Demonstrate your knowledge by giving clear, concise solutions to each problem. Be sure to include all
relevant drawings and justify your answers (show all your work). You may show your solution in more
than one way to investigate beyond the requirements of the problem. (21 pts)
1. a. Tell how you know that a system of linear equations has no solutions. (3 pts)
b. Write a system of two linear equations that has no solution. (3 pts)
c. Tell how you know that a system of two linear equations has more than one solution. How
many solutions will such a system have? Justify your answer. (3 pts)
d. Write a system of two linear equations that has more than one solution. (3 pts)
e. Write a system of linear equations that has only one solution. (3 pts)
f. Solve the system of equations in part e in at least two ways. (3 pts)
g. Write a word problem that can be solved using a system of two linear equations. Solve the
problem and give the meaning of the answer. (3 pts)

Given:-

To find:-

Answer:-

Here we can assume the shape of the cup to be cylindrical . And the volume of a cylinder is given by;

[tex]\implies V = \pi r^2 h \\[/tex]

where ,

Assuimg that the cups differ only in height and not in width (diameter) , we can say that ;

[tex]\implies V \propto h \\[/tex]

So , we can write;

[tex]\implies \dfrac{V_s}{V_l}=\dfrac{h_s}{h_l}\\[/tex]

[tex]\implies \dfrac{V_s}{18\ oz}=\dfrac{6\ in.}{8 \ in.}\\[/tex]

[tex]\implies V_s = \dfrac{6(18)}{8} \\[/tex]

[tex]\implies \underline{\underline{ V_s =13.5 \ oz}} \\[/tex]

Hence the volume of the smaller cup is 13.5 oz .

and we are done!

(a) The probability that the student is a woman is 0.7381. (b) The probability that the student received an A is 0.3095. (c) The probability that the student is a woman or received an A is 0.9405. (d) The probability that the student did not receive an A is 0.6905.

(a) Probability that the student is a woman:

P(woman) = number of women / total number of students

P(woman) = 31 / 42

P(woman) = 0.7381

Answer: 0.7381

(b) Probability that the student received an A:

P(A) = number of students who received an A / total number of students

P(A) = (6 + 7) / 42

P(A) = 0.3095

Answer: 0.3095

(c) Probability that the student is a woman or received an A:

P(woman or A) = P(woman) + P(A) - P(woman and A)

To find P(woman and A), we note that there are 7 women who received an A, so:

P(woman and A) = 7 / 42

P(woman or A) = 0.7381 + 0.3095 - (7 / 42)

P(woman or A) = 0.9405

Answer: 0.9405

(d) Probability that the student did not receive an A:

P(not A) = 1 - P(A)

P(not A) = 1 - 0.3095

P(not A) = 0.6905

Answer: 0.6905

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Calculated t-value (-3) is less than the critical t-value (-1.69), we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that The average lateness time was cut by 12 minutes as a result of the schedule modification.

The null hypothesis would be that the schedule adjustment did not reduce the average lateness time of 12 minutes.

[tex]H0:[/tex][tex]\mu = 12[/tex]

The alternative hypothesis would be that the schedule adjustment did reduce the average lateness time of 12 minutes.

[tex]Ha:[/tex] [tex]\mu < 12[/tex]

Here, we are testing whether the population mean arrival time is less than 12 minutes, based on a sample of 36 runs that had an average of 8 minutes late. We can use a one-sample t-test to test this hypothesis, with the following test statistic:

[tex]t = (\bar x - \mu) / (s / \sqrt{n} )[/tex]

where[tex]\bar x[/tex] is the sample mean (8 minutes),[tex]\mu[/tex] is the population mean (12 minutes), s is the population standard deviation (12 minutes), and n is the sample size (36 runs).

Using these values, we get:

[tex]t = (8 - 12) / (12 / \sqrt{(36)} ) = -3[/tex]

The degrees of freedom for this test are n - 1 = 35. Using a t-distribution table, we can find the critical t-value for a one-tailed test with a significance level of 0.05 and 35 degrees of freedom to be approximately -1.69.

