The sum of two numbers is 30. Determine the two numbers of their product is a maximum.

Step-by-step explanation:

f(x)  =  2x^2 -12x + 16       to get to vertex form you will need to complete the square for 'x'....to do THAT you will need the x^2 coefficient to be '1'

Start like this:

2 (x^2 - 6x)    + 16        now complete the square

2 (x^2 - 6x +9)  - 18    + 16

f(x) =  2 (x-3)^2  - 2        Done.

The expression that represents the side length of the square pool is 2x² + 36x - 1120 = 0. The side length of the square pool is x = 16.31 feet.

Let's denote the side length of the square pool as x.

The walkway on two opposite sides of the pool is 8 feet, which means that the overall length of the pool and walkway on those sides is x + 8 + 8 = x + 16.

Similarly, x + 10 + 10 = x + 20.

The total area covered by the pool and walkway is given as 1,440 square feet.

Total Area = Pool Area + Walkway Area

The area of the square pool is x², and the area of the walkway is the difference between the total area and the pool area:

Walkway Area = Total Area - Pool Area

Substituting the values, we have:

1440 = x² + (x + 16)(x + 20)

1440 = x² + (x² + 36x + 320)

Combining like terms:

1440 = 2x² + 36x + 320

2x² + 36x + 320 - 1440 = 0

2x² + 36x - 1120 = 0

After solving, we get:

[tex]x=-9+\sqrt{641},\:x=-9-\sqrt{641}[/tex]

Take positive value:

[tex]x=-9+\sqrt{641},\\x = -9+25.31\\x = 16.31[/tex]

So, the side length of the square pool is 16.31 feet.

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The value of the expression when x=8 is 56/9.

To find the value of an expression, follow these steps:

To find the value of the expression when x=8, we substitute 8 for x in the expression:

(8^2 - 8) / (8+1)

= (64 - 8) / 9

= 56/9

Therefore, the value of the expression when x=8 is 56/9.

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The probability of selecting a red bead is 3/20 when the probability of selecting a white bead, replacing it, and then selecting a red bead.

We need to find the probability of selecting a red bead when first a white bead is selected and then it is replaced and then selected a red bead. The formula to find the probability is,

P(A) = f / N

Where,

f = number of outcomes

N = total number of outcomes

Given data:

Total number of beads =  10

Number of blacks beads = 2

Number of white beads = 3

Number of red beads = 5

The probability of selecting a white bead is given as,

P(A) = f / N

P(W) = 3/10

When the bead is replaced, the probability of selecting a red bead is  P(R) = 5/10

The probability of selecting a white bead and then a red bead is the product of the probabilities of each event:

P(white and red) = P(white) × P(red)

= (3/10) × (5/10)

= 3/20

Therefore, the probability of selecting a white bead, replacing it, and then selecting a red bead is 3/20.

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Step-by-step explanation:

If u want to find the volume

V= width × length × height

V= 8.5 cm × 12 cm × 29 cm

V= 2958 cm3

If the photo’s current height is 314 inches, the enlarged height should be approximately 547.6 inches (rounded to one decimal place), which is 80.0% larger than the current height.

To calculate the enlarged height, you need to find out the percentage increase in height from the current height to the desired height.

The desired height is 234 inches, which is 234/314 = 0.744 or 74.4% of the current height.

To find the actual enlarged height, you need to increase the current height by 74.4%.

Enlarged height = current height + (current height x percentage increase)
Enlarged height = 314 + (314 x 0.744)
Enlarged height = 314 + 233.616
Enlarged height = 547.616

Therefore, the enlarged height should be approximately 547.6 inches (rounded to one decimal place), which is 80.0% larger than the current height.

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Tim and Bella working together can paint the room in 2.4 hours.

We can use the formula:

1/Total Work = 1/Time1 + 1/Time2

where Total Work is the work to be done (in this case, painting the room), Time1 is the time taken by Tim to complete the work, and Time2 is the time taken by Bella to complete the work.

Let x be the time taken by Tim and Bella working together to complete the work. Then we can write:

1/1x = 1/6 + 1/4

Simplifying this equation, we get:

1/1x = 5/12

Multiplying both sides by x, we get:

x = 12/5 = 2.4 hours.

Therefore, Tim and Bella working together can paint the room in 2.4 hours.

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The basketball coach can order approximately 34 shooting shirts for $300.

To determine the number of shooting shirts the basketball coach can order for $300, we need to solve the equation C = 0.2x^2 + 1.6x + 15, where C represents the cost and x represents the number of shooting shirts.

The equation is given as C = 0.2x^2 + 1.6x + 15.

