last week Alexander was paid $56 for 7 hours of work. how much money does Alexander's job pay her hour??

Ryan makes a profit of £240.  the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.

Ryan spends £190 to buy 80 jumpers. He sells 50% of the jumpers, i.e. 40 jumpers, at £12 each. This brings the total sales to £480. Then, he puts the remaining 40 jumpers on a Buy one get one half price offer. He sells 20 of the remaining jumpers using this offer. Therefore, the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.

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he tree nearest the house is our starting point. Our point person is taking the clinometer reading 15.24 meters from the tree's base and they get a reading of 237`. Our point person is 1.83 meters in height.

How tall is the tree, rounded to the nearest meter?The height of the tree can be determined by using the tangent formula. The tangent formula is tan θ = h/d where θ = angle of elevation, h = height of the object, and d = horizontal distance.

The clinometer reading is the angle of elevation. Hence, we can use the given data to determine the height of the tree.The point person is standing at 15.24 m from the base of the tree. Therefore, the horizontal distance (d) is 15.24 m. The angle of elevation (θ) is 237 degrees (given in the question).

Convert the degrees to radians as tan function uses radians. Convert degrees to radians:[tex]237 × (π/180) = 4.135[/tex]radians.Now we can use the tangent formula to determine the height of the tree:tan θ = h/dtan 4.135 = [tex]h/15.24h = 15.24 × tan 4.135h = 15.24 × 0.07311h ≈ 1.1132[/tex] metersThe height of the tree is 1.1132 meters. But, we have to round the answer to the nearest meter. Therefore, the height of the tree, rounded to the nearest meter, is 1 meter.

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The condition that must be true in order for a confidence interval based on de Moivre's equation to be valid is:

d. The underlying distribution of the data must be normally distributed.

A confidence interval is an interval estimate of a population parameter that specifies a range of values within which the parameter is likely to lie with a certain level of confidence. In other words, it represents the degree of uncertainty associated with the estimate.

De Moivre's equation is a formula for approximating the probability of a specific number of successes in a series of independent Bernoulli trials. This formula is only relevant if the sample size is large enough such that the normal approximation to the binomial distribution is valid. Thus, this formula can be used to calculate confidence intervals for binomial proportions when the sample size is large enough to apply the normal approximation.

Answers to other options:

a. We must be forming a confidence interval for a coefficient in a multiple regression model - This statement is incorrect. De Moivre's equation is not related to multiple regression models.

b. All of these answers are correct - This statement is incorrect because not all of the options are correct. Only one option is correct.

c. We must be forming a confidence interval for a population mean based on a sample mean - This statement is incorrect. De Moivre's equation is not relevant for calculating confidence intervals for population means. The Central Limit Theorem is used instead.

Hence, option "d" only is true.

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The length of AD in the triangular prism is 9.8 cm.

The length of AD in the triangular prism can be found using Pythagoras' theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The base of the prism is a right triangle, so:

AD² = AE² + ED²

AD² = 4² + 9²

AD² = √97

AD = √97

AD = 9.8 cm (rounded to 1 decimal)

Therefore, the length of AD in the triangular prism is 9.8 cm.

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Complete Question:

Use Pythagoras' theorem to work out the length of AD in this triangular prism. Give your answer in centimeters (cm) to 1 decimal point.

[tex] \Large{\boxed{\sf C = 31.40 \: inches}} [/tex]

[tex] \\ [/tex]

The circumference of a circle can be calculated using the following formula:

[tex] \Large{\sf C = 2 \pi r } [/tex]

Where:

[tex] \\ [/tex]

Since "a segment drawn from the center of a circle to a point on the circle" is actually the definition of the radius of said circle, we can take r = 5 inches.

[tex] \\ [/tex]

Applying our formula and using 3.14 for π, we get:

[tex] \sf C = 2 \times 3.14 \times 5in \\ \\ \implies \boxed{\boxed{\sf C = 31.4 \: inches = 31.40 \: inches}} [/tex]

Answer:

31.40 inches

Step-by-step explanation:

The circumference of a circle can be calculated using the formula:

Given:

Substitute the given value into the formula:

Simplifying the expression:

[tex]\therefore[/tex] The circumference of the circle is 31.40 inches.

Since only 25% of the variability in y is explained, Julie's actual score could have been much higher than the predicted value.

The term "standard deviation" (or "") refers to a measurement of the data's dispersion from the mean. When the standard deviation is low, the data are concentrated around the mean, and when it is large, the data are widely dispersed.

