

Since the loan has a payment grace period of six months, the student will not need to make any payments during this time, but interest will still be accruing. The interest rate is 6.2% per year, compounded monthly. To calculate the monthly interest rate, we divide 6.2% by 12:

i = 6.2%/12 = 0.00517

The amount of interest that accrues during the six-month grace period can be calculated using the formula for compound interest:

A = P(1 + r/n)^(nt)

where:

A is the amount of money accrued after time t

P is the principal amount (the amount borrowed)

r is the annual interest rate

n is the number of times the interest is compounded per year

t is the time (in years)

In this case, we have:

P = $7,685

r = 6.2%

n = 12 (since interest is compounded monthly)

t = 0.5 (since the grace period is six months)

Plugging in these values, we get:

A = $7,685(1 + 0.00517/12)^(12*0.5) = $7,997.63

So the interest that accrues during the grace period is:

I = $7,997.63 - $7,685 = $312.63

Step 2: Calculate the monthly payment amount

After the grace period, the student will make fixed monthly payments of $198.80 for four years. To calculate the total number of payments, we multiply the number of years by 12:

n = 4*12 = 48

Step 3: Calculate the total amount paid

The total amount paid will be equal to the sum of the principal amount (the amount borrowed) plus the total interest paid. The principal amount is $7,685, and we can use an annuity formula to calculate the total interest paid:

PV = (PMT/i)(1 - 1/(1+i)^n)

where:

PV is the present value of the loan (the amount borrowed)

PMT is the monthly payment

i is the monthly interest rate

n is the total number of payments

In this case, we have:

PV = $7,685

PMT = $198.80

i = 6.2%/12 = 0.00517

n = 48

Plugging in these values, we get:

PV = ($198.80/0.00517)(1 - 1/(1+0.00517)^48) = $10,129.94

So the total interest paid is:

Total interest = $10,129.94 - $7,685 - $312.63 = $2,131.31

Rounding this value to two decimal places, we get $2,131.30, which is closest to the answer option: $2,190.75. Therefore, the correct answer is $2,190.75.

Answer:

A

Step-by-step explanation:

To find the total amount of interest the student will pay, we need to calculate the interest that accrues during the grace period and the interest that accrues after the grace period during the repayment phase.

First, let's calculate the interest that accrues during the six-month grace period.

The loan amount is $7,685, and the interest rate is 6.2% per year compounded monthly. To figure out the monthly interest rate, we divide 6.2% by 12 to get 0.00517.

So, during the six-month grace period, the interest that accrues is:

$7,685 x 0.00517 x 6 = $237.14

Now let's calculate the total amount of interest that will accrue during the repayment phase. The student will be making fixed monthly payments of $198.80 for four years, which is a total of:

$198.80 x 12 x 4 = $9,545.60

The total amount repaid over the four years is the sum of the principal and interest, so we can subtract the original loan amount from the total amount repaid to get the total interest paid:

$9,545.60 - $7,685 = $1,860.60

However, we need to subtract the interest that accrued during the grace period from this total, so:

$1,860.60 - $237.14 = $1,623.46

So the total amount of interest the student will pay is $1,623.46, which is closest to option A ($1,221.13).

(Note: i am into the conservatism or prudence principle in accounting that's why i chose option A)

To construct a histogram of the estimates and compare it to the above estimates, follow these steps:
1. Compile the data and organize it into groups or bins.
2. Assign each group a height or frequency, usually the number of occurrences in the group.
3. Represent each group by drawing a bar with the corresponding height.
4. Connect the tops of the bars to form a histogram.

Comparing the histogram to the above estimates, if the histogram is similar in shape and spread, then the estimates are consistent.

The best estimate of the random modifier without examining the value is the mode of the data, which is the value that occurs most often.