Since our calculated t-value (-3) is less than the critical t-value (-1.69), we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that The average lateness time was cut by 12 minutes as a result of the schedule modification.

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The total price of the ring after increment is $15900

A percentage is a term used in mathematics to describe a value or ratio that may be expressed as a fraction of 100. If we need to calculate a percentage of a number, we should first divide it by 100 before multiplying it by the remainder. As a result, the proportion denotes a component per 100. Percent is what the word "percent" means. Its symbol is the letter "%".

The current price of the ring= $ 15,000

Increase in percentage=6

New price:

15000*(1+0.06)=15900

The total price of the ring after increment is $15900

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We can say that after answering the offered question Therefore, the equation exact value of[tex]$\sin A$ i[/tex]n simplest radical form using a rational denominator is [tex]{-\frac{4\sqrt{261}}{65}}[/tex]

An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.

[tex]$\sec A = \frac{1}{\cos A}$ to find the cosine of angle $A$.[/tex]

[tex]$\sec A = -\frac{65}{7}$, we have $\frac{1}{\cos A} = -\frac{65}{7}$.\\$$\cos A \cdot \frac{1}{\cos A} = \cos A \cdot \left(-\frac{65}{7}\right)$$\\$$[/tex]

Since angle A is in Quadrant III, we know that [tex]$\cos A < 0$ and $\sin A < 0$.[/tex]

Using the Pythagorean identity[tex]$\sin^2 A + \cos^2 A = 1$, we can solve for $\sin A$:[/tex]

[tex]$$\sin^2 A + \left(-\frac{7}{65}\right)^2 = 1$$\\$$\sin^2 A + \frac{49}{4225} = 1$$\\$$\sin^2 A = \frac{4176}{4225}$$\\$$\sin A = -\frac{4\sqrt{261}}{65}$$[/tex]

Therefore, the exact value of[tex]$\sin A$ i[/tex]n simplest radical form using a rational denominator is [tex]${-\frac{4\sqrt{261}}{65}}$[/tex]

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The values of x are x = 13 and x = 1. The other two options, x = -29 and x = 42, are not correct.

What is expression ?

To solve for x, we need to take the square root of both sides of the equation:

[tex](x - 7)^2 = 36[/tex]

Taking the square root of both sides, we get:

x - 7 = ±6

Adding 7 to both sides, we get:

x = 7 ± 6

So, the values of x are:

x = 7 + 6 = 13 or x = 7 - 6 = 1

Therefore, the values of x are x = 13 and x = 1. The other two options, x = -29 and x = 42, are not correct.

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The solution to the equation √(2 + 2x) - 6 = 2 when evaluated is x = 31

From the question, we have the following parameters that can be used in our computation:

√(2 + 2x) - 6 = 2

So, we have

√(2 + 2x) = 8

Take the square of both sides

so, we have the following representation

2 + 2x = 64

Evalyate the like terms

2x = 62

Divide by 2

This means that

x = 31

Hence, the solution is x = 31

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Complete question

What is √(2 + 2x) - 6 = 2

The coordinates of the fourth vertex of the parallelogram are given as follows:

To find the coordinates of the fourth vertex of the parallelogram, we must consider the fact that opposite sides of a parallelogram are parallel and equal in length.

Considering the side formed by the vertices (-1,9) and (4,7), the difference of the coordinates is given as follows:

Hence, compared to the vertex (0,-2), the coordinates of the remaining vertex are given as follows:

Then the coordinates of the fourth vertex are given as follows:

(-5,0).

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the corresponding angles are A. 7 and 2, C. 5 and 7, and E. 6 and 8. The correct answer is A, C, and E.

Corresponding angles are angles that are in the same position at each intersection where a straight line crosses two others. In the given list of angles, we can see that:

Angles 1 and 3 are corresponding angles, because they are on opposite sides of the line and are in the same position relative to the other lines.

