Solve the following equation using the zero product property. Enter one solution per box. No brackets {} are needed.

This strategy ensures that every multiple of gcd(a, b) up to max(a, b) is eventually on the board, and since the player who cannot make a move loses, you will always win.

In mathematics, a linear combination is a sum of scalar multiples of one or more variables.

(a) To show that every number on the board at the end of the game is a multiple of gcd(a, b), we will use mathematical induction.

First, note that any number that is a multiple of gcd(a, b) can be written as a linear combination of a and b. That is, for any positive integer k, there exist integers x and y such that k*gcd(a,b) = xa + yb.

Now, suppose that after some number of turns, the numbers on the board are c and d, where c is a multiple of gcd(a, b) and d is some other number. Then, we can write c = xa + yb and d = wa + zb for some integers x, y, w, and z.

On the next turn, a player must choose a number that is the difference of two numbers already on the board. Thus, the only possible choice is |c - d| = |xa + yb - wa - zb|.

We can rewrite this as |(x-w)a + (y-z)b|. Note that (x-w) and (y-z) are integers, so this number is a linear combination of a and b, and therefore a multiple of gcd(a, b). Thus, the new number on the board is a multiple of gcd(a, b).

By induction, every number on the board at the end of the game is a multiple of gcd(a, b).

(b) To show that every positive multiple of gcd(a, b) up to max(a, b) is on the board at the end of the game, we will again use induction.

First, note that gcd(a, b) itself must be on the board, since it is a multiple of gcd(a, b) and can be written as a linear combination of a and b.

Now, suppose that after some number of turns, all multiples of gcd(a, b) up to k are on the board, where k is a positive integer less than or equal to max(a, b).

Consider the next turn. The player must choose a number that is the difference of two numbers already on the board. Let c and d be the two numbers chosen. Then, we know that c - d is a multiple of gcd(a, b) by part (a).

Thus, every multiple of gcd(a, b) up to k + (c - d) is on the board. If k + (c - d) is greater than max(a, b), then we are done, since all multiples of gcd(a, b) up to max(a, b) are on the board.

Otherwise, we can continue the game and use induction to show that all multiples of gcd(a, b) up to max(a, b) will eventually be on the board.

(c) To win the game every time, always start by choosing gcd(a, b). This is a legal move, since it can be written as a linear combination of a and b.

From then on, always choose a number that is the difference of the two numbers on the board, except when that number is already on the board. In that case, choose any other number that is a multiple of gcd(a, b) that is not already on the board.

This strategy ensures that every multiple of gcd(a, b) up to max(a, b) is eventually on the board, and since the player who cannot make a move loses, you will always win.

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Since the arrival times are independent and uniformly distributed, the probability that a single person arrives in the first hour is 1/10^6. Therefore, the number of people N that arrive in the first hour follows a binomial distribution with parameters n=10^6 and p=1/10^6.

The probability that exactly i people arrive in the first hour is then given by the binomial probability mass function:

P{N=i} = (10^6 choose i) * (1/10^6)^i * (1 - 1/10^6)^(10^6 - i)

Using the normal approximation to the binomial distribution, we can approximate this probability as:

P{N=i} ≈ φ((i+0.5 - np) / sqrt(np(1-p)))

where φ is the standard normal probability density function. Plugging in the values of n=10^6 and p=1/10^6, we get:

P{N=i} ≈ φ((i+0.5 - 1) / sqrt(1*0.999999)) = φ(i - 0.5)

Therefore, an approximation for P{N=i} is given by the standard normal density function evaluated at i-0.5.

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Finally, using the commutative and associative properties of Boolean addition, we can group the terms to get AB + AC + B.

Q1:

Using De Morgan's theorem, we have:

F = XYZ + XYZ = XYZ(1 + 1) = XYZ

Taking the complement of F, we get:

F' = (XYZ)'

= (X'+Y'+Z')

= X'Y'Z'

Now, let's find the complement of F';

F' = X(YZ + Y'Z')

Taking the complement of F', we get:

F'' = (X(YZ + Y'Z'))'

= (X(YZ)')(Y(Y')Z')'

= (X'(Y'+Z))(YZ)

= X'YZ + XYZ'

Therefore, the complement of F is X'Y'Z', and the complement of F'; is X'YZ + XYZ'.

