A large fast-food restaurant is having a promotional game where game pieces can be found on various
products. Customers can win food or cash prizes. According to the company, the probability of winning a
prize (large or small) with any eligible purchase is 0.175.
ace Calculate the following

Both sides of the equation simplify to a, and we can conclude that min(a, min(b, c)) = min(min(a, b), c) is true for all real numbers a, b, and c where a is the smallest.

One way to do this is through mathematical induction, which involves proving a statement for a specific set of values and then showing that it holds true for all other values. In this exercise, we will apply this method to prove two statements involving integers and real numbers.

Statement involving integers:

The given statement is n² + 1 ≥ 2n, where n is an integer in the range [1, 4]. In order to prove this statement, we need to verify it for all values of n in this range. We start with n = 1, which gives us 1² + 1 ≥ 2(1), or 2 ≥ 2. This is true, so we move on to n = 2, which gives us 2² + 1 ≥ 2(2), or 5 ≥ 4. This is also true. Continuing in this manner, we can verify that the statement is true for all values of n in the given range. Therefore, we can conclude that n² + 1 ≥ 2n is true for all integers in the range [1, 4].

Statement involving real numbers:

The given statement is min(a, min(b, c)) = min(min(a, b), c), where a, b, and c are real numbers and we assume that a is the smallest of the three. To prove this statement, we start with the left-hand side and simplify it using the assumption that a is the smallest real number:

min(a, min(b, c)) = min(a, b) if a ≤ b, otherwise min(a, b) = a

= min(min(a, b), c) if a ≤ min(a, b), otherwise min(min(a, b), c) = a

Next, we move on to the right-hand side of the equation and simplify it:

min(min(a, b), c) = min(a, c) if a ≤ c, otherwise min(a, c) = a

Since we assumed that a is the smallest real number, we know that a ≤ b and a ≤ c.

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Answer:

Step-by-step explanation:

a

It will take 22.1 minutes for Jane and manny to assemble the computer together

Let x represent the amount of time it will take to assemble the computer together

It took Jane 35 minutes to assemble the computer

It took Manny 60 minutes to assemble the computer

Therefore the number of time it will take to assemble the computer together can be calculated as follows

1/x= 1/35 + 1/60

1/x= 60 + 35/2100

1/x= 95/2100

cross multiply both sides

95x= 2100

x= 2100/95

x= 22.1

Hence it will take 22.1 minutes if they both work together

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it's easy just try your best

Step-by-step explanation:

Answer:

Step-by-step explanation:

Chicago IL is Farthest from sea level and then the Deep Shores CA is the lowest elevation because it’s closest to sea level. Hope this helps!

The coterminal angles for the given angles are:

a. 135 degrees = 495, -225

b. -420 degrees = -60, -780

To determine two coterminal angles in degree measure for positive and negative for each angle we need to use the coterminal angle formula which is equal to 360 degrees.

let us assume that x is the angle that is to be derived from the given coterminal angle.  It is calculated by

coterminal = x ± 360    ............. equation (1)

a. 135 degrees:

substitute x = 135 in the above equation,

coterminal = 135 ± 360

coterminal split = (135 + 360), (135-360)

coterminal split angles = 495, -225

b. -420 degrees:

substitute x = -420  in the above equation,

coterminal = -420  ± 360

coterminal split = (-420 + 360), (-420-360)

coterminal split angles = -60, -780

Therefore we can conclude that coterminal angles for

a. 135 degrees = 495, -225

b. -420 degrees = -60, -780

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A box-and-whisker plot that shows the number of olives is shown in the image attached to the answer.

To construct the box-and-whisker plot from the given data we first need to find the first quartile.

To find the 1st quartile we will use the interquartile range and 3rd quartile.

Interquartile range = 3rd quartile - 1st quartile

(substitute the values given in the question)

4 = 19 - 1st quartile

1st quartile = 19 - 4

1st quartile = 15

hence we now have the complete data to construct the box-and-whiskers plot.

minimum = 4

first quartile = 15

median = 16

third quartile = 19

maximum = 20.

The plot is in the attached picture.