To find the number of shooting shirts for $300, we set the cost C equal to 300 and solve for x:

0.2x^2 + 1.6x + 15 = 300

0.2x^2 + 1.6x + 15 - 300 = 0

0.2x^2 + 1.6x - 285 = 0

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For this equation, a = 0.2, b = 1.6, and c = -285. Plugging in these values into the quadratic formula:

x = (-1.6 ± sqrt(1.6^2 - 4 * 0.2 * -285)) / (2 * 0.2)

Simplifying the equation further:

x = (-1.6 ± sqrt(2.56 + 228)) / 0.4

x = (-1.6 ± sqrt(230.56)) / 0.4

x = (-1.6 ± 15.18) / 0.4

Now we have two solutions:

x1 = (-1.6 + 15.18) / 0.4 = 33.95

x2 = (-1.6 - 15.18) / 0.4 = -44.95

Since the number of shooting shirts cannot be negative, we discard the negative solution.

Therefore, the basketball coach can order approximately 34 shooting shirts for $300.

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Let's assume that the cylindrical can has a height of "h" and a radius of "r". We want to find the values of "h" and "r" that minimize the amount of material needed to manufacture the can.

The amount of material needed to manufacture the can can be represented by the surface area of the can, which is the sum of the area of the top and bottom circles and the lateral area of the cylinder.

The area of the top and bottom circles can be calculated using the formula for the area of a circle:

A_top = A_bottom = πr^2

The lateral area of the cylinder can be calculated using the formula for the lateral surface area of a cylinder:

A_lateral = 2πrh

Therefore, the total surface area of the cylindrical can can be calculated as:

A_total = A_top + A_bottom + A_lateral

= 2πr^2 + 2πrh

Now, we need to express "h" in terms of "r" and the volume of the can, which is given as 180 cm^3. The formula for the volume of a cylinder is:

V = πr^2h

Substituting the given value of the volume and solving for "h", we get:

h = 180/(πr^2)

Substituting this expression for "h" in the equation for the total surface area, we get:

A_total = 2πr^2 + 2πr(180/(πr^2))

= 2πr^2 + 360/r

To find the values of "r" and "h" that minimize the surface area, we need to take the derivative of "A_total" with respect to "r", set it equal to zero, and solve for "r".

dA_total/dr = 4πr - 360/r^2 = 0

Solving for "r", we get:

r = (360/(4π))^(1/3) ≈ 4.35 cm

Substituting this value of "r" in the expression for "h", we get:

h = 180/(π(4.35)^2) ≈ 3.9 cm

Therefore, the height and radius that minimize the amount of material needed to manufacture the can are approximately 3.9 cm and 4.35 cm, respectively.

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The probability of getting strawberry is 11/60. The correct option is d.

To determine the probability of getting strawberry, we need to consider the probabilities of all the possible flavors and calculate the probability of strawberry using the information given.

Given probabilities:

Probability of getting cherry = 1/5

Probability of getting orange = 1/4

Probability of getting lemon = 1/3

Since there are only four flavors in total, we can calculate the probability of getting strawberry by subtracting the sum of the probabilities of cherry, orange, and lemon from 1.

Probability of getting strawberry = 1 - (1/5 + 1/4 + 1/3)

To simplify the calculation, we find a common denominator for 5, 4, and 3, which is 60.

Probability of getting strawberry = 1 - (12/60 + 15/60 + 20/60)

                                 = 1 - 47/60

                                 = 13/60

Therefore, the probability of getting strawberry is indeed 13/60, which corresponds to option d) 11/60 in the given list of options.

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Paul has 18 interest-free days for the $300 stereo purchase.

He will need to pay the full balance of his June billing statement.

If Paul's credit card closes on the 6th of the month and his payment is due on the 24th, then he has 18 days between the close of the billing cycle and the due date of his payment.

If Paul purchases a stereo for $300 on June 8th, then the transaction will be included in his billing cycle for the month of June. Since his billing cycle closes on the 6th, the $300 charge will appear on his June billing statement.

If Paul pays off his credit card in full each month, then he will need to pay the full balance of his June billing statement by the due date of June 24th to avoid finance charges. This means he will need to pay $300 for the stereo, plus any other charges that may have been included on his billing statement for the month of June.

Therefore, the number of interest-free days that Paul will have for the $300 stereo purchase is 18 days, which is the number of days between the billing cycle close date (June 6th) and the payment due date (June 24th).

To summarize:

Paul has 18 interest-free days for the $300 stereo purchase.

Paul will need to pay the full balance of his June billing statement, including the $300 stereo charge, by June 24th to avoid finance charges.