Q2:

ABC + ABC + ABC + ABC + ABC = ABC + ABC + ABC = ABC

Explanation: Using the associative property of Boolean addition, we can group the terms to get ABC + ABC + ABC = ABC.

AB + A(B+C) + B(B+C) = AB + AB + AC + BB + BC

= AB + AC + B

Explanation: Using the distributive property of Boolean multiplication over addition, we can expand the second and third terms to get AB + AC + BB + BC. Using the identity law, BB can be simplified to B. Finally, using the commutative and associative properties of Boolean addition, we can group the terms to get AB + AC + B.

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The 95% confidence interval for the mean amount (in milligrams) of nicotine in the sampled brand of cigarettes is C.39.2 to 40.8

We can use a t-table or a calculator to calculate the t-score. The t-score for a 95% confidence interval with 22 degrees of freedom (n-1) is around 2.074.

When we plug in the values, we get:

CI = 40 ± 2.074 * 1.8/√23 = 40 ± 0.763 = (39.237, 40.763)

As a result, the 95% confidence interval for the mean nicotine content of the studied cigarette brand is (39.237, 40.763) mg.

Because 39.2 to 40.8 is the closest response choice, the answer is 39.2 to 40.8.

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The calculated value of the expression when a = 3 and b = -2 is 1

The expression is given as

(StartFraction 3 a Superscript negative 2 Baseline b Superscript 6 Baseline Over 2 a Superscript negative 1 Baseline b Superscript 5 Baseline EndFraction) squared

Mathematically, this can be expressed as

(3a^-2b^6/2a^-1b^5)^2

The values of a and b are given as

a =3 and b = -2

substitute the known values in the above equation, so, we have the following representation

(3(3)^-2(-2)^6/2(3)^-1(-2)^5)^2

Evaluate

So, we have

1

Hence, the solution is 1

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Answer:   4.

Step-by-step explanation: y = kx

36 = k(9)

k = 4

Santiago used 185 cell phone minutes over the course of a month.

To determine the total number of cell phone minutes that Santiago used over the course of a month, you would use the operation of addition.

You would add up the number of minutes that Santiago spoke with his mom, dad, and friends:

Total cell phone minutes = 45 minutes + 30 minutes + 110 minutes

Total cell phone minutes = 185 minutes

Therefore, Santiago used 185 cell phone minutes over the course of a month.

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The value of the iterated integral is [tex]7/3[/tex] √2 in the given case

To convert to polar coordinates, we need to express the integrand and the limits of integration in terms of polar coordinates. Let's start by finding the limits of integration:

0 ≤ y ≤ √2 - y[tex]^2[/tex]

0 ≤ x ≤ 1

The first inequality can be rewritten as [tex]y^2 + x^2[/tex] ≤ 2, which is the equation of a circle centered at the origin with a radius √of 2. Therefore, the limits of integration in polar coordinates are:

0 ≤ r ≤ √2

0 ≤ θ ≤ π/2

Now, let's express the integrand in polar coordinates:

7xy = 7r cos(θ) sin(θ)

And the differential area element in polar coordinates is:

dA = r dr dθ

Therefore, the integral becomes:

= [tex]7/3[/tex] √2

Therefore, the value of the iterated integral is [tex]7/3[/tex] √2.

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Answer:

Have a Nice Best Day : )

The domain of this relation is the set of all non-negative real numbers that can be expressed as: A weight from 0 to 5 ounces A weight from 5 to 20 ounces that is a multiple of 0.2 ounces A weight greater than 20 ounces is a multiple of 0.15 ounces.

The domain of this relation is the set of all possible weights of packages that can be shipped using the postal service.

Since the postal service charges $2 for packages up to 5 ounces, the domain includes all weights from 0 to 5 ounces. For packages weighing between 5 and 20 ounces, the domain includes all weights from 5 to 20 ounces, with each weight being a multiple of 0.2 ounces.