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Answers:

1.2x10

2.72 cubic cm

The conversion of the statement of the expression into inequality form in terms of x is given by 1/2 × 3^x + 2^(2 - x) - 2 ≤ x ≤ 3

Using the details of the inequality,

1 half left parenthesis 3 right parenthesis to the power of x is,

1/2 × 3^x

Plus space 2 end exponent minus space 2 , negative 1 space  is written as ,

+ 2^(2 - x) - 2

less or equal than space x space less or equal than space 3

≤ x ≤ 3

Now, combine all to get the required inequality,

1/2 × 3^x + 2^(2 - x) - 2 ≤ x ≤ 3

Therefore, the required inequality for the given expression is equal to

1/2 × 3^x + 2^(2 - x) - 2 ≤ x ≤ 3

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The given question is incomplete, I answer the question in general according to my knowledge:

If left parenthesis x right parenthesis space equals space 1 half left parenthesis 3 right parenthesis to the power of x space plus space 2 end exponent minus space 2 , negative 1 space less or equal than space x space less or equal than space 3.

Represent the expression in the inequality form of x.

Answer:

Clockwise it would be (3,-5)

Step-by-step explanation:

Counterclockwise it would be (-3,-5)

hope this helps!

A circle is a shape formed by a point moving in a plane while keeping the same distance from a fixed point. The measurement of angle EDF is 23 degrees.

To start, a circle is a shape that is created by a point moving in a plane while its distance from a fixed point remains constant. This fixed point is called the center of the circle.

Since we know that angles created by the same chord in a circle are equal, we can use this information to find the measure of angle EDF.

In this case, EF is the chord of the circle, and we are given that angle EGF measures 23°. Therefore, we can conclude that angle EDF must also measure 23° since it is created by the same chord EF.

Thus, we can say that the size of angle EDF is 23°.

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Complete question is the image attached below

Answer:

y=2x+4

Step-by-step explanation:

m is slope.

Slope is change in y over change in x.

Line goes up (+y) two for every move to the right (+x), so the slope is 2.

B is the y-intercept, where the line crosses the y-axis. It crosses at 4, so B is 4.

Plug those into y=mx+b to get

y=2x+4.

Answer: 31.8 years

Step-by-step explanation:

The formula to calculate the simple interest is:

I = Prt

where:

I = Interest

P = Principal amount

r = Rate of interest

t = Time (in years)

We want to know how long it will take for the investment to double in value. That means the interest earned should be equal to the principal amount. So, we can write:

2P = P + I

Simplifying:

P = I

Now, we can substitute the values given in the problem:

1400 = 14000.044t

Simplifying:

t = 1400/(1400*0.044) = 31.82 years

Therefore, it will take approximately 31.8 years for the investment to double in value.

A researcher testing the hypothesis that men and women differ in the amount of time they spend exercising weekly may use the Repeated Measures ANOVA (Analysis of Variance).

A researcher tests the hypothesis that men and women differ in the amount of time (number of hours) they spend exercising weekly. What statistical test might the researcher use assuming a repeated-measures design?

The repeated measures ANOVA is a statistical test that can be used to assess the significance of the differences between the group means when there is a within-subjects factor in a study. The researcher can also use a t-test to compare the means of two independent groups.

The repeated measures ANOVA requires the following assumptions:

Independence of observations: There should be no relationship between the observations.

Independent variable is categorical and dependent variable is continuous. The data should be normally distributed. Homogeneity of variance: The variances of each group should be the same.

Therefore, a researcher testing the hypothesis that men and women differ in the amount of time they spend exercising weekly may use the Repeated Measures ANOVA (Analysis of Variance).

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The answer of the given question based on the assuming that the spins are independent, the pmf of x is P(x = k) = (2/3)^(k-1) * (1/3).

Probability is a measure of likelihood or chance that specific event will occur. It is  branch of mathematics that deals with study of random events and their associated outcomes.

In probability theory, events are typically represented as sets of possible outcomes, and probabilities are assigned to these events based on likelihood of their occurrence. The probability of  event is  number between 0 and 1, where 0 represents  impossible event and 1 represents certain event.