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The measure of the smallest angle is 37.1 degrees

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

The legs of a right triangle measure 11.4 meters and 15.1 meters

So, the measure of one of the acute angles is

tan(x) = 11.4/15.1

Evaluate

tan(x) = 0.7550

Take the arc tan of both sides

So, we have

x = 37.1

This means that the measure of the smallest angle is 37.1 degrees

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If Mr Louis gets a 6. 5% discount on the price of the oil, Mr. Louis gets a discount of £65.41 on this purchase.

First, let's find out how many litres of oil Mr. Louis needs to fill his tank:

Maximum capacity of the tank: 1,400 litres
Current oil in the tank: 150 litres
Litres needed to top up: 1,400 - 150 = 1,250 litres

Next, we'll calculate the original price of the oil without the discount:

Price per litre: £0.805
Cost of oil needed: 1,250 litres * £0.805 = £1,006.25

Now, let's calculate the discount amount:

Discount percentage: 6.5%
Discount amount: £1,006.25 * 0.065 = £65.41

So, Mr. Louis gets a discount of £65.41 on this purchase.

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Answer:

Each car cost $0.81, you would need to do 0.81 times 25 and you would get $20.25

Step-by-step explanation:

Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A). To determine the convergence of the series, we can use the Ratio Test. The series in question is:

Σ (3n² + 3) / (3n³ + 1)
First, we find the ratio between consecutive terms:
R = (a_(n+1)) / a_n = ((3(n+1)² + 3) / (3(n+1)³ + 1)) / ((3n² + 3) / (3n³ + 1))
R = (3(n+1)² + 3)(3n³ + 1) / ((3n² + 3)(3(n+1)³ + 1))
Now, as n goes to infinity:
lim (n -> ∞) R = lim (n -> ∞) (9n² + 6n + 3)(3n³ + 1) / (9n² + 3)(9n³ + 3n² + 3n + 1)
In this case, the highest power of n in the numerator and denominator is n⁵. To simplify, we can divide both the numerator and denominator by n⁵:
lim (n -> ∞) (9 + 6/n + 3/n²)(3 + 1/n²) / (9 + 3/n)(9 + 3/n² + 3/n + 1/n³)
As n approaches infinity, the terms with n in the denominator approaches zero:
lim (n -> ∞) (9)(3) / (9)(9) = 27 / 81 = 1/3
Since the limit of the ratio is less than 1 (1/3 < 1), the series is absolutely convergent (Option A).

To determine whether the series 3n² +  3n³ +1 is absolutely convergent, conditionally convergent, or divergent, we can use the ratio test.
Using the ratio test, we have:
lim n→∞ |(3(n+1)² + 3)/(3n² + 3n³ +1)
= lim n→∞ |(3n² + 6n + 3)/(3n³ + 3n² +1)
= lim n→∞ |(n² + 2n + 1)/(n³ + n²)
= 0
Since the limit is less than 1, the series is absolutely convergent. Therefore, the answer is A. Absolutely convergent.

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The next of the sequence 1, 2, 6, 22 is equal to 86.

First term of the sequence is equal to 1

Second term of the sequence is 2

Which can be written as

1 + 2⁰ = 2

Third term is 6

which can be written as

2 + 2² = 2 + 4

          = 6

Fourth term is 22

which can be written as

6 + 2⁴ = 6 + 16

          = 22

Next term using the above pattern is equal to

Pattern is add the previous term with increment of the even square of 2.

22 + 2⁶ = 22 + 64

            = 86

Therefore, the next term of the given sequence is equal to 86.

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The given question is incomplete, I answer the question in general according to my knowledge:

What comes next in the sequence: 1, 2, 6, 22, ____ ?

The probability of picking a number greater than 1 or a divisor of 2 is 3/4.

The numbers greater than 1 or divisor of 2 in the set {1, 2, 3, 4} are {2, 3, 4}.

Therefore, the probability of picking a number greater than 1 or a divisor of 2 is:

P(greater than 1 or divisor of 2) = P(2 or 3 or 4) = P(2) + P(3) + P(4)

Since there are four equally likely outcomes, the probability of each individual outcome is 1/4. Thus, we have:

P(greater than 1 or divisor of 2) = 1/4 + 1/4 + 1/4 = 3/4

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The ratios that have a unit rate of 3 include the following:

A. 15/2 cups: 2 1/2 cups

C. 2 cups: 2/3 cups

F. 2 1/2 cups: 5/6 cups

In Mathematics, the unit rate is sometimes referred to as unit ratio and it can be defined as the quantity of material that is equivalent to a single unit of product or quantity.

15/2 cups : 5/2 cups

15/2 ÷ 5/2 : 5/2 ÷ 5/2

15/2 × 2/5  : 1

3 : 1  (True)

1 cup: 1/4 cups

1 × 4 : 1/4 × 1

4 : 1   (False)

2/3 cups: 1 cup

2/3 × 3/2  : 1 × 3/2

1 : 3/2   (False)

3 3/4 cups: 2 cups

(4 × 3 + 3)/4 : 2

15/4 : 2

15/8  : 2/2

15/8 : 1    (False).

2 cups: 2/3 cups

2 × 3/2 : 2/3 × 3/2

3  : 1   (True).

2 1/2 cups: 5/6 cups

(2 × 2 + 1)/2 : 5/6

5/2  : 5/6

5/2 × 6/5 : 5/6 × 6/5

3 : 1  (True).

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Complete Question:

Which ratios have a unit rate of 3? Choose all that apply.

15/2 cups: 2 1/2 cups

1 cup: 1/4 cups

2/3 cups: 1 cup

3 3/4 cups: 2 cups

2 cups: 2/3 cups

2 1/2 cups: 5/6 cups

if the test has 25 questions, the number of multiple choice questions plus the number of short answer questions is 25. The correct answer is c.

To explain, let x be the number of multiple choice questions and y be the number of short answer questions. We know that there are 25 questions in total, so x + y = 25.

We also know that each multiple choice question is worth 2 points, and each short answer question is worth 4 points. If we let M be the total number of points from the multiple choice questions, and S be the total number of points from the short answer questions, we can set up the equation:

M + S = 80

We can also express M and S in terms of x and y:

M = 2x
S = 4y

Substituting these equations into the first equation, we get:

2x + 4y = 80

Dividing both sides by 2:

x + 2y = 40

Now we have two equations with two variables:

x + y = 25
x + 2y = 40

Subtracting the first equation from the second, we get:

y = 15

Substituting this into the first equation, we get:

x + 15 = 25

x = 10

Therefore, there are 10 multiple choice questions on the test.

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For the function f(x, y) = 1 - xy + y²= -y = -1, fy(-2,1) = 2y - x = 3.

When y is stationary, the rate of change of f with respect to x is determined by the partial derivative fx. Meanwhile, fy measures the rate of change of f in relation to y while holding x constant.

At the point (-2,1), the fx partial derivative symbolizes the incline of the level curve tangent line that follows the x-direction in relation to f. Its counterpart fy denotes the incline of the level curve tangent line that follows the y-direction regarding f at (-2,1).

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Complete answer - Let f(x, y) = 1 - xy + y² Compute fx(-2, 1) and fy(-2,1), and interpret these partial derivatives geometrically."

Answer: B, C and E

Step-by-step explanation: the term growing means the value is increasing, and a number multiplied by 1 gives the same number; therefore in order for that number to increase, it need to be multiplied by a bigger number which is greater than 1, in this case it can not be a number less than 0 like 0.85, so all the number which are greater than 1 in the bracket are correct.

the number before the bracket is the population before increasing, so you can substitute any value for t and try out different outcomes, you will get only B, C and E as number greater than the original number

The algebraic expression is equivalent to the expression is 125/4x + 34. Option B

Algebraic expressions are simply defined as expressions that are composed of terms, variables, constants, coefficients and factors.

These algebraic expressions are also composed of mathematical operations, such as;

From the information given, we have;

25/4(5x-4/5)+29

expand the bracket, we get;

125x - 100/5 + 29

Find the LCM, we get;

125x - 100 + 145/5

Divide the values

125/4x + 34

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You can use one of the following properties to prove that a triangle is isosceles without knowing the measure of each side.

Angle-Angle (AA): A triangle is equal if two angles of one triangle meet (equal in measure) two angles of another triangle. For an isosceles triangle, if you can prove that two angles are equal, then the triangle must be an isosceles triangle. This is because two right angles are the defining feature of an isosceles angle, as opposed to two congruent sides.

Base Angles: If the base angles of a triangle are equal (equal in measure), then the triangle is isosceles. This is because in an isosceles triangle, the base angles are always congruent.

Vertex angle bisection: If the vertex angle bisector of a triangle is also a perpendicular bisector of the opposite side (base), then the triangle is isosceles. This is because in an isosceles triangle, the two sides of the vertex angle always bisect the base and are perpendicular.

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The equation of the straight line connecting these points in the form of y = mx + c is given as y = (1/2)x + 1.