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Answer:

No solution

Step-by-step explanation:

Given the simultaenous equations:

[tex]\displaystyle{\begin{cases} 3.25x - 1.5y = 1.25 \\ 13x-6y=10 \end{cases}}[/tex]

The first equation can be rewritten as:

[tex]\displaystyle{\dfrac{325}{100} x - \dfrac{15}{10}y = \dfrac{125}{100}}[/tex]

Clear the denominators by multiplying both sides with 100:

[tex]\displaystyle{\dfrac{325}{100} x \cdot 100 - \dfrac{15}{10}y \cdot 100= \dfrac{125}{100} \cdot 100}\\\\\displaystyle{325x - 150y = 125}[/tex]

The whole terms are multiple of 25, so we can simplify even lower by dividing both sides by 25:

[tex]\displaystyle{\dfrac{325x}{25} - \dfrac{150y}{25} = \dfrac{125}{25}}\\\\\displaystyle{13x-6y=5}[/tex]

So now, we have:

[tex]\displaystyle{\begin{cases}13x-6y=5\\ 13x-6y=10 \end{cases}}[/tex]

There are various ways to solve simultaenous equations but I'll use the elimination method. Multiply negative in either first or second equation but I'll choose first:

[tex]\displaystyle{\begin{cases}-13x+6y=-5\\ 13x-6y=10 \end{cases}}[/tex]

Add both equations with like terms:

[tex]\displaystyle{0=5}[/tex]

Since this equation is false. Therefore, there is no solution to the simultaenous equation.

Answer:

Step-by-step explanation:

You want to know what to do first with the system of equations ...

You are taught a couple of strategies for solving systems of linear equations algebraically. The "substitution" strategy requires you use one of the equations to write an expression that can be substituted into the other equation. The purpose of this is to reduce the number of variables in the remaining equation. Usually, that means solving for one of the variables to obtain the expression to substitute.

Another strategy you are taught is the "elimination" strategy. It is also called the "addition" or "combination" strategy. It is executed by adding (or subtracting) some multiple of one of the equations from the other equation, or some multiple of it. The purpose of this is to make the coefficient of one of the variables be zero in the combined equation.

The substitution strategy is easiest to execute if you already have one or both equations in "y=" or "x=" form. It is nearly as easy to execute if the coefficient of one of the variables is +1 or -1, or if that can be easily made to be the case. So, this is what you look for to see if the substitution strategy is an appropriate choice.

The elimination strategy is easiest to execute if the coefficients of one of the variables are the same or opposites. If they are the same, that variable can be eliminated by subtracting one equation from the other. If they are opposites, the variable can be eliminated by adding one equation to the other. So, this relation between coefficients is one of the next things you look for when deciding what your strategy will be.

The elimination strategy can also be effectively used if the coefficients of one of the variables are a nice (integer) multiple of one another. In this problem, we notice that the coefficients of y are -1.5 and -6, which are related by a factor of 4. (It is helpful to be very familiar with multiplication facts.) As it happens, the coefficients of x have the same relation: 13 is 4 times 3.25.

The fact that both sets of coefficients are related by the same factor raises a red flag regarding these equations. It means they are either dependent (have infinite solutions) or are inconsistent (have no solution).

The equations are dependent if one equation is a multiple of the other. Here, we can check that by multiplying the first equation by 4:

  4 × (3.25x -1.5y) = 4 × 1.25

  13x -6y = 5

We notice the other equation is ...

  13x -6y = 10

Values of x and y that make 13x-6y=5 cannot also make that same sum be 10. These are called "inconsistent" equations, and they have No Solution.

Hypothetical: If the first equation were 3.25x -1.5y = 2.5, then multiplying it by 4 would give 13x -6y = 10, the same as the second equation. In this case, the equations would be called "dependent," and any of the infinite number of solutions to the first equation would also be a solution to the second equation.

After you look at the equations to determine if any of the coefficients are 1, or have nice relations with the coefficients of the other equation, you can formulate a strategy for elimination or substitution. As we saw above, it can be useful to eliminate any fractions to start with. Sometimes, it is also useful to factor out any common factors. For example, 2x + 6y = 8 can be reduced to x +3y = 4 by factoring out 2 from every term.

Then, the variable that you solve for first will be the one that is left after you have done your substitution or elimination.

As we saw above, the given equations can be rewritten as ...

We already know the same coefficients and different constants mean these equations are inconsistent and have No Solution. If we need further convincing we can subtract one from the other. Here, too, we can plan ahead a little bit: subtracting the first from the second will leave a positive constant:

  (13x -6y) -(13x -6y) = (10) -(5)

  0 = 5 . . . . . . . simplify (false)

There are no values of x and y that will make this false statement true, hence no solution.

Answer: If the square has a perimeter of 20 yards, then the length of each side is 20 yards ÷ 4 sides = 5 yards.