Let's denote the probability of getting a 3 on any given spin as p. Since the spinner only has three numbers, the probability of getting a 3 on any given spin is 1/3.

Now, let's consider the probability distribution of x, the number of spins until the first 3 occurs. The probability that x = 1 is simply the probability of getting a 3 on the first spin, which is p = 1/3.

The probability that x = 2 is the probability of not getting a 3 on the first spin (which is 2/3), multiplied by the probability of getting a 3 on the second spin (which is p = 1/3). So:

P(x = 2) = (2/3) * (1/3) = 2/9

Similarly, the probability that x = 3 is the probability of not getting a 3 on the first two spins (which is (2/3)^2), multiplied by the probability of getting a 3 on the third spin (which is p = 1/3). So:

P(x = 3) = (2/3)²* (1/3) = 4/27

In general, the probability that x = k (where k is a positive integer greater than or equal to 1) is the probability of not getting a 3 on the first k-1 spins (which is (2/3)^(k-1)), multiplied by the probability of getting a 3 on the kth spin (which is p = 1/3). So:

P(x = k) = (2/3)^(k-1) * (1/3)

This is the pmf (probability mass function) of x, the number of spins until the first 3 occurs.

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The answer of the given question based on the assuming that the spins are independent, the pmf of x is P(x = k) =[tex](2/3)^{k-1}[/tex] × (1/3).

Probability is a measure of likelihood or chance that specific event will occur. It is  branch of mathematics that deals with study of random events and their associated outcomes.

In probability theory, events are typically represented as sets of possible outcomes, and probabilities are assigned to these events based on likelihood of their occurrence. The probability of  event is  number between 0 and 1, where 0 represents  impossible event and 1 represents certain event.

Let's denote the probability of getting a 3 on any given spin as p. Since the spinner only has three numbers, the probability of getting a 3 on any given spin is 1/3.

Now, let's consider the probability distribution of x, the number of spins until the first 3 occurs. The probability that x = 1 is simply the probability of getting a 3 on the first spin, which is p = 1/3.

The probability that x = 2 is the probability of not getting a 3 on the first spin (which is 2/3), multiplied by the probability of getting a 3 on the second spin (which is p = 1/3). So:

P(x = 2) = (2/3) × (1/3) = 2/9

Similarly, the probability that x = 3 is the probability of not getting a 3 on the first two spins (which is (2/3)^2), multiplied by the probability of getting a 3 on the third spin (which is p = 1/3). So:

P(x = 3) = (2/3)² × (1/3)

= 4/27

In general, the probability that x = k (where k is a positive integer greater than or equal to 1) is the probability of not getting a 3 on the first k-1 spins (which is  [tex](2/3)^{k-1}[/tex]), multiplied by the probability of getting a 3 on the kth spin (which is p = 1/3). So:

P(x = k) = [tex](2/3)^{k-1}[/tex] × (1/3).

This is the pmf (probability mass function) of x, the number of spins until the first 3 occurs.

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The points x ≤ 1 2/3 and x<-3/4 has plotted in the number line

A number line is a visual representation of the real number system. It is a straight line on which numbers are marked at equal intervals, with zero in the center and positive numbers to the right of zero and negative numbers to the left of zero

The line represents the set of all real numbers, with negative numbers to the left of zero and positive numbers to the right. The point marked with a circle (o) represents the value of x that satisfies both conditions, which is x ≤ 1 2/3 and x < -3/4. Since -3/4 is less than 1 2/3, any value of x that satisfies x ≤ 1 2/3 must also satisfy x < -3/4, so the two conditions are equivalent in this case.

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Since both terms are perfect squares, factor using the difference of squares formula, [tex]a^2-b^2=(a+b)(a-b)[/tex] where [tex]a=b[/tex] and [tex]b=3[/tex]