If one card is drawn from a standard 52 card playing deck, determine the probability of getting a jack, a three, a club or a diamond. Round to the nearest hundredth.

Answer: 2.71

Step-by-step explanation:

We know that the formula to calculate the standard error is given by :-

[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex], where [tex]\sigma[/tex] is the standard deviation and 'n' is the sample size.

Given : Standard deviation : [tex]\sigma=\$9[/tex]

Sample size : [tex]n=11[/tex]

Then , the standard error of the mean is given by :-

[tex]S.E.=\dfrac{9}{\sqrt{11}}=2.7136021012\approx2.71[/tex]

Hence, the standard error of the mean = 2.71

Final answer:

The standard error of the mean for the size of individual customer orders with a standard deviation of $9 and a sample size of 11 is approximately $2.71.

Explanation:

The Ransin Sports Company is looking to calculate the standard error of the mean for the size of individual customer orders. The standard error of the mean (SEM) is found by dividing the standard deviation by the square root of the sample size. Given a standard deviation of $9 and a sample size of 11 players (the soccer team), the standard error of the mean can be calculated using the formula SEM = σ / √n, where σ is the standard deviation and n is the sample size.

SEM = $9 / √11
SEM = $9 / 3.316...
SEM = approximately $2.71.

Therefore, the standard error of the mean is $2.71.

The solution to the system of equations is [tex]\(x = -1\) and \(y = 2\)[/tex].

To solve the system of equations:

[tex]\[ \begin{cases} -5x + 2y = 9 \\ 3x + 5y = 7 \end{cases} \][/tex]

We can use either the substitution method or the elimination method. Let's use the elimination method here.

First, let's rewrite the equations in standard form:

Equation 1: [tex]\( -5x + 2y = 9 \)[/tex]

Equation 2: [tex]\( 3x + 5y = 7 \)[/tex]

To eliminate one of the variables, let's multiply Equation 1 by 3 and Equation 2 by 5 to make the coefficients of x the same:

[tex]\[ \begin{cases} -15x + 6y = 27 \\ 15x + 25y = 35 \end{cases} \][/tex]

Now, let's add the two equations:

[tex]\[ (-15x + 6y) + (15x + 25y) = 27 + 35 \]\[ -15x + 6y + 15x + 25y = 62 \]\[ 31y = 62 \][/tex]

Now, let's solve for y:

[tex]\[ y = \frac{62}{31} \][/tex]

y=2

Now that we have found the value of y, let's substitute it back into one of the original equations to find x. Let's use Equation 1:

[tex]\[ -5x + 2(2) = 9 \]\[ -5x + 4 = 9 \]\[ -5x = 9 - 4 \]\[ -5x = 5 \]\[ x = \frac{5}{-5} \][/tex]

[tex]\[ x = -1 \][/tex]

Complete question: Find the solution of the following system of equations

-5x+2y=9

3x+5y=7

\[ x = -\frac{28}{3} \] and \( y = 7 \) are the solutions to the system of equations.

To solve the system of equations:

1. -5 + 2y = 9

2. 3x + 5y = 7

We can start by solving equation 1 for [tex]\( y \):[/tex]

[tex]\[ -5 + 2y = 9 \][/tex]

Add 5 to both sides:

[tex]\[ 2y = 9 + 5 \]\[ 2y = 14 \][/tex]

Divide both sides by 2:

[tex]\[ y = \frac{14}{2} \]\[ y = 7 \][/tex]

Now that we have the value of [tex]\( y \)[/tex], we can substitute it into equation 2 and solve for [tex]\( x \):[/tex]

[tex]\[ 3x + 5(7) = 7 \]\[ 3x + 35 = 7 \][/tex]

Subtract 35 from both sides:

[tex]\[ 3x = 7 - 35 \]\[ 3x = -28 \][/tex]

Divide both sides by 3:

[tex]\[ x = \frac{-28}{3} \][/tex]

So, the solution to the system of equations is [tex]\( x = -\frac{28}{3} \) and \( y = 7 \).[/tex]

Answer:

i cant see the picture

Step-by-step explanation:

Answer:

this is the answer with steps

hope it helps!