The line in the slope-intercept equation is given as,

y = mx + c

where:

m = the slope of the line

c = the y-intercept

The slope m of the line connecting two points (x1, y1) and (x2, y2) is given by the formula

= (y₂ - y₁) / (x₂ - x₁)

Substituting the coordinates values of P and Q into this formula, we get

[tex]m = (3 - 0) / (4 - (-2))[/tex]

= 3/6

= 1/2

Therefore, the value of the m is 1/2

We can find the value of c by substituting the m and x values in the equation. using the point P and  Substituting x and y values in the equation we get

x = - 2

y = 0  

[tex]0 = (1/2) × (-2) + c[/tex]

c = 1

Therefore, the value of c is 1.

By substuting the m and c values in the standard slope-intercept formula we get y = (1/2)x + 1.

Therefore, the equation of the line connecting points P and Q is y = (1/2)x + 1.

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The critical value z a/2 that corresponds to a 97% level of confidence is 1.96, and the critical value z a/2 that corresponds to an 80% level of confidence is 1.28.

In statistical hypothesis testing, the critical value is the point beyond which we reject the null hypothesis. To find the critical values, we need to use the standard normal distribution table or calculator.

For a 97% level of confidence, the significance level (α) is 0.03, and the critical value can be found by dividing α by 2 to get 0.015 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.96.

Similarly, for an 80% level of confidence, the significance level (α) is 0.20, and the critical value can be found by dividing α by 2 to get 0.10 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.28.

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Answer:

Step-by-step explanation:

To use a number line to find the number that is 7 more than -9, we start by finding -9 on the number line. Then, we move 7 units to the right (because we want to find a number that is 7 more), staying on the number line.

Starting from -9 on a number line, we can count 7 units to the right, which brings us to:

-9 + 7 = -2

Therefore, the number that is 7 more than -9 is -2.

Using Rule of thumb,

1) A common rule of thumb is to spend no more than 30% of your income on rent. Therefore, the maximum amount you should spend on rent is:

$2,800 x 0.3 = $840

2) Using the same rule of thumb, the maximum amount you should spend on rent is:

$1,900 x 0.3 = $570

3) To afford an $820/month apartment, you should aim to spend no more than 30% of your income on rent:

$820 = 0.3x

x = $2,733.33

So your monthly income must be at least $2,733.33 to afford the apartment.

4) Following the same reasoning, your monthly income must be at least:

$900 = 0.3x

x = $3,000

5) To start renting, you need to pay first and last month's rent, plus a $650 security deposit, for a total of:

$665 x 2 + $650 = $1,980

So you would need $1,980 to start renting the apartment.

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The probability that Sam will get a parking ticket given that he has to park in a no-parking zone on Monday is approximately 0.1308 or 13.08%.

We can use Bayes' theorem to solve this problem. Let A be the event that Sam gets a parking ticket and B be the event that Sam parks in a no-parking zone. Then, we want to find P(A|B), which is the conditional probability of A given B.

Bayes' theorem states that P(A|B) = P(B|A)*P(A)/P(B), where P(B|A) is the probability of B given A, P(A) is the prior probability of A, and P(B) is the prior probability of B.

From the problem, we know that P(A) = 0.07, P(B|A) = 1 (since if Sam parks in a no-parking zone, he will definitely get a parking ticket), and P(B|not A) = 0.50 (since if Sam finds a legal parking space, he has a 0.50 probability of parking in a no-parking zone).

To find P(B), we can use the law of total probability, which states that P(B) = P(B|A)*P(A) + P(B|not A)*P(not A), where P(not A) = 1 - P(A).

Therefore, P(B) = 10.07 + 0.50(1-0.07) = 0.5351.

Finally, we can use Bayes' theorem to find P(A|B):

P(A|B) = P(B|A)P(A)/P(B) = 10.07/0.5351 ≈ 0.1308.

Therefore, the probability that Sam will get a parking ticket given that he has to park in a no-parking zone on Monday is approximately 0.1308 or 13.08%.

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Answer:

18x^5 - 7x^2y - 2xy^2 + 17y^4

Step-by-step explanation:

Standard form means that the variables are in alphabetical order, and the degree of the variable closer to the start of the alphabet decreases while the degree of the variable closer to the end of the alphabet increases.

The chances of success in cycling across the United States is higher on the central route because more than half of the people who started on this route finished it (52%), while on the other routes the chances of successful completion is lower (route north (0%) south (0.02%).

According to the above, to find the probability of success in crossing the United States from coast to coast, we must divide the number of successful attempts by the total number of attempts as shown below:

North Route

0 ÷ 2 = 0

Center Route

10 ÷ 19 = 0.52

South Route

1 ÷ 39 = 0.025

According to the above, the highest probability of success is found in the central route with 0.52, that is to say that more than half of those who tried this route finished.

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