Angles 5 and 7 are corresponding angles, for the same reason.

Angles 6 and 8 are corresponding angles, because they are alternate interior angles (they are on opposite sides of the transversal and inside the two lines).

Angles 7 and 2 are vertical angles, not corresponding angles.

Angles 6 and 3 are alternate interior angles, not corresponding angles.

Angles 1 and 6 are alternate interior angles, not corresponding angles.

Therefore, the corresponding angles are A. 7 and 2, C. 5 and 7, and E. 6 and 8. The correct answer is A, C, and E.

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Answer:

49.7 mm

Step-by-step explanation:

the circumference (C) of a circle (2πr) is also the perimeter of a circle.  The figure shown is a half-circle.  To get the perimeter of the shape shown, find the C/2 (half the circumference) and add the diameter:

P = 2π(12/2) + 12 = 49.7 mm

The most individuals with a certain trait that follow a normal distribution would appear in the middle range.

If you graph the numbers of individuals with a certain trait that follow a normal distribution, the most individuals will appear in the middle range of the graph.

When graphing a normal distribution, it creates a bell-shaped curve.

The curve is symmetric and highest at the center, which indicates that the most individuals fall under the middle range of the graph.

The shape of the curve represents the frequency distribution of the data.

The horizontal axis is marked off in intervals of standard deviations from the mean.

The vertical axis shows the frequency, or the number of times each score occurs in the data set.

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Answer:

really it's 4

Step-by-step explanation:

Answer:4

Step-by-step explanation:

The Slope of the line whose coordinates are given that (10 -1) and (-8 6) is the change in y coordinate with respect to the change in x coordinate is (-7/18).

Here the coordinates are (10 -1) and (-8 6) that means :-

X1 = 10

X2 = -8

Y1 = -1

Y2 = 6

So slope of line = (Y2-Y1)/(X2-X1)

     slope = 6-(-1)/-8-10

    slope = 7/-18

Hence the slope of the line containing (10 -1) and (-8 6) would be 7/-18.

In Mathematics, a slope of a line is the change in y coordinate with respect to the change in x coordinate. The net change in y-coordinate is represented by Δy and the net change in x-coordinate is represented by Δx.

Hence, the change in y-coordinate with respect to the change in x-coordinate is given by,

m = change in y/change in x = Δy/Δx

Where “m” is the slope of a line.

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By answering the above question, we may state that As a result, the arc's  expressions PB length is around 20.45 cm.

In mathematics, you can multiply, divide, add, or take away. The following is how an expression is put together: Numeric value, expression, and math operator The elements of a mathematical expression include numbers, parameters, and functions. It is feasible to use contrasting words and expressions. Any mathematical statement containing variables, numbers, and a mathematical action between them is known as an expression, often known as an algebraic expression. As an example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

Area of sector OAP = (1/12) x r2 x (30/360)

OAP's area is equal to (1/2) x OA, PA, and sin (AOP)

Knowing that OA = PA = r and that AOP = 30°, we can calculate the area of the triangle OAP as (1/2) x r x r x sin(30°) = (1/4) x r2.

As a result, the darkened region's area is:

Area of the darkened zone is equal to (1/12) x r2 - (1/4) x r2 (462 cm2).

When we simplify this equation, we obtain:

[tex](\pi/3 - 1) x r^2 = 1848[/tex]

The result of dividing both sides by (/3 - 1) is:

[tex]r^2 = 1848 / (π/3 - 1) = 378.9[/tex]

When we square the two sides, we obtain:

[tex]r ≈ 19.47\\PB = (1/6) x 2r = (1/3) x r = 20.45 cm[/tex], where

As a result, the arc's PB length is around 20.45 cm.

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Answer: 1.60

Step-by-step explanation:

ask siri

Answer:

Step-by-step explanation:

Greetings!!