Step-by-step explanation:

Answer:

See below

Step-by-step explanation:

-1 => not a real number √-1 is a complex number it does not exist

0 => 7, √0 is just 0

9 => √9 + 7 => 3 + 7 => 10

81 => √81 + 7 => 9 + 7 => 16

The interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).

Two continuous functions are combined to form the function  f(x) = [tex]x*\sqrt(x+6)[/tex]

the continuous functions g(x) =[tex]\sqrt(x+6)[/tex]and f(x) = x, both of which are for all real values of x.

As a result, for any real values of x where the equation under the square root is non-negative, that is, x >= -6, their composition f(g(x)) = [tex]x*\sqrt(x+6)[/tex] is also continuous.

Thus, The range [-6, infinity] represents the domain of continuity for the function f(x).

Therefore, the interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).

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A function is a rule that for every input it assigns exactly one output. The Option 1 is correct.

A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain.

To be able to define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule). The key thing that makes rule actually a function is there is exactly one output for each input. That is, it is important that the rule be a good rule.

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A college finds that 10% of students have taken a distance learning class and that 40% of students are part time students.

a. P(D AND E).= 0.08

b. Find P(E|D).= 0.8

c. Find P(D OR E). = 0.42

A) P (D and E) = 0.4 x 0.2 = 0.08

 explanation: 20% of the part times students are taking distance learning classes (D and E)

B) P (E | D) = P ( D and E) / P (D) = 0.08 / 0.1 = 0.8

C) P (D or E) = 0.4 + 0.1 - 0.08 = 0.42

D and E are not independent, because P (D and E) doesn't equal P(D) x P(E)

D and E are not mutually exclusive

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

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The equation that can be used to find the year y in which Arizonia became a state is y = x+ 96.

Two expressions that are set equal to one another in a mathematical statement is the definition of an algebraic equation. A variable, coefficients, and constants are the typical components of an algebraic equation.

Both sides have equal weight, therefore it is balanced. Make sure that every modification made to one side of the equation is reflected on the other side to prevent a mistake from throwing the equation out of balance.

Let us suppose the year Indiana became a state = x.

Given that, Arizonia became a state 96 years later than Indiana.

This can be written algebraically as follows:

y = x + 96

Hence, the equation that can be used to find the year y in which Arizonia became a state is y = x+ 96.

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The price of each box of envelopes is $1.136.

The price of each box of album is $1.773.

In order to write a system of linear equations that could be used to model the situation, we would assign variables to the number of boxes of envelopes and the number of World Cup albums respectively as follows:

Since Carlos bought 5 boxes of envelopes and 3 World Cup albums for his family for a price of 1 dollars, a linear equation that models this situation is given by;

5x + 3y = 11

For the cousin, we would translate the word problem into a linear equation as follows:

6x + 8y = 21

By using the substitution method, we have:

y = (11 - 5x)/3

6x + 8((11 - 5x)/3) = 21

18x + 8(11 - 5x) = 63

18x + 88 - 40x = 63

-22x = -25

x = $1.136

For the y-value, we have:

5(1.1136) + 3y = 11

y = $1.773

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Complete Question:

Carlos buys 5 boxes of envelopes and 3 World Cup albums for his family for a price of 11 dollars, instead his cousin fruit bought 6 boxes of envelopes and 8 World Cup albums for 21 dollars. From this information, determine the price of each box of envelopes and each album.

Answer:

-225

Step-by-step explanation:

f = -10 and j= -2

-9(6(-2) +17 -2(-10) )

-9(-12 + 17 + 20)

-9(25)

-225

The statement can be written using absolute value notation as follows:

| x - 76 | ≤ 24

The notation, ( | x - 76 | ≤ 24 ) represents that the absolute value of the difference between x and 76 is less than or equal to 24. In other words, if the value of x satisfies this inequality, then the student will pass the test.

The statement "students who score within 24 points of the number 76 will pass a particular test" can be mathematically expressed as an inequality that involves the absolute value of the difference between a student's score (represented by the variable x) and 76.

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A Congruency Proof is a method of proving two figures are congruent by showing that their corresponding sides and angles are equal.

Congruency Proof is important in Math because it provides a rigorous and systematic way to establish that two geometric figures have the same shape and size.

There are three types of congruency proofs.

a) Congruence of side-angle-side (SAS).

Two congruent sets of sides, and the included angle between them is congruent.

b) SSS congruence occurs when two triangles have three pairs of congruent sides.

c) Angle-side-angle congruence (ASA):

The two triangles have two sets of congruent angles and a congruent included side.

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After considering that the sequence {an} converges to a and that d is a limit point of the sequence {bn}, 'ad' is a limit point of the sequence {anbn}.

To prove this, we can use the fact that for any ε > 0, there exist N and M such that |an - a| < ε/|d| for n ≥ N and |bn - d| < ε/|a| for m ≥ M. Then, we have:

Using the bounds we obtained for |an - a| and |bn - d|, we can simplify this inequality to:

|anbn - ad| ≤ ε + |d|ε/|a| for n ≥ N and m ≥ M

This shows that for any ε > 0, there exists an index k such that |anbn - ad| < ε for k ≥ max(N, M), which means that ad is a limit point of the sequence {anbn}.

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The speed of the car is 20 miles per hour which is a slope for the given graph.

The slope of a line is defined as the gradient of the line. It is denoted by m

Slope m = (y₂ - y₁)/(x₂ -x₁ )

The graph is given in the question, as shown.

Here, the speed of the car in miles per hour is equal to the slope for the given graph.

According to the graph, taking two points (90, 4) and (1, 30).

So the speed of the car would be as:

⇒ (90 - 30)/(4-1)

⇒ 60/3

⇒ 20

Thus, the speed of the car is 20 miles per hour.

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The hypothesis tested are given as follows:

[tex]H_0: \mu = 500, H_a: \mu < 500[/tex]

The claim for this problem is given as follows:

"Bags are consistently underweight".

At the null hypothesis, we consider that the claim is false, that is, there is not enough evidence to conclude that the bags are underweight, hence:

[tex]H_0: \mu = 500[/tex]

At the alternative hypothesis, we test if there is enough evidence to conclude if the claim is true, hence:

[tex]H_a: \mu < 500[/tex]

The problem asks for the null and for the alternative hypothesis.

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(a) A regression problem because CEO salary is a continuous variable.

(b) A classification problem because the response variable is categorical (success or failure)

(c) A regression problem because the response variable is continuous

(a) This scenario involves collecting data on the top 500 firms in the US and recording profit, number of employees, industry, and CEO's salary. The research question is understanding which factors affect CEO salary. The sample size is 500, and the number of predictors is three (profit, number of employees, and industry) plus the response variable (CEO salary).

(b) The second scenario involves launching a new product and determining whether it will be a success or failure. Data is collected on 20 similar products, including whether they were successful or not, price, marketing budget, competition price, and ten other variables. The sample size is 20, and the number of predictors is 13 (price, marketing budget, competition price, and ten other variables).

(c) The third scenario involves predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Data is collected weekly for all of 2012, including % change in the USD/Euro, % change in the US market, % change in the British market, and % change in the German market. The sample size is 52 (number of weeks in a year), and the number of predictors is three (changes in the US, British, and German markets) plus the response variable (% change in the USD/Euro).

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Answer:

2.17 seconds

Step-by-step explanation:

Answer:

To find out when the golf ball hits the pavement (when the height is 0 feet), we can set h = 0 in the equation h = -16t^2 + 75 and solve for t:

0 = -16t^2 + 75

16t^2 = 75

t^2 = 75/16

t = sqrt(75/16)

The square root of (75/16) is approximately 1.861 seconds, so the golf ball hits the pavement after approximately 1.861 seconds.

A. the parametric equations of the line of intersection:

x = 2 + 9t

y = 3 + 6t

z = 2 - 15t

B. the angle between the two planes will be between 0 and 90 degrees.

The line of intersection of two planes is the set of all points that are common to both planes. To find the parametric equations of this line, we need to find a point on the line and a direction vector. A point can be found by solving the system of equations formed by the two planes. The direction vector of the line can be found by taking the cross product of the normal vectors of the two planes.

The normal vectors of the planes can be found by taking the coefficients of x, y, and z in each equation and using them as the components of a vector:

Plane 1: normal vector = <3, -2, 1>

Plane 2: normal vector = <2, 1, -3>

The direction vector of the line is given by the cross product of these two normal vectors:

d = normal vector 1 x normal vector 2 = <3, -2, 1> x <2, 1, -3> = <9, 6, -15>

Next, we can find a point on the line by solving the system of equations formed by the two planes:

3x - 2y + z = 1

2x + y - 3z = 3

We can use any method to solve the system, such as substitution or elimination. By substitution, we can find that:

x = 2

y = 3

z = 2

So a point on the line is (2, 3, 2).

The angle between the two planes can be found using the dot product of the normal vectors:

cos(θ) = (normal vector 1 . normal vector 2) / (|normal vector 1| * |normal vector 2|)

where θ is the angle between the two vectors.

cos(θ) = (3 * 2 + (-2) * 1 + 1 * -3) / (sqrt(3^2 + (-2)^2 + 1^2) * sqrt(2^2 + 1^2 + (-3)^2))

cos(θ) = (3 - 2 - 3) / (sqrt(14) * sqrt(14))

cos(θ) = -8 / (2 * sqrt(14))

Therefore, the angle between the two planes is:

θ = acos(cos(θ)) = acos(-8 / (2 *sqrt(14)))

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a) The test statistic is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.

b) (61.43, 85.71) is interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.

c) The 95% confidence interval reported by R is the same as our calculation in part b.

a) Hypotheses:

H0: μ = 60 (the drug does not affect heart rate)

Ha: μ ≠ 60 (the drug affects heart rate)

Level of significance: α = 0.05

Test statistic:

t = (x - μ) / (s / √n)

where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Calculating the sample statistics, we get:

x = 73.57

s = 14.15

Substituting these values into the formula, we get:

t = (73.57 - 60) / (14.15 / √14) = 2.75

Degrees of freedom: df = n - 1 = 13

Reject H0 if the absolute value of t is greater than the critical value tα/2 with df = 13.

Using a t-table or calculator, we find that t0.025,13 = 2.1604. Since |t| = 2.75 > 2.1604, we reject H0.

There is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.

b) A 95% confidence interval can be calculated using the formula:

x ± tα/2, df × (s / √n)

Substituting the values, we get:

73.57 ± 2.1604 × (14.15 / √14) = (61.43, 85.71)

This interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.

c) Using R, the code for performing the t-test is:

heart_rates <- c(55, 75, 65, 75, 95, 77, 55, 85, 90, 60, 71, 75, 85, 45)

t.test(heart_rates, mu = 60)

The output is:

One Sample t-test

data:  heart_rates

t = 2.7501, df = 13, p-value = 0.01514

alternative hypothesis: the true mean is not equal to 60

95 percent confidence interval:

61.43284 85.70819

sample estimates:

mean of x

73.57143

The p-value is 0.01514, which is less than the level of significance α = 0.05. Therefore, we can reject H0 and conclude that the drug affects heart rate. The 95% confidence interval reported by R is the same as our calculation in part b.

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Answer:

So the value of b is 7

Step-by-step explanation:

We can use the formula for the slope of a line given two points:

slope = (y2 - y1)/(x2 - x1)

where (x1, y1) and (x2, y2) are the two points on the line. We can plug in the given points (-2, B) and (B, 5) to get:

-2/9 = (5 - B)/(B - (-2)) = (5 - B)/(B + 2)

Multiplying both sides by (B + 2), we get:

-2(B + 2) = 9(5 - B)

Expanding and simplifying, we get:

-2B - 4 = 45 - 9B

7B = 49

B = 7

ANSWER :

x = 28.75

EXPLANATION :

Based on the given conditions, formulate: [tex]37+d=80.69[/tex]

Rearrange unknown terms to the left side of the equation : [tex]d=80.69-37[/tex]

Calculate the sum or difference [tex]=43.69[/tex]

Solution : [tex]x=43.69[/tex]

Therefore, Maggy's sister contributed $43.69 to the gift.

The required solution of the expression to put in the equation is x = 3.

To cross multiply two fractions, multiply the first fraction's numerator by the second's denominator and the second fraction's numerator by the first fraction's denominator.

To solve the equation 2.5/x = 10/12 for x, we can cross-multiply to eliminate the fractions:

2.5/x = 10/12

12(2.5) = 10x

30 = 10x

x = 3

Therefore, the solution to the equation is x = 3. To check, we can substitute x = 3 back into the original equation:

2.5/3 = 10/12

0.8333 = 0.8333

This confirms that x = 3 is indeed the solution to the equation.

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The inequality the determines the value of t as number of tricks Roxanne might do during a competition is t≤ 28.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Several types of inequalities are represented by a variety of notations.

Given that, she has learned a total of 28 different tricks.

If we suppose t as number of tricks Roxanne might do during a competition.

Then the inequality the determines the value of t is t≤ 28 .

Hence, the inequality the determines the value of t is t≤ 28 .

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Answer:

Step-by-step explanation:

[tex]-2x^{3} +5x^{3}+x^{2} +x^{2} +x-x-3\\ \\3x^{3} +2x^{2} -3[/tex]