(a) Slope, b = r*(sy/sx) = 0.5*(8/40) = 0.1

Intercept = y - bx = 75 - 0.1*280 = 47

The regression equation is:

y = 47 + 0.1*x

(b) Put x = 300

y = 47 + 0.1*300

y = 77

(c) r2 = 0.5*0.5 = 0.25

Since only 25% of the variability in y is explained, Julie's actual score could have been much higher than the predicted value.

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The probability of randomly drawing a penny is 6/24 or 1/4, since there are 6 pennies out of a total of 24 coins.

How to solve and What is Probability?

The probability of randomly drawing a quarter is 10/24 or 5/12, since there are 10 quarters out of a total of 24 coins. The probability of randomly drawing a coin that is not a penny is 18/24 or 3/4, since there are 18 coins that are not pennies out of a total of 24 coins.

Probability is the branch of mathematics that deals with measuring the likelihood or chance of an event or outcome occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

Probability theory is used to make predictions and informed decisions based on available data in various fields, including statistics, finance, engineering, and science.

It involves understanding and analyzing random events, and determining the likelihood of specific outcomes. Probability is an essential tool for decision-making in various applications, such as risk analysis, game theory, and quality control.

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Answer:

25%

Step-by-step explanation:

The experimental probability of drawing a white marble can be found by dividing the number of times a white marble was chosen by the total number of trials:

Experimental probability of drawing a white marble = number of times a white marble was chosen / total number of trials

In this case, the number of times a white marble was chosen is 18, and the total number of trials is 80, so:

Experimental probability of drawing a white marble = 18/80 = 0.225 or 22.5%

To compare the experimental probability with the theoretical probability, we need to know the total number of marbles in the bag and the number of white marbles in the bag. Let's assume that there are 4 colors of marbles in the bag (red, white, black, and green), and that each color has an equal number of marbles. This means that there are a total of 4 x 18 = 72 marbles in the bag, and 18 of them are white.

The theoretical probability of drawing a white marble can be found by dividing the number of white marbles by the total number of marbles:

Theoretical probability of drawing a white marble = number of white marbles / total number of marbles

In this case, the number of white marbles is 18, and the total number of marbles is 72, so:

Theoretical probability of drawing a white marble = 18/72 = 0.25 or 25%

Comparing the two probabilities, we can see that the experimental probability (22.5%) is slightly lower than the theoretical probability (25%). This could be due to chance or sampling error in the experiment, or it could indicate that the actual probability of drawing a white marble is slightly lower than the theoretical probability.

The range of the quadratic function above is (-∞, 7].

In mathematics, a quadratic prοblem is οne that invοlves multiplying a variable by itself, alsο knοwn as squaring. In this language, the area οf a square is equal tο the length οf its side multiplied by itself. The term "quadratic" cοmes frοm the Latin wοrd fοr square, quadratum.

Tο determine the quadratic functiοn's range, we must first determine the functiοn's minimum and maximum pοints. The given functiοn is in vertex fοrm, with the vertex at the pοint (h, k), where h is the vertex's x-cοοrdinate and k is the vertex's y-cοοrdinate.

We can see frοm the given equatiοn that the vertex is at the pοint (1, 7). Because the cοefficient οf the x² term is pοsitive, the parabοla οpens upwards and the vertex is the functiοn's minimum pοint.

Thus, The range of the quadratic function above is (-∞, 7].

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a) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

Exponents are mathematical notation used to indicate that a quantity is being multiplied by itself a certain number of times. The exponent is usually written as a superscript to the right of the base number.For example, in the expression 2³, the base number is 2 and the exponent is 3. This means that 2 is being multiplied by itself three times, resulting in a value of 8. Exponents can also be negative or fractional, indicating that the base number is being divided by itself a certain number of times.

In the given question,

a) The value of -42 is different from the value of (-4)². The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The value of -23 is different from the value of (-2)³. The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

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The statement "the equations of motion of a two-degree-of-freedom system can be expressed in terms of the displacement of either of the two masses" is TRUE.

The equations of motion refer to a set of mathematical equations that describe the behavior of a physical system over time. These equations define how the position, velocity, and acceleration of an object are related. The equations of motion are applicable to both single and multi-degree-of-freedom systems.

A two-degree-of-freedom system is a physical system with two independent modes of motion or two degrees of freedom. It is defined by two generalized coordinates that completely define the system's state.

A two-degree-of-freedom system can be either linear or nonlinear, depending on the nature of the force. It is used in the study of structural dynamics, mechanical vibrations, and control engineering.

In a two-degree-of-freedom system, the equations of motion can be expressed in terms of the displacement of either of the two masses. The equations of motion are usually derived using Lagrange's equations, which are a set of equations that describe the dynamics of a mechanical system in terms of its energy. They are given as follows:

Where q₁ and q₂ are the generalized coordinates, m₁ and m₂ are the masses, k₁ and k₂ are the spring constants, and c₁ and c₂ are the damping coefficients.

These equations of motion are nonlinear and can be solved analytically or numerically using various techniques.

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Table values are -3, -1, 3, 5, 13

A function is a mathematical object that maps each element from one set to a unique element in another set. Functions are represented using symbols and can be described using graphs, tables, or equations.

Given function is,

[tex]f(x)=2x +3[/tex]

Solve for x = -3,   f(-3) = 2×(-3) + 3 = -6 + 3 = -3

f(-3) = -3

Solve for x = -2,   f(-2) = 2×(-2) + 3 = -4 + 3 = -1

f(-2) = -1

Solve for x = 0,   f(0) = 2×(0) + 3 = 0 + 3 = +3

f(0) = 3

Solve for x = 1,   f(1) = 2×(1) + 3 = 2 + 3 = 5

f(1) = 5

Solve for x = 5,   f(5) = 2×(5) + 3 = 10 + 3 = 13

f(5) = 13

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The vector equation and the parametric equations in t for the line through the point and parallel to the given line are:

x= 2t - 4

y= -4t + 7

z= 8t + 5

To find x, y, and z in the given scenario, the following steps can be followed:

1: Vector Equation of Line

To find the vector equation, use the given line and its coefficients:

x = -4 + 2t

y = 7 - 4t

z = 5 + 8t

Take the coefficients of x, y, and z, and place them in a 3 by 1 matrix:

Column Matrix= [-4 7 5]

Add the parameter t and place it in a column matrix to get the vector equation:

Vector Equation= [-4 7 5] + t[2 -4 8]

2: Parametric Equation.

To find the parametric equations, write the components of the vector equation in terms of the parameters:

x= -4 + 2t

y= 7 - 4t

z= 5 + 8t

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The probability that a randomly selected finishing time is greater than 87 seconds is 14.08%. This can be calculated using the empirical rule.

The empirical rule states that for any data that is normally distributed, about 68% of the data will fall within one standard deviation of the mean (in this case, within 69 ± 6 seconds). Approximately 95% of the data will fall within two standard deviations (in this case, within 69 ± 12 seconds), and about 99.7% of the data will fall within three standard deviations (in this case, within 69 ± 18 seconds).Given the mean and standard deviation given, we can calculate the probability that a randomly selected finishing time is greater than 87 seconds.

We can do this by subtracting the area under the curve from the mean to the value we are interested in (in this case, 87 seconds). Since the total area under the curve is 1, subtracting the area from the mean to 87 seconds will give us the desired probability.To calculate the area under the curve, we need to calculate the Z-score, which is the number of standard deviations away from the mean a particular value is. In this case, the Z-score is (87 - 69) / 6, which is 2.16. Using a Z-table, the probability of a Z-score of 2.16 or higher is 0.8592. Therefore, the probability that a randomly selected finishing time is greater than 87 seconds is 1 - 0.8592, which is 0.1408. Rounding to two decimal places, this is 14.08%.

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The correct answer is: (4, 3) is a member. Its cardinality is 4.

Matching the sets on the left with a true statement about the Cartesian product of those sets on the right:{1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}{1, 2, 3, 4} × {3, 4, 5, 6} = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 3), (4, 4), (4, 5), (4, 6)}{4, 5, 6, 7} × {4, 5, 6, 7} = {(4, 4), (4, 5), (4, 6), (4, 7), (5, 4), (5, 5), (5, 6), (5, 7), (6, 4), (6, 5), (6, 6), (6, 7), (7, 4), (7, 5), (7, 6), (7, 7)}{a, e, i, o, u} × {b, g, t, d} = {(a, b), (a, g), (a, t), (a, d), (e, b), (e, g), (e, t), (e, d), (i, b), (i, g), (i, t), (i, d), (o, b), (o, g), (o, t), (o, d), (u, b), (u, g), (u, t), (u, d)}{1, 2, 3} × {1, 2, 4} = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (3, 4)}The following are true statements about the Cartesian product of these sets:its cardinality is 4. (4, 3) is a member.

Therefore, the correct answer is: (4, 3) is a member. Its cardinality is 4.

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To conclude, the probability of the Acme Drug Company seeing a percent difference of -.13 or less if the true difference is 0 is quite low and is equal to 0.0934.

The probability that the Acme Drug Company would see a percent difference of -.13 or less if the true difference is 0 is quite low. This is because a difference of -.13 is a very small percentage in comparison to a true difference of 0.

Mathematically, the probability of this happening would be equal to the area under the standard normal distribution curve for values between -0.13 and 0. In other words, the probability that the Acme Drug Company would see a percent difference of -.13 or less if the true difference is 0 is equal to the area from the left tail of the standard normal distribution curve up to the mean (0) of the curve.

Using a standard normal distribution calculator, we can see that the probability of the Acme Drug Company seeing a percent difference of -.13 or less is 0.0934. This probability is extremely low and it is not likely that the Acme Drug Company would experience such a small percent difference.

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Therefore, you would be more likely to win the game by choosing option (2) - winning if the spinners do not match.

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is used in many areas of mathematics, science, engineering, finance, and other fields to model and analyze uncertain situations. It helps to make predictions, to assess risks and opportunities, and to make informed decisions based on available information. Probability theory provides a foundation for statistical inference, which is used to draw conclusions from data and to test hypotheses about the underlying population.

Here,

In the "Two Spinner Game", there are two possible outcomes for each spin - a match or a non-match. The probability of the spinners matching is the probability of both spinners landing on the same color. Let's say that there are 3 red sections, 3 blue sections, and 2 green sections on each spinner.

The probability of the first spinner landing on red is 3/8, and the probability of the second spinner landing on red is also 3/8. Therefore, the probability of both spinners landing on red (a match) is (3/8) x (3/8) = 9/64.

Similarly, the probability of both spinners landing on blue (another match) is (3/8) x (3/8) = 9/64, and the probability of both spinners landing on green (a match) is (2/8) x (2/8) = 4/64.

The probability of the spinners not matching is the probability of them landing on different colors. There are 3 different pairs of colors that are not a match: red-blue, red-green, and blue-green. The probability of each of these pairs is (3/8) x (3/8) = 9/64.

So, there are 6 possible outcomes, and the probability of winning by a match is 9/64 + 9/64 + 4/64 = 22/64, or about 34.4%. The probability of winning by a non-match is 3 x 9/64 = 27/64, or about 42.2%.

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The equation of the line that passes through the points (1,3) and (2,2) is y = -x + 4.

To find the equation of the line, we can use the slope-intercept form of a linear equation, y = mx + c, where m is the slope and c is the y-intercept.

First, we need to find the slope of the line. The slope is given by:

where (x1, y1) and (x2, y2) are the coordinates of the two given points. Plugging in the values, we get:

Next, we can use one of the given points and the slope to find the y-intercept. Using the point (1,3), we get:

Simplifying this equation gives us:

c = 4

Therefore, the equation of the line in slope-intercept form is:

y = -x + 4.

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The sales of marshmallow is 5 million and the sales of graham cracker is 5 million is 0.5648 and the probability that 3X1-1X2 + 3X3 > 20 is 0.000005.

The multivariate normal distribution is a probability distribution which describes the joint behavior of multiple random variables. In the given case, the profit (in millions) for selling chocolate (Xi), marshmallow (X2) and graham cracker (X3) follows a multivariate normal distribution with parameters 1, 0.3, 0.3 and Σ = 0.31 0 0.3 01.

1. To calculate the probability that the profit for selling chocolate is greater than 6 millions, we need to calculate the probability that X1>6. Using the given parameters, we can use the formula for calculating the cumulative probability of a standard normal distribution: [tex]P(X1>6) = 1-P(X1≤6) = 1-0.9999994 = 0.000006.[/tex]

2. To calculate the probability that the profit for selling chocolate is greater than 6 millions, given the sales of marshmallow is 5 million and the sales of graham cracker is 5 million, we need to calculate the conditional probability [tex]P(X1>6|X2=5, X3=5)[/tex]. Using the given parameters, we can calculate this probability using the formula for conditional probability:[tex]P(X1>6|X2=5, X3=5) = P(X1>6 ∩ X2=5 ∩ X3=5) / P(X2=5 ∩ X3=5) = 0.002207 / 0.003915 = 0.5648.[/tex]

3. To calculate the probability that, we need to calculate the probability that[tex]X1>7-X2/3-X3/3[/tex]. Using the given parameters, we can calculate this probability using the formula for cumulative probability of a standard normal distribution: [tex]P(3X1-1X2 + 3X3 > 20) = 1-P(3X1-1X2 + 3X3 ≤ 20) = 1-0.9999995 = 0.000005.[/tex]

In conclusion, the probability that the profit for selling chocolate is greater than 6 millions is 0.000006, the probability that the profit for selling chocolate is greater than 6 millions

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The maximum sum of the products taken two at a time is 180, and this can be achieved by choosing 60, 60, and 60 as the three numbers.

In order to write the number 180 as a sum of three numbers so that the sum of the products taken two at a time is a maximum, one way to do it is to use the formula:

[tex](x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz)[/tex]

Let the three numbers be x, y, and z.

Then the product of the numbers taken two at a time is: [tex]xy + xz + yz[/tex]

If we want to maximize the sum of the products taken two at a time, we need to maximize [tex]xy + xz + yz[/tex].

In the formula: [tex](x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz)[/tex]

We can see that the first three terms on the right-hand side are fixed since they depend on x, y, and z. Therefore, to maximize the sum of the products taken two at a time, we need to maximize 2(xy + xz + yz). Since we have the number 180, we can let: [tex]x + y + z = 180[/tex]

Then, we need to maximize: 2(xy + xz + yz) Using calculus, we can find that the maximum value of 2(xy + xz + yz) is attained when: [tex]x = y = z = 60[/tex]

Therefore, the three numbers that can be used to write the number 180 as a sum of three numbers so that the sum of the products taken two at a time is a maximum are: 60, 60, 60.

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Answer:

[tex]the \: two \: numbers \: are \: 14 \: and \: 14.[/tex]

Step-by-step explanation:

let x, y be the two numbers

:

x + y = 28

:

if the two numbers are 1 and 27, then

:

1) x + y = 28

:

2) xy = 27

:

solve equation 1 for y, then substitute for y in equation 2

:

3) y = 28 -x

:

x(28-x) = 27

:

4) -x^2 +28x -27 = 0

:

the graph of equation 4 is a parabola that curves downward, so the coordinates of the vertex is the maximum values for x and y

:

x coordinate = -b/2a = -28/2(-1) = 14

:

substitute for x in equation 3

:

y = 28 -14 = 14

:

*****************************************************

the maximum product occurs when x=14 and y=14

:

Note 14 * 14 = 196

Answer: Yes, the Mean Value Theorem can be applied to f(x) = 9x^3 on the closed interval [1, 2].

To find all values of c in the open interval (1, 2) such that f'(c) = (f(b) - f(a))/(b - a), we first find the derivative of f(x):

f'(x) = 27x^2

Then, we can use the Mean Value Theorem to find a value c in the open interval (1, 2) such that:

f'(c) = (f(2) - f(1))/(2 - 1)

27c^2 = 9(2^3 - 1^3)

27c^2 = 45

c^2 = 5/3

c = +/- sqrt(5/3)

Therefore, the values of c in the open interval (1, 2) such that f'(c) = (f(b) - f(a))/(b - a) are:

c = sqrt(5/3), -sqrt(5/3)

Note that these values are not in the closed interval [1, 2], as they are not between 1 and 2, but they are in the open interval (1, 2).

Step-by-step explanation:

[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{"y" varies inversely with "x"}}{y = \cfrac{k}{x}}\hspace{5em}\textit{we also know that} \begin{cases} x=25\\ y=100 \end{cases} \\\\\\ 100=\cfrac{k}{25}\implies 2500=k\hspace{12em}\boxed{y=\cfrac{2500}{x}} \\\\\\ \textit{when x = 10, what's "y"?}\qquad y=\cfrac{2500}{10}\implies y=250[/tex]

The valid symbolized arguments is HS-valid. (option e).

Now let's focus on the symbolized arguments presented in this question. To determine their validity, we need to identify the form of each argument. In some cases, we may need to rewrite the argument using double negation or commutativity before assigning a name to its form.

The argument form known as Hypothetical Syllogism (HS) is represented as follows:

P⊃Q

Q⊃R

P⊃R

The given argument can be rewritten using double negation as follows:

H⊃~M

M⊃H

H⊃~H

Then, we can see that the argument follows the form of HS, where if H implies not M, and not M implies not H, then we can conclude that H implies not H. This conclusion is always true because it is a contradiction. Therefore, the argument is valid.

Hence the correct option is (e).

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Complete Question:

Determine whether the following symbolized arguments are valid of invalid by identifying the form of each. In some cases the argument must be rewritten using double negation or commutativity before it has a named for. Those arguments without a specific name are valid.

H⊃~M

M

______

~H

a. DA-Invalid.

b. MP-valid.

c. AC-invalid.

d. MT-valid.

e. HS-valid.

A patient weighing 220 pounds needs 293 milligrams of medicine.

We have given that,

patient weighing 162

pounds requires 216 milligrams of medicine

We have to calculate the amount of medicine required by a patient weighing 220 pounds

Consider the value of amount of medicine is x.

Set up a proportion.

pounds / milligrams of medicine

What is the proportion we get?

[tex]162/216=220/x[/tex]

So,

[tex]162/216=220/x[/tex]

[tex]162x=216\times220[/tex]

[tex]162x=47,520[/tex]

[tex]x=47,520/162[/tex]

[tex]x=293[/tex]

A patient weighing 220 pounds needs 293 milligrams of medicine.

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Solving a system of equations we can see that he cost of a corn dog is $1.25 and the cost of the fries is $3.50

We can define two variables here:

x = cost of a corn dog.

y = cost of the fries.

With the information in the question we can write two equations, these two equations form a system of equations that we can solve, the system of equations is the one below:

2x + 3y = 13

4x + y = 8.50

We can isolate y in the second equation to get:

y = 8.50 - 4x

Replace it in the other one:

2x + 3*(8.5 - 4x) = 13

2x + 25.5 - 12x = 13

-10x = 13 - 25.5

x = -12.5/-10 = 1.25

Then:

y = 8.5 - 4*1.25 = 3.5

The cost of a corn dog is 1.25 dollars and the cost of the fries is 3.50 dollars.

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a) The probability that A wins if A answers the first question is α / (α + β).

b) The probability that A wins if B answers the first question is βα+β.

a) If A answers the first question, then the game will be over only if A answers the first question correctly, which has a probability of α. If A answers incorrectly, then B gets to answer the next question, and we are in the same situation but with A's and B's roles reversed. Therefore, the probability that A wins in this case is given by:

P(A wins | A answers first) = α + (1 - α)β P(A wins | B answers first)

b) If B answers the first question, then we are in the same situation as in part a) but with A's and B's roles reversed. Therefore, the probability that A wins in this case is given by:

P(A wins | B answers first) = β P(A wins | A answers first) + (1 - β)β P(A wins | B answers first)

We can solve these equations for P(A wins | A answers first) and P(A wins | B answers first) using algebraic manipulation. For example, we can rearrange the equation in part a) to get:

P(A wins | A answers first) = α / (1 - (1 - α)β)

Similarly, we can rearrange the equation in part b) to get:

P(A wins | B answers first) = β / (1 - (1 - β)α)

Note that these probabilities depend on the values of α and β, which represent the abilities of A and B to answer questions correctly.

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Since 7 is the smallest integer that divides 173/168, we can conclude that the sum is not a prime number.

How to solve?

To add the fractions 4/7, 1/8, and 1/3, we need to find a common denominator.

The prime factorization of 7 is 7, the prime factorization of 8 is 2²3, and the prime factorization of 3 is 3. The least common multiple (LCM) of these three numbers is 7× 2²3× 3 = 168.

So, we can rewrite the fractions with the common denominator of 168:

4/7 = 96/168

1/8 = 21/168

1/3 = 56/168

Now we can add these fractions:

96/168 + 21/168 + 56/168 = 173/168

To check if this sum is a prime number, we can use trial division by checking all the integers between 2 and the√ of 173/168 (which is approximately 1.053):

2 does not divide 173/168

3 does not divide 173/168

4 does not divide 173/168

5 does not divide 173/168

6 does not divide 173/168

7 divides 173/168 (24 times)

8 does not divide 173/168

9 does not divide 173/168

...

Since 7 is the smallest integer that divides 173/168, we can conclude that the sum is not a prime number.

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Complete question:

What is the result of adding 4/7, 1/8, and 1/3, and is the sum a prime number?