[tex]step1[/tex]

20%=100%/5

[tex]step2[/tex]

8/5=1.6

[tex]step3[/tex]

5= 10/2

[tex]step4[/tex]

Divide the numbers by 10:

8/10=0.8

[tex]step5[/tex]

Multiply the numbers from step4 by 2.

0.8×2= 1.6

If you have any questions tag it on comments

Hope it helps!!!

Part A:

For Column 1 : Line AB = Line BC: always true

For Column 2: Line AD = Line CD; sometimes true

For Column 3: Line BD is always vertical; never true

Part B: m< CBD is 20 degrees. Option A

It is important to note the properties of a circle, they are;

From the information given, we have that;

m< BDA = 20 degrees

To determine the angle of m< CBD, we need to understand that alternating angles are angles that on  a transversal and are equal to each other.

Then, if m<BDA = 20

m<CBD = 20 degrees

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An inverse variatiοn includes the pοints (8, 3) and (2, n), the value οf n is 12.

Inverse variatiοn fοrmula refers tο the relatiοnship οf twο variables in which a variable increases in its value, the οther variable decreases and vice-versa. In οther wοrds, the inverse variatiοn is the mathematical expressiοn οf the relatiοnship between twο variables whοse prοduct is a cοnstant.

Using the given pοints, we can set up a prοpοrtiοn tο find k:

8 * 3 = k

2 * n = k

Setting the twο expressiοns fοr k equal tο each οther, we get:

8 * 3 = 2 * n

24 = 2n

Dividing bοth sides by 2, we get:

n = 12

Therefοre, the value οf n is 12.

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The jοint prοbability οf A and B οccurring is 0.12.

Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.

Here the given P(A)=0.3 and P(B│A) = 0.4

The οbjective is tο find the jοint prοbability οf A and B οccurring.

A and B are twο events.

The cοnditiοnal prοbability fοrmula is given by,

P(B│A) =[tex]\frac{P(A\cap B)}{P(A)}[/tex]

=> 0.4 = [tex]\frac{P(A\cap B)}{0.3}[/tex]

=> P(A∩B) = 0.12.

Hence the joint probability of A and B occurring is 0.12.

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Answer:

Step-by-step explanation:

First, take 1 coin away and subtract .5 from the total of 85 cents to get 9 coins for 35 cents. Then, you must figure out any amount of coins that will fit that value for 9 coins; A penny = 1 cent, nickel = 5 cents and dime = 10 cents

You can do 2 nickels 2 dimes and 5 pennies for 9 coins total.

You can also do an equation for more intricate problems:

.10x + .05y + .01z = .35

where you would solve for a variable and input the new equation in that variable to solve for your variables. I hope this helps :)

The process of counting from 0 to 10 using 4 fours and the four basic operations can be explained mathematically by the following equation: 4 + 4 - 4 x 4 / 4 + 4 + 4 - 4 = 12

The result when using 4 fours and the four basic operations (addition, subtraction, multiplication, division) to count from 0 to 10 is as follows:

0 = 4 ,1 = 4 + 4 = 8 ,2 = 8 - 4 = 4 ,3 = 4 x 4 = 16 ,4 = 16 / 4 = 4 ,5 = 4 + 4 + 4 = 12 ,6 = 12 - 4 = 8 ,7 = 8 x 4 = 32, 8 = 32 / 4 = 8 ,9 = 8 + 4 + 4 = 16 ,10 = 16 - 4 = 12 .The process of counting from 0 to 10 using 4 fours and the four basic operations can be explained mathematically by the following equation: 4 + 4 - 4 x 4 / 4 + 4 + 4 - 4 = 12. This equation is made up of four parts, each representing an operation in the correct order. The first part of the equation is 4 + 4 = 8, which adds the two fours together to get 8. The second part of the equation is 8 - 4 = 4, which subtracts the four from the 8 to get 4. The third part of the equation is 4 x 4 = 16, which multiplies the two fours together to get 16. The fourth part of the equation is 16 / 4 = 4, which divides the 16 by 4 to get 4. The fifth part of the equation is 4 + 4 + 4 = 12, which adds the three fours together to get 12. The sixth part of the equation is 12 - 4 = 8, which subtracts the four from the 12 to get 8. Finally, the last part of the equation is 8 + 4 + 4 = 16, which adds the three fours together to get 16.