For packages weighing more than 20 ounces, the domain includes all weights greater than 20 ounces, with each weight being a multiple of 0.15 ounces.

The domain does not include negative numbers or numbers that are not expressible using the above criteria.

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Answer:

Assuming that those are fraction the simplest form would be --> 7/4

Hope this helps

The total change to his score as an integer is -115

The total change in Juan score is calculated as follows;

Let Juan's initial score = x

when Juan finished the next level of his video game, he lost 10 points for each of the two targets he missed.

total points deducted = 20 points.

New score = x - 20

He was also penalized 95 points for taking too long.

His final score;

(x - 20) - 95

= x - 115.

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The percent of container A that is empty after the pumping is complete is approximately 35.4%.

We have,

The volume of water in container A is given by:

V(A) = πr²h

= π(6 ft)²(7 ft)

= 882π cubic feet

The volume of water in container B is given by:

V(B) = πr²h

= π(5 ft)²(10 ft)

= 250π cubic feet

When container A is emptied into container B, the total volume of water becomes:

= V(A) + V(B)

= 882π + 250π

= 1132π cubic feet

The volume of container B is 250π cubic feet, so the remaining volume of water in container A is:

= 1132π - 250π

= 882π cubic feet

The percent of container A that is empty after the pumping is complete is:

= (882π / (πr²h)) x 100%

= (882 / (6² x 7)) x 100%

= 35.4%

Therefore,

The percent of container A that is empty after the pumping is complete is approximately 35.4%.

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Part(a),

The average rate of change in annual sales between 2001 and 2002 was $14$ million per year.

Part(b),

The average rate of change in annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

a) Compute the difference in sales between 2001 and 2002 and divide it by the total number of years in order to determine the rate of change between those two years:

Rate of change = (Sales in 2002 - Sales in 2001) / (2002 - 2001)

Rate of change = (255 - 241) / 1 = 14 million/year

Therefore, the average rate of change in annual sales between 2001 and 2002 was $14$ million per year.

b) Calculate the difference in sales between 2001 and 2004 and divide it by the total number of years to determine the rate of change between those two years:

Rate of change = (Sales in 2004 - Sales in 2001) / (2004 - 2001)

Rate of change = (231 - 241) / 3 = -3.33 million/year

Therefore, the average rate of change of annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

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Since 504 > 100, Town A had a greater average rate of change.

The given function is P(x)=10x²+8x+600 represents the population of Town A.

Here, x represent the number of years.

In year 0, Town B had a population of 400 people. Town B's population increased by 100 people each year.

P(x)=400+100x

We can calculate the average rate of change for each town by finding the difference between the population in year 8 and the population in year 4 and dividing by the number of years (4).

For Town A, we have:

P(8) - P(4) = 10(8² + 8(8) + 600 - (10(4²) + 8(4) + 600) = 2016

Average rate of change = 2016/4 = 504

For Town B, we have:

400 + (100×4) = 800

Average rate of change = 400/4 = 100

Since 504 > 100, Town A had a greater average rate of change.

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There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

The probability that exactly two kittens will be chocolate brown is 1/12.

The probability that at least two kittens will be chocolate brown is 7/12.

Priya can increase the accuracy of the simulation by increasing the number of trials.

We have,

To determine if a trial was a success, we need to count the number of chocolate brown kittens in each trial.

If a trial has at least two chocolate brown kittens, it is considered a success.

Now,

Using the table provided, we can count the number of chocolate brown kittens in each trial:

trial number outcome count of chocolate brown kittens

1 ggggg 0

2 gggwg 1

3 wgwgw 0

4 gwggg 1

5 gggwg 1

6 wwggg 0

7 gwggg 1

8 ggwgw 1

9 wwwgg 0

10 ggggw 2

11 wggwg 1

12 gggwg 1

So,

There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

To estimate the probability that exactly two kittens will be chocolate brown, we need to count the number of trials where exactly two chocolate brown kittens were born and divide it by the total number of trials.

From the table, we can see that there is only one trial where exactly two chocolate brown kittens were born (trial 10).

The estimated probability.

=  1/12

= 0.0833.