Answer:

The solution is:

[tex](0, 1)[/tex]

Step-by-step explanation:

We have the following equations

[tex]9x + 4y = 4[/tex]

[tex]-5x + 7y = 7[/tex]

To solve the system multiply by [tex]\frac{9}{5}[/tex] the second equation and add it to the first equation

[tex]-5*\frac{9}{5}x + 7\frac{9}{5}y = 7\frac{9}{5}[/tex]

[tex]-9x + \frac{63}{5}y = \frac{63}{5}[/tex]

[tex]9x + 4y = 4[/tex]

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

[tex]\frac{83}{5}y=\frac{83}{5}[/tex]

[tex]y=1[/tex]

Now substitute the value of y in any of the two equations and solve for x

[tex]9x + 4(1) = 4[/tex]

[tex]9x +4 = 4[/tex]

[tex]9x = 4-4[/tex]

[tex]9x = 0[/tex]

[tex]x=0[/tex]

The solution is:

[tex](0, 1)[/tex]

To calculate the time required for an investment of $1000 at 3% interest compounded monthly to grow to $1500, use the compound interest formula. Solve for 't' using natural logarithms and rounding to the nearest month.

To determine how long it takes for $1000 invested at 3% interest compounded monthly to grow to $1500, we use the formula for compound interest:

Final Amount = Principal (1 + (Interest Rate / Number of Compounding Periods in a Year))^(Total Number of Compounding Periods)

Plugging in the values we have:

$1500 = $1000 (1 + 0.03/12)^(12t)

Where 't' is in years. To find 't', we need to isolate it in the equation:

1.5 = (1 + 0.03/12)^(12t)

Take the natural logarithm of both sides:

ln(1.5) = 12t * ln(1 + 0.03/12)

Then, solve for 't' by dividing both sides by 12 * ln(1 + 0.03/12), and round to the nearest month:

t = ln(1.5) / (12 * ln(1 + 0.03/12))

Answer:

[tex]F(x)=(x-11)(x-3)[/tex]

Step-by-step explanation:

we have

[tex]F(x)=x^{2} -14x+33[/tex]

Find the zeros of the function

F(x)=0

[tex]0=x^{2} -14x+33[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex]-33=x^{2} -14x[/tex]

Complete the square. Remember to balance the equation by adding the same constants to each side

[tex]-33+49=x^{2} -14x+49[/tex]

[tex]16=x^{2} -14x+49[/tex]

Rewrite as perfect squares

[tex]16=(x-7)^{2}[/tex]

square root both sides

[tex](x-7)=(+/-)4[/tex]

[tex]x=(+/-)4+7[/tex]

[tex]x=(+)4+7=11[/tex]

[tex]x=(-)4+7=3[/tex]

so

The factors are

(x-11) and (x-3)

therefore

[tex]F(x)=(x-11)(x-3)[/tex]

Question 1:

For this case we have the following expression:

[tex]\frac {\frac {y-1} {y ^ 2-36}} {\frac {1-7y} {y + 6}} =\\\frac {(7y-1) (y + 6)} {(y ^ 2-36) (1-7y)} =[/tex]

We have to:

[tex]y ^ 2-36 = (y + 6) (y-6)[/tex]

Rewriting:

[tex]\frac {(7y-1) (y + 6)} {(y + 6) (y-6) (1-7y)} =\\\frac {7y-1} {(y-6) (1-7y)} =[/tex]

We take common factor "-" in the denominator:

[tex]\frac {7y-1} {(y-6) * - (- 1 + 7y)} =\\\frac {7y-1} {- (y-6) * (7y-1)} =\\- \frac {1} {(y-6)}[/tex]

ANswer:

[tex]- \frac {1} {(y-6)}[/tex]

Question 2:

For this case we must simplify the following expression:

[tex](3n ^ 4 + 1) + (- 8n ^ 4 + 3) - (- 8n ^ 4 + 2) =[/tex]

We eliminate parentheses keeping in mind that:

[tex]+ * - = -\\- * - = +\\3n ^ 4 + 1-8n ^ 4 + 3 + 8n ^ 4-2 =[/tex]

We add similar terms:

[tex]3n ^ 4-8n ^ 4 + 8n ^ 4 + 1 + 3-2 =\\3n ^ 4 + 2[/tex]

Answer:

[tex]3n ^ 4 + 2[/tex]

Answer:

The value of x = -1 makes the denominator of the function equal to zero. That is why this value is not included in the domain of f(x)

Step-by-step explanation:

We have the following expression

[tex]f(x) = \frac{x^2+4x+3}{x^2-x-2}[/tex]

Since the division between zero is not defined then the function f(x) can not include the values of x that make the denominator of the function zero.

Now we search that values of x make 0 the denominator factoring the polynomial [tex]x^2-x-2[/tex]

We need two numbers that when adding them get as a result -1 and when multiplying those numbers, obtain -2 as a result.

These numbers are -2 and 1

Then the factors are:

[tex](x-2) (x + 1)[/tex]

We do the same with the numerator

[tex]x^2+4x+3[/tex]

We need two numbers that when adding them get as a result 4 and when multiplying those numbers, obtain 3 as a result.

These numbers are 3 and 1

Then the factors are:

[tex](x+3)(x + 1)[/tex]

Therefore

[tex]f(x) = \frac{(x+3)(x+1)}{(x-2)(x+1)}[/tex]

Note that [tex]\frac{(x+1)}{(x+1)}=1[/tex] only if [tex]x \neq -1[/tex]

So since [tex]x = -1[/tex] is not included in the domain the function has a discontinuity in [tex]x = -1[/tex]

Final answer:

The function f(x) = (x²+4x+3)/(x²-x-2) is not continuous at x = -1 because the denominator becomes zero at that point, rendering the function undefined.

Explanation:

The function f(x) = (x²+4x+3)/(x²-x-2) is not continuous at x = -1 primarily because the denominator of the function becomes zero at x = -1.

Specifically, the denominator factors as (x-2)(x+1), and when x equals -1, the denominator equals zero, which makes the function undefined at that point.

Therefore, the function has a discontinuity at x = -1, and by definition, a function is not continuous at points where it is not defined.

A) 3x²-5x-28. The area of the rectangle  with length 3x+7 and width x-4 can be represented as 3x²-5x-28.

The equation to find the area of ​​the rectangle is simply A = l * w. This means that the area of ​​a rectangle is equal to the product of its length (l) by its width (w), or of its length by its width.

A = w*l

A = (3x + 7)(x -4) = (3x)(x) + (3x)(-4) + (7)(x) + (7)(-4)

A = 3x² - 12x + 7x - 28

A = 3x² -5x - 28

The area of a square that measures 3.1 m on each side will be 9.61 m².

The area of the square is found as the square of the length of its side. If the length of a side is a;

Area of a square = side²

Given data;

S is the length of the side= 3.1 m

Area of a square = a²

A=a²

A= (3.1 m)²

A = 9.61 m²

Hence, the area of a square will be 9.61 m².

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The area of a square With each side measuring 3.1 m is 9.61 m², and this answer is provided with three significant figures.

The area of a square is calculated as the product of its side lengths. Since all sides of a square are equal, if a square measures 3.1 m on each side, the area will be:

Area = side × side = 3.1 m × 3.1 m

To find this product, you multiply 3.1 by itself:

3.1 m × 3.1 m = 9.61 m²

To report this area, we express it in square meters (m²) and use the correct number of significant figures, which in this case is three, based on the given measurements of the sides of the square.

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The functions F(x) and G(x) are not inverses of each other.

The correct answer is B. No, because the functions contain different operations.

Given are composition of functions, F(x) = √(x) -6G(x) = (x+6)²

We need to determine if the two functions are inverses of each other.

To determine if the functions F(x) = √(x) - 6 and G(x) = (x + 6)² are inverses of each other using composition of functions, we need to check if their compositions result in the identity function.

Let's calculate the composition:

F(G(x)) = F((x + 6)²) = √((x + 6)²) - 6 = |x + 6| - 6

Now, let's calculate the composition in the reverse order:

G(F(x)) = G(√(x) - 6) = (√(x) - 6 + 6)² = (√(x))² = x

Since F(G(x)) = |x + 6| - 6 and G(F(x)) = x, we can see that they are not equal for all values of x.

Therefore, the functions F(x) and G(x) are not inverses of each other.

The correct answer is B. No, because the functions contain different operations.

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Two functions are inverses if both (f o g)(x) and (g o f)(x) are equal to x. If they are, their composition will yield x, indicating that the two functions are indeed inverses.

To determine if two functions are inverses of each other using composition of functions, you should perform the operation (f o g)(x) and (g o f)(x). If f and g are inverse functions, both of these compositions will yield x.

Let's take the example of functions f(x) = 2x + 3 and g(x) = (x - 3) / 2. To check if they are inverses:

Since both (f o g)(x) and (g o f)(x) equals x, so f(x) and g(x) are inverses of each other.

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b. An increase in expected inflation, combined with a constant real risk-free rate and a constant market risk premium, would lead to identical increases in the required returns on a riskless asset and on an average stock, other things held constant.

Hope this helps :)

Answer: 0.21

Step-by-step explanation:

We know that if two events A and B are independent , then the probability of A and B is given by :-

[tex]\text{P and B}=P(A)\times P(B)[/tex]

Given: The probability that an adult possesses a credit card P(A)= 0 .70

The probability that an adult  does not possess a credit card[tex]P(B)= 1-P(A)=0 .30[/tex]

By assuming the independence, the probability that the first adult possesses a credit card and the second adult does not possess a credit card is given by :-

[tex]0.70\times0.30=0.21[/tex]

Hence, the probability that the first adult possesses a credit card and the second adult does not possess a credit card is 0.21.

Final answer:

To find the probability that the first adult selected at random has a credit card and the second does not, multiply the probability of the first event (0.70) by the probability of the second event (0.30), which yields 0.21 or 21%.

Explanation:

The subject of this question is Mathematics, specifically dealing with probability. The question is at a High School level, focusing on the concept of independent events in probability. To calculate the probability that the first adult possesses a credit card and the second adult does not possess a credit card, we use the rule of independent events:

The probability of the first adult having a credit card is 0.70 (given).

The probability of the second adult not having a credit card is 1 - 0.70 = 0.30.

Since these two events are independent, we multiply the probabilities of each event occurring:

P(First has a credit card AND Second does not have a credit card) = P(First has a credit card) * P(Second does not have a credit card) = 0.70 * 0.30

The answer is therefore 0.21 or 21%

The probability distribution of the number of female children in a family with 4 children, assuming male and female children are equally likely, is calculated by enumerating combinations for each possible number of girls and dividing by the total number of outcomes.

This problem involves understanding the concept of probability distribution. Let's denote 'G' for girl and 'B' for boy. In a family with 4 children, every child can be either a boy or a girl which gives us 2*2*2*2 = 16 possible combinations which we see listed in the problem.

Let's represent 'X' as the number of girls in the family. X could be 0, 1, 2, 3 or 4. For each of these values of X, we need to calculate the probability, i.e., the number of combinations which satisfy each X, divided by 16 (the total possibilities).

So the probability distribution of X is: P(X=0) = 1/16, P(X=1) = 1/4, P(X=2) = 3/8, P(X=3) = 1/4, P(X=4) = 1/16.

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The probability distribution of the number of girls in a family with four children is as follows: P(X = 0) = 1/16, P(X = 1) = 4/16, P(X = 2) = 6/16, P(X = 3) = 4/16, P(X = 4) = 1/16.

The probability distribution of the number of girls in a family with four children can be determined by analyzing the possible outcomes. There are 16 possible outcomes, ranging from all boys (BBBB) to all girls (GGGG) and various combinations in between. To find the probability distribution, we need to calculate the probability of each outcome. Since all outcomes are equally likely, the probability of each outcome is 1/16. Therefore, the probability distribution is as follows:

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Answer:

(a)Two lines parallel to a third line are parallel.

(c) Two planes parallel to a third plane are parallel.

(f) Two lines perpendicular to a plane are parallel.

(h) Two planes perpendicular to a line are parallel.

(i) Two planes either intersect or are parallel.

(k) A plane and a line either intersect or are parallel

Step-by-step explanation:

(b) Two lines perpendicular to a third line are parallel. -- No. The y-, and z-axes are perpendicular to the x-axis, but are not parallel.

(d) Two planes perpendicular to a third plane are parallel. -- No. The x-y and y-z coordinate planes are both perpendicular to the x-z coordinate plane, but are at right angles to each other.

(e) Two lines parallel to a plane are parallel. -- No. Two intersecting lines in the plane z=0 are both parallel to the plane z=1, but are not parallel to each other.

(g) Two planes parallel to a line are parallel. -- No. Both the x-z plane and the y-z plane are parallel to the line (x, y, z) = (1, 1, z), but those coordinate planes are perpendicular to each other.

(j) Two lines either intersect or are parallel. -- No. The lines may be skew, running different directions in parallel panes.

Final answer:

The student's inquiry into the truth of various geometric statements has been addressed, confirming the true relationships and correcting the false ones, based on the principles of Euclidean geometry which govern lines and planes.

Explanation:

When it comes to geometry in the context of Euclidean space, which is the setting for high school mathematics, the rules governing the behavior of lines and planes can be understood through the principles of parallel and perpendicular relationships. Now, let's assess each of the statements given by the student:

These principles form the basis for understanding the complex relations of geometric shapes and objects which are important for most geometrical problems and real-world applications.

Final answer:

To solve the problem, use the system of inequalities: x + y ≥ 0, x ≥ 0, y ≥ 0, 6x + 6y ≥ 60, and x + y ≤ 12.

Explanation:

To solve the given real-world problem, the system of inequalities you would use is:

These inequalities represent the conditions that need to be met: x and y (representing the number of hours worked as a cashier and as a babysitter, respectively) must be greater than or equal to 0, and the total income from both jobs (6x + 6y) must be greater than or equal to $60, and the total number of hours worked (x + y) must be less than or equal to 12.

Answer:

10,000

Step-by-step explanation:

With a bike lock, you can usually use the same number more than one... you can have them the same (7777) if you want.

So, you have 10 possibilities for the first digit, 10 again for the second digit, 10 for the third digit... and also 10 for the last digit.  So...

10 * 10 * 10 * 10 = 10,000

These combinations range from 0000 to 9999

Final answer:

The total number of possible combinations for a 4-digit bike lock with each digit ranging from 0-9 is 10,000, as determined by the fundamental counting principle and calculated by multiplying 10 choices for each digit.

Explanation:

To find the total number of possible combinations for a 4-digit bike lock where each digit can range from 0-9, you apply the fundamental counting principle. This principle states that if there are n ways to perform one task and m ways to perform another task, then there are n × m ways to perform both tasks in sequence.

For the bike lock, each of the 4 digits can be chosen in 10 ways (0 through 9). Since the choice of each digit is independent of the others, you multiply the number of choices for each digit together:

Therefore, the total number of combinations is 10 × 10 × 10 × 10 = 10,000.

Answer:

  175

Step-by-step explanation:

The rate of change of the goose population is a function of the population:

  G'(x) = (x/5)(350 -x)

This function describes a downward-opening parabola with zeros at x=0 and x=350. The value of x halfway between these zeros, at x = 175, is where the maximum value of G'(x), hence the maximum rate of change, is located.

The goose population is increasing most rapidly when it is 175.

Answer: 0.1093

Step-by-step explanation:

Given: Mean : [tex]\mu=93[/tex]

Standard deviation : [tex]\sigma = 22[/tex]

Sample size : [tex]n=30[/tex]

The formula to calculate z-score is given by :_

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x= 97.95, we have

[tex]z=\dfrac{97.95-93}{\dfrac{22}{\sqrt{30}}}\approx1.23[/tex]

The P-value = [tex]P(z>1.23)=1-P(z<1.23)=1-0.8906514=0.1093486\approx0.1093[/tex]

Hence, the  probability that the sample mean would be greater than 97.95 WPM =0.1093

Answer:

[tex]\large\boxed{f^{-1}(x)=9x+18}[/tex]

Step-by-step explanation:

[tex]f(x)=\dfrac{1}{9}x-2\to y=\dfrac{1}{9}x-2\\\\\text{Exchange x to y and vice versa}\\\\x=\dfrac{1}{9}y-2\\\\\text{solve for}\ y:\\\\\dfrac{1}{9}y-2=x\qquad\text{add 2 to both sides}\\\\\dfrac{1}{9}y=x+2\qquad\text{multiply both sides by 9}\\\\9\!\!\!\!\diagup^1\cdot\dfrac{1}{9\!\!\!\!\diagup_1}y=9x+(9)(2)\\\\y=9x+18[/tex]

Answer:

y = 17x + 130

For 9 days, you would pay $283.

Step-by-step explanation:

y = 17x + 130

Total cost = 17$ a day, plus the 130$ fee.

x = 9

y = (17)(9) + 130

y = 153 + 130

y = 283

The required cost of renting a car for 9d days with the company is $283.

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

here,
At The Car rental Company, You must pay a rate of $ 130 and then a daily fee of $ 17 Per day.
Let the number of days be x for renting a car,
According to the question,
Total cost(y) = 130 + 17x
Put x = 9

Total cost = 130 + 17×9
               = $283

Thus, the required cost of renting a car for 9d days with the company is $283.

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Answer: 0.02

Step-by-step explanation:

Given: Sample size : [tex]n= 525[/tex]

The population proportion [tex]P=0.3[/tex]

Then, [tex]Q=1-P=1-0.3=0.7[/tex]

The formula to calculate the standard error is given by :-

[tex]S.E.\sqrt{\dfrac{PQ}{n}}[/tex]

[tex]\Rightarrow\ S.E.=\sqrt{\dfrac{0.3\times0.7}{525}}=0.02[/tex]

Hence, the standard deviation of the sample proportions (i.e., the standard error of the proportion) is 0.02.

Triangles with height [tex]h[/tex] and base [tex]b[/tex], with [tex]b=h[/tex] have area [tex]\dfrac{b^2}2[/tex].

Such cross sections with the base of the triangle in the disk [tex]x^2+y^2\le25[/tex] (a disk with radius 5) have base with length

[tex]b(x)=\sqrt{25-x^2}-\left(-\sqrt{25-x^2}\right)=2\sqrt{25-x^2}[/tex]

i.e. the vertical (in the [tex]x,y[/tex] plane) distance between the top and bottom curves describing the circle [tex]x^2+y^2=25[/tex].

So when [tex]x=4[/tex], the cross section at that point has base

[tex]2\sqrt{25-16}=6[/tex]

so that the area of the cross section would be 6^2/2 = 18.

In case it's relevant, the entire solid would have volume given by the integral

[tex]\displaystyle\int_{-5}^5\frac{b(x)^2}2\,\mathrm dx=4\int_0^5(25-x^2)\,\mathrm dx=\frac{1000}3[/tex]

The question is about finding the volume of a solid with a circular base and equilateral triangular cross-sections, and the area of a cross section at x = 4. The base is defined by the circle equation x2 + y2 = 25 and the height and base of triangles are equal.

The question relates to the calculation of the volume of a solid object and the area of its cross section. The base of the solid is a circle defined by x2 + y2 = 25, which is a circle of radius 5. As the cross sections perpendicular to the x-axis are equal in height and base, they form equilateral triangles.

So the area A of the triangle at x = 4 is given by A = 1/2 * Base * Height. But in an equilateral triangle, the base and height are equal, so A = 1/2 * b2. From the equation of circle, the value of 'b' at x = 4 can be calculated as √(25 - 42) = 3. To get the volume we integrate the area A over the x domain of [-5,5].

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Answer:

D. Incomplete randomization

Step-by-step explanation:

ANSWER

Extraneous solution: x=6

Real solution: x=11

EXPLANATION

The given expression is

[tex] \sqrt{x - 2} + 8 = x[/tex]

Add -8 to both sides:

[tex]\sqrt{x - 2} + 8 + - 8= x + - 8[/tex]

[tex] \implies\sqrt{x - 2} = x - 8[/tex]

Square both sides.

[tex]\implies(\sqrt{x - 2} )^{2} =( x - 8)^{2} [/tex]

[tex]x - 2=( x - 8)^{2} [/tex]

We expand the to get

[tex]x - 2 = {x}^{2} - 16x + 64[/tex]

Write in standard quadratic form.

[tex] {x}^{2} - 16x - x + 64 + 2 = 0[/tex]

[tex] {x}^{2} - 17x + 66 = 0[/tex]

Factor to get:

[tex](x - 6)(x - 11) = 0[/tex]

[tex]x = 6 \: or \: \: x = 11[/tex]

We check for extraneous solutions by substituting each value of x into the original equation.

When x=6

[tex]\sqrt{6 - 2} + 8 = 6[/tex]

[tex]\sqrt{4} + 8 =6[/tex]

[tex]2 + 8 = 10 \ne8[/tex]

Hence x=6 is an extraneous solution.

When x=11

[tex]\sqrt{11- 2} + 8 = 11[/tex]

[tex]\sqrt{9} + 8 = 11[/tex]

[tex]3 + 8 = 11[/tex]

This statement is true.

Hence x=11 is the only solution.

Answer: 0.7021

Step-by-step explanation:

Let D be the event that team plays in day , N be the event that the team plays in night and W be the event when team wins.

Then , [tex]P(D)=0.40\ \ \ P(N)=0.60[/tex]

[tex]P(W|D})=0.35\ \ \ \ P(W|N)=0.55[/tex]

Using the law of total probability , we have

[tex]P(W)=P(D)\timesP(W|D)+P(N)\timesP(W|N)\\\\\Rightarrow\ P(W)=0.40\times0.35+0.60\times0.55=0.47[/tex]

Using Bayes theorem ,

The required probability :[tex]P(N|W)=\dfrac{P(N)P(W|N)}{P(W)}[/tex]

[tex]=\dfrac{0.60\times0.55}{0.47}=0.702127659574\approx0.7021[/tex]

The probability that exactly one customer dines on the first floor is:

                     0.26337  

We need to use the binomial theorem to find the probability.

The probability of k success in n experiments is given by:

       [tex]P(X=k)=n_C_k\cdot p^k\cdot (1-p)^{n-k}[/tex]

where p is the probability of success.

Here p=1/3

( It represents the probability of choosing first floor)

k=1 ( since only one customer has to chose first floor)

n=6 since there are a total of 6 customers.

This means that:

[tex]P(X=1)=6_C_1\times (\dfrac{1}{3})^1\times (1-\dfrac{1}{3})^{6-1}\\\\\\P(X=1)=6\times (\dfrac{1}{3})\times (\dfrac{2}{3})^5\\\\\\P(X=1)=0.26337[/tex]

Using the binomial distribution, it is found that there is a 0.2634 = 26.34% probability that exactly one customer dines on the first floor.

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For each customer, there are only two possible outcomes, either they dine on the first floor, or they do not. The probability of a customer dining on the first floor is independent of any other customer, which means that the binomial probability distribution is used to solve this question.

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Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of a success on a single trial.

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The probability that exactly one customer dines on the first floor is P(X = 1), thus:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 1) = C_{6,1}.(0.3333)^{1}.(0.6667)^{5} = 0.2634[/tex]

0.2634 = 26.34% probability that exactly one customer dines on the first floor.

A similar problem is given at https://brainly.com/question/13036444

Let assume that [tex]\sqrt3[/tex] is rational. Therefore we can express it as [tex]\dfrac{a}{b}[/tex] where [tex]a,b\in \mathbb{Z}[/tex] and [tex]\text{gcd}(a,b)=1[/tex].

[tex]\dfrac{a}{b}=\sqrt3\\\dfrac{a^2}{b^2}=3\\a^2=3b^2[/tex]

It means that [tex]3|a^2[/tex] and so also [tex]3|a[/tex].

Therefore [tex]a=3k[/tex] where [tex]k\in\mathbb{Z}[/tex].

[tex](3k)^2=3b^2\\9k^2=3b^2\\b^2=3k^2[/tex]

It means that [tex]3|b^2[/tex] and so also [tex]3|b[/tex].

If both [tex]a[/tex] and [tex]b[/tex] are divisible by 3, then it contradicts our initial assumption that [tex]\text{gcd}(a,b)=1[/tex]. Therefore [tex]\sqrt3[/tex] must be an irrational number.

Answer: 0.2420

Step-by-step explanation:

Given: Mean : [tex]\mu = 250 \text{ feet}[/tex]

Standard deviation : [tex]\sigma =50\text{ inch}[/tex]

Sample size : [tex]n=49[/tex]

The formula to calculate z is given by :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x= 245

[tex]z=\dfrac{245-250}{\dfrac{50}{\sqrt{49}}}=-0.7[/tex]

The P Value =[tex]P(Z<245)=P(z<-0.7)=0.2419637\approx0.2420[/tex]

Hence, the probability that the 49 touchdowns traveled an average of less than 245 feet= 0.2420

The probability that the 49 touchdowns traveled an average of less than 245 feet is approximately 0.2438, or 24.38%.

Step 1:

To find the probability that the 49 touchdowns traveled an average of less than 245 feet, we can use the Central Limit Theorem (CLT) since we have a large enough sample size (49) to assume that the sample mean follows a normal distribution.

The CLT states that the distribution of sample means of a sufficiently large sample size will be approximately normal, regardless of the distribution of the original population, as long as the sample size is large enough.

Given:

- Population mean mu = 250 feet

- Population standard deviation [tex](\( \sigma \))[/tex] = 50 feet

- Sample size n = 49

Step 2:

The standard error of the sample mean SE is given by:

[tex]\[SE = \frac{\sigma}{\sqrt{n}}\][/tex]

Substituting the given values:

[tex]\[SE = \frac{50}{\sqrt{49}} = \frac{50}{7} \approx 7.14\][/tex]

Step 3:

Now, we can calculate the z-score for the sample mean of 245 feet using the formula:

[tex]\[z = \frac{\bar{x} - \mu}{SE}\][/tex]

Where:

- [tex]\( \bar{x} \)[/tex] is the sample mean

- [tex]\( \mu \)[/tex] is the population mean

- [tex]\( SE \)[/tex] is the standard error of the sample mean

Step 4:

Substituting the given values:

[tex]\[z = \frac{245 - 250}{7.14} \approx -0.6993\][/tex]

Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to this z-score.

The probability that the sample mean is less than 245 feet can be found by finding the area to the left of the z-score on the standard normal distribution curve.

From the standard normal distribution table, we find that the probability corresponding to a z-score of -0.6993 is approximately 0.2438.

Therefore, the probability that the 49 touchdowns travelled an average of less than 245 feet is approximately 0.2438, or 24.38%.

Answer:

Step-by-step explanation:

-2.7(13)^2 + 72.4(13) + 1911

1,232.01 + 941.2 + 1911 = 4084.21