By combining these six operations together, we can count from 0 to 10 using 4 fours and the four basic operations.

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What is the result when using 4 fours and the four basic operations (addition, subtraction, multiplication, division) to count from 0 to 10?

The probability that more than 175 cable homes will assess cable firms as providing outstanding or exceptional transmission quality is roughly 0.0287.

The word "probability" comes directly from the Latin word "probabilitas," which means "credibility, likelihood," in Old French probability (14th century) (see probable). The word first appeared in mathematics around 1718.

Let X be the proportion of the 300 randomly chosen cable homes that are satisfied or excellent with their provider's transmission quality.

Let p be the probability that a cable household will rate cable firms as providing good or exceptional quality transmission, which in this instance is 53% or 0.53.

Let q represent the likelihood that a cable household does not consider cable companies to be providing good or exceptional quality transmission, which in this instance is 47% or 0.47.

Let 300 * 0.53 = 159, or the mean of the binomial distribution, be denoted by = n * p.

The binomial distribution's standard deviation, = sqrt(n * p * q), is sqrt(300 * 0.53 * 0.47) = 8.38. (rounded to two decimal places).

The binomial distribution can be approximated by the normal distribution:

P(X > 175) = P(Z > (175 - μ) / σ)

where Z is a standard normal variable.

P(Z > (175 - 159) / 8.38) = P(Z > 1.91)

Using a standard normal table or calculator, we can find that P(Z > 1.91) is approximately 0.0287.

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Answer:

  A, D, E

Step-by-step explanation:

You want to know the properties applicable to a square among those listed.

Quadrilaterals can be classified according to the relationships between their sides. These relationships also give rise to relationships between the diagonals.

The diagonals have the same midpoint in a parallelogram.

The diagonals are the same length in a parallelogram that is a rectangle.

A square is a rectangle that has adjacent sides of the same length (A). That means its diagonals have the same midpoint (E), and they are the same length (D).

__

Additional comment

When segments have the same slope, they are parallel. Consecutive sides intersect, so cannot have the same slope (E). Likewise, the diagonals intersect, so cannot have the same slope (C).

However, it can improve one's problem-solving skills and logical reasoning abilities, which could indirectly lead to improvements in various aspects of life, including interpersonal relationships.

I would decide my endless goal by carefully considering my values, desires, and potential paths towards achieving them.

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Answer:

To simplify the expression, you can start by using the distributive property of multiplication over addition/subtraction and then combine like terms.

1/3(3m+6) - 2(2m+6)

= 1/33m + 1/36 - 22m - 26 (distribute the multiplication)

= m + 2 - 4m - 12 (simplify by multiplying and adding the terms)

= -3m - 10 (combine like terms)

Therefore, 1/3(3m+6)-2(2m+6) simplifies to -3m - 10.

Step-by-step explanation:

Question 1

This question has two parts. First, answer Part A. Then, answer Part B.

Part A

Fill in the blank question.

There are many cylinders with a height of 18 meters. Let r represent the radius in meters and V represent the volume in cubic meters.

Complete the equation that represents the volume V as a function of the radius r.

V =

πr2

Complete the table.

r V

1

π

2

π

4

π

Part B

Select the correct choices to complete the sentence.

Complete the statement about the radius and volume of a cylinder.

If the radius of a cylinder is doubled, the volume 1 of 2.

Select Choice

.

Complete the statement about the graph of this function.

The graph of this function 2 of 2.

Select Choice

a line.