To estimate the probability that at least two kittens will be chocolate brown, we need to count the number of trials where at least two chocolate brown kittens were born and divide it by the total number of trials.

From the table, we can see that there are 7 successful trials.

The estimated probability.

= 7/12

= 0.5833.

Another question Priya could answer using this simulation is:

Question:

What is the probability that all five kittens will be white?

Answer:

We need to count the number of trials where all five cubes drawn were white (trial 6 and trial 9) and divide it by the total number of trials.

The estimated probability.

= 2/12

= 0.1667.

To increase the accuracy of the simulation, Priya could increase the number of trials conducted.

The more trials conducted, the more accurate the estimated probabilities will be.

Thus,

There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

The probability that exactly two kittens will be chocolate brown is 1/12.

The probability that at least two kittens will be chocolate brown is 7/12.

Priya can increase the accuracy of the simulation by increasing the number of trials.

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The number of people that are expected to go to the youth wing on Sunday would be = 543.

For Saturday;

The number of people that went to the youth wing = 497

The number of people that went to social issues = 369

The number of people that went to Fiction and Literature = 416

The total number of people that went to the central library = 497+369+416 = 1,282

For Sunday, the number of people that would visit the youth wing ;

= 497/1282× 1400/1

= 695800/1282

= 543 (to the nearest whole number)

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The quantity that maximizes total revenue is 40. The correct option is d. 40.

To maximize total revenue, we need to find the quantity where marginal revenue (MR) equals zero.

Given the MR equation:

MR = 40 - Q

Set MR to zero and solve for Q:

0 = 40 - Q

Q = 40

So, the quantity that maximizes total revenue is 40. The correct option is d. 40.

Maximizing total revenue requires optimizing prices, bundling products, cross-selling/upselling, improving customer retention, expanding the customer base, and reducing costs. Make sure to regularly check on revenue performance and modify strategies accordingly.

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Suppose a monopolist has the following equations: P=40-0.5Q MR=40-Q What is the quantity that maximizes total revenue? a. 25 10 30 d.40 e. 20 T1e MC=10

The time taken for the smokestack A is 22.5 hours.

The time taken for the smokestack B is 45 hours.

The time taken for the smokestack is calculated as follows;

Let's the rate at which smokestack B emits pollution = r

Then smokestack A = 2r

Their total rate of pollution combined;

= r + 2r

= 3r

The total amount of pollution they emitted after 15 hours;

= 3r x 15

= 45r pollution

The time taken for each to emit the same amount;

rate of B = r pollution/hr

time of B = 45r/r = 45 hours

rate of A = 2r

time of A = 45r/2r = 22.5 hours

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Algebra is used to solve the mathematical problems, the total amount spent by Andy on lunches in this week is equals to $195.

Algebra is the branch of mathematics that use in the representation of problems or situations in the form of mathematical expressions. Mathematical ( arithmetic) operations say multiplication (×), division (÷), addition (+), and subtraction (−) are used to form a mathematical expression.

We have, a data of amount spent by Andy on lunches in a week. Let the total amount spent by him in this week be "x dollars". Using algebra of mathematics, we can written as x = sum of amounts spent by him in whole week so, x = $50 + $20 + $10 + $25 + $25 + $15 + $50 = $195

Hence, required total amount value is $195.

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Complete question:

Andy spent the following amounts on lunches this week,

day. amount

Sunday $50

Monday. $20

Tuesday $10

Wednesday $25

Thursday $25

Friday $15

Saturday $50

Calculate total amount he spent in this week.

The choice that matches the graph of the function as is defined to us is:  Graph A.

We are given a function f(x) as:

    f(x)=   2x    if x < 3

 and        4     if  x ≥ 3

This means that in the region (-∞,3) the graph of a function is a straight line that passes through the origin and has a open circle at x=3.

Also, in the region [3,∞) the graph is a straight horizontal line i.e. y=4.

Hence, the graph of this function is Graph A.

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On a piece of paper, graph f(x)={2x if x <3

{4 if x >3. Then determine which answer choice matches the graph you drew

Answer: (10.4, 0).

Step-by-step explanation: In order to determine the coordinate where the projectile would land, we need to find the parabolic path followed by the projectile. Since the catapult is launched from the origin (0,0), and has a maximum height at (5.2, 6.3), we can find the vertex of the parabola, which is also (5.2, 6.3).

The standard form of a parabola is y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola. In this case, h = 5.2 and k = 6.3.

We can use the information from the origin to determine the value of 'a'. Substituting x = 0 and y = 0, we get:

0 = a(0 - 5.2)^2 + 6.3

Solving for 'a':

a(5.2)^2 = -6.3

a = -6.3 / (5.2)^2

Now that we have 'a', we can rewrite the equation of the parabola:

y = a(x - 5.2)^2 + 6.3

To find the x-coordinate where the projectile would land, we need to find the other x-intercept (the other point where y = 0). Since the parabola is symmetric, the other x-intercept will be equidistant from the vertex:

x = 5.2 * 2 = 10.4

Now, we can plug in x = 10.4 into the equation to find the y-coordinate:

y = a(10.4 - 5.2)^2 + 6.3

However, since we are looking for the landing coordinate, which is an x-intercept, we know the y-coordinate will be 0.

Thus, the coordinate where the projectile would land is (10.4, 0).

The sample mean of the population is 3/4 and the variance is 3/80. Using the central limit theorem, P( > 0.8) can be simplified as 0.003.

The mean of the population can be computed as follows:

µ = ∫x f(x) dx from 0 to 1

  = ∫x (3x²) dx from 0 to 1

  = 3/4

The variance of the population can be computed as follows:

σ² = ∫(x-µ)² f(x) dx from 0 to 1

    = ∫(x-(3/4))² (3x²) dx from 0 to 1

    = 3/80

By the Central Limit Theorem, as the sample size n = 80 is large, the distribution of the sample mean  can be approximated by a normal distribution with mean µ and variance σ²/n.

Therefore, P( > 0.8) can be approximated by P(Z >0.8- 0.75)/(sqrt(3/80)/(sqrt(80))), where Z is a standard normal random variable.

Simplifying, we get P( > 0.8) ≈ P(Z > 2.73) ≈ 0.003.

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Answer:

The expected profit from the game can be calculated as the revenue from ticket sales minus the investment cost:

Expected profit = $89,399 - $23,197 = $66,202

The expected loss if it rains can be calculated as the investment cost:

Expected loss = $23,197

To find the net result, we need to use the probability of the game happening (1 - 0.28 = 0.72) and the probability of it raining (0.28):

Net result = (0.72) * (Expected profit) + (0.28) * (Expected loss)

Net result = (0.72) * ($66,202) + (0.28) * ($23,197)

Net result = $47,683.44

Since the net result is positive, the expected outcome is a profit of $47,683.44.

There is an expected profit of $57,963.32.

To calculate the expected profit or loss, we need to consider two possible scenarios:

Scenario 1: It doesn't rain on the day of the game, and the club is able to sell tickets worth $89,399.

Scenario 2: It rains on the day of the game, and the club loses the entire investment of $23,197.

To calculate the expected profit, we need to multiply the revenue from scenario 1 by the probability of it happening, which is (1 - 0.28) = 0.72 (since there's a 28% chance of rain). So, the expected profit is:

Expected profit = 0.72 x $89,399 = $64,451.28

To calculate the expected loss, we need to multiply the investment from scenario 2 by the probability of it happening, which is 0.28 (since there's a 28% chance of rain). So, the expected loss is:

Expected loss = 0.28 x $23,197 = $6,487.96

To find the net result, we subtract the expected loss from the expected profit:

Net result = Expected profit - Expected loss = $64,451.28 - $6,487.96 = $57,963.32

Therefore, there is an expected profit of $57,963.32.

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The number of those games won in that season are: 22 games

Percentage is defined a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. It is given by:

Percentage = (value / total value) * 100%

We are given:

Total number of games played through the season = 40 games

Percentage of games won = 40%

Thus:

Number of games won = 40% * 55

= 22

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(a) The joint probability mass function of X and Y x1=0 x2=2 x3=4

y1=0 1/8 0 PY(y1)=1/2

y2=1 1/8 1/4 PY(y2)=1/4

y3=2 0 0 PY(y3)=1/4

(b) X and Y are uncorrelated. (c) The random variables X and Y are not independent with each other.

(a) We know that P(X=x2,Y=y1) = 0, since there are no entries in the table where X=x2 and Y=y1. Therefore,

x1=0 x2=2 x3=4

y1=0 1/8 0 PY(y1)=1/2

y2=1 1/8 1/4 PY(y2)=1/4

y3=2 0 0 PY(y3)=1/4

(b) The covariance of X and Y is :

Cov(X,Y) = E[XY] - E[X]E[Y]

where E[XY] is the expected value of the product XY,

E[X] = x1P(X=x1) + x2P(X=x2) + x3P(X=x3) = 0(1/4) + 2(1/2) + 4(1/4) = 2

E[Y] = y1P(Y=y1) + y2P(Y=y2) + y3P(Y=y3) = 0(1/2) + 1(1/4) + 2(1/4) = 1

Now,

E[XY] = x1y1P(X=x1,Y=y1) + x2y1P(X=x2,Y=y1) + x2y2P(X=x2,Y=y2) + x3y2P(X=x3,Y=y2) = 0(1/8) + 2(0) + 2(1/8) + 4(1/4) = 1.5

Therefore,

Cov(X,Y) = E[XY] - E[X]E[Y] = 1.5 - 2(1) = -0.5

Since the covariance is negative, hence X and Y are negatively correlated.

(c) To determine whether X and Y are independent, check whether:

P(X=x,Y=y) = P(X=x)P(Y=y)

for all possible values of x and y.

Using the joint probability mass function we determined in part (a), we can check this condition:

P(X=0,Y=0) = 1/8 ≠ (1/4)(1/2) = P(X=0)P(Y=0)

P(X=2,Y=1) = 1/8 ≠ (1/2)(1/4) = P(X=2)P(Y=1)

P(X=4,Y=2) = 1/4 ≠ (1/4)(1/4) = P(X=4)P(Y=2)

Therefore, X and Y are not independent.

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The number of miles is 7 while the number of minutes for this trip is 21.

Let x represent the number of minutes and y represent the number of miles. According to the given information, we have two equations:

1) 0.20x + 1.40y = $14
2) x = 3y

First, we will solve equation (2) for x:

x = 3y

Next, substitute this value of x into equation (1):

0.20(3y) + 1.40y = $14

Now, simplify and solve for y:

0.60y + 1.40y = $14
2.00y = $14
y = 7

Now that we have the value for y (number of miles), we can find the value for x (number of minutes) using equation (2):

x = 3y
x = 3(7)
x = 21

So, the number of minutes for this trip is 21, and the number of miles is 7.

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The sample space for choosing one card is, S={I, L, L, M, P, T, U, Y}

Since, We know that;

A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.

Given that,

A set of eight cards were labeled with M, U, L, T, I, P, L, Y.

Here, the sample space is;

{ S, U, B, T, R, A, C, T}

Now, Elements in order is,

⇒ S = {I, L, L, M, P, T, U, Y}

Therefore, the sample space for the given cards is,

S = {I, L, L, M, P, T, U, Y}

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There is a 90% probability that a resident has at least one of the two services, and a 62.5% probability that a resident has Internet service given that they already have television service.

To answer your question, we will use the formula for the probability of the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where A represents having Internet service and B represents having television service.

The probability of having at least one of the two services is:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
= 0.60 (Internet) + 0.80 (television) - 0.50 (both)
= 1.40 - 0.50
= 0.90 or 90%

Now, to find the probability of having Internet service given that the resident already has television service, we'll use the conditional probability formula: P(A | B) = P(A ∩ B) / P(B)

P(Internet | Television) = P(Internet ∩ Television) / P(Television)
= 0.50 (both) / 0.80 (television)
= 0.625 or 62.5%

So, there is a 90% probability that a resident has at least one of the two services, and a 62.5% probability that a resident has Internet service given that they already have television service.

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