The can be done by running the following command in R:

The solution to this problem requires you to load the “rpart” package from CRAN and load the “car90” dataset. The last 20% of the observations should be held out as the test set. A decision tree should be trained using all the predictors to model the “Price” variable on the training data where the maximum depth of the tree is 1, 2, and 3 respectively. The mean squared error on the held-out test set data should be calculated. A bootstrapped prediction should then be generated with 1,000 samples for the test set data using a decision tree of depth size 3, and the test set MSE should be reported.

Here are the steps to follow:

Step 1: Install the “rpart” package from CRAN

This can be done by running the following command in R: install.packages("rpart")

Step 2: Load the “car90” dataset

This can be done by running the following command in R: data(car90)

Step 3: Hold out the last 20 percent of the observations as the test set

This can be done by running the following command in R: test_set <- car90[round(nrow(car90) * 0.8) + 1:nrow(car90), ]

Step 4: Train a decision tree using all the predictors to model the “Price” variable on the training data where the maximum depth of the tree is 1, 2, and 3 respectively

This can be done by running the following commands in R:

train_set <- car90[1:round(nrow(car90) * 0.8), ]
library(rpart)
set.seed(123)
depths <- 1:3
models <- lapply(depths, function(depth) rpart(Price ~ ., data = train_set, method = "anova", maxdepth = depth))

Step 5: Calculate the mean squared error on the held-out test set data

This can be done by running the following commands in R:

library(Metrics)
set.seed(123)
mse <- lapply(models, function(model) mse(predict(model, newdata = test_set), test_set$Price))

Step 6: Generate a bootstrapped prediction with 1,000 samples for the test set data using a decision tree of depth size 3

This can be done by running the following commands in R:

set.seed(123)
boot_preds <- replicate(1000, {
 sampled_test_set <- test_set[sample(nrow(test_set), replace = TRUE), ]
 predict(models[[3]], newdata = sampled_test_set)
})

Step 7: Report the test set MSE

This can be done by running the following command in R:

mse(predict(models[[3]], newdata = test_set), test_set$Price)

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The term to term rule in the sequence is specific formula which describes the each term of sequence. So, term by term formula for sequence 1,2,4,7,11.. is equals to the [tex] x_n = \frac{n(n-1)}{2} + 1[/tex].

A sequence, or a series, is a chain of numbers obtained by following a particular common factor. To complete a series or sequence, we need to determine the common factor that guides the series . We have a sequence of number defined as 1, 2,4,7, 11, ... We have to determine term rule for this sequence. Each number in the sequence is called a term. Now, check the known terms of sequence, first term = 1.

Difference between first two terms =1

difference between second and third=2

difference between third and fourth terms = 3

difference between fourth and fifth terms = 4

So, the difference between two successive terms increases by 1. Thus the term by term rule is written as [tex] x_n = \frac{ n(n-1)}{2} + 1[/tex], where n is nᵗʰ term.

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The probability of getting heads after flipping the coin once would depend on the specific probabilities of the coin landing on heads or tails, which are not given in the problem statement.

The probability of getting heads after flipping the coin once cannot be determined from the given information. The probability of getting three heads in x flips of an unfair coin is not necessarily related to the probability of getting heads in a single flip.

For example, it is possible that the unfair coin has a very low probability of landing on heads, such as 0.1. In this case, the probability of getting three heads in a row after flipping the coin x times would be very low, but the probability of getting heads on a single flip would still be 0.1.

Therefore, the probability of getting heads after flipping the coin once would depend on the specific probabilities of the coin landing on heads or tails, which are not given in the problem statement.

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Answer:

Let x be the number of hot dogs sold.

Then the number of sodas sold is 3x.

The total number of sodas and hot dogs sold is 132, so we can write:

x + 3x = 132

4x = 132

x = 33

Therefore, the number of hot dogs sold is 33, and the number of sodas sold is 3x = 3(33) = 99.

Step-by-step explanation: