Exercises : Find a solution for the following an (1 а. a = 1 an = n 2 anni +1 (2) a = 1, 9, = 2, 11 Van an-z 4 a n- n 2 2 (3) Hard Problem *te a = 6, 0,= 17, a +5na, +6nen-ida n-1 M-2

Find the gradient of a line perpendicular to the longest side of the triangle formed by A(-3,4),B(5,2) and C(0,-3)

The equivalent distance travelled by Mia is 2.3333... and 2[tex]\frac{1}{3}[/tex] miles.

The distance run by Mia is equivalent to 7/3 miles. We can express the fraction as -

7/3 = 2.3333

7/3 = (3 x 2 + 1)/3 = 2[tex]\frac{1}{3}[/tex]

So, the equivalent distance travelled by Mia is 2.3333... and 2[tex]\frac{1}{3}[/tex] miles.

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Answer: 2.33 and 2 1/3

Step-by-step explanation:

Answer:

Let's assume that the company invites x people to the dinner party.

The cost of the function room is a fixed cost of $280.

The cost of dinner is $40 per person. Therefore, the total cost of dinner for x people is 40x.

The total cost of the dinner party is the sum of the cost of the function room and the cost of dinner:

Total cost = 280 + 40x

The problem states that the company has a budget of no more than $1500. Therefore, we can write:

280 + 40x ≤ 1500

Subtracting 280 from both sides gives:

40x ≤ 1220

Dividing both sides by 40 gives:

x ≤ 30.5

Since we cannot invite a fraction of a person, the company can invite at most 30 people to the dinner party.

Therefore, the greatest number of people they can invite to the dinner is 30.

Step-by-step explanation:

The calculated equation of the line is y = -2x

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

The linear graph

The points on the graph are

(-4,8) (0,0)

It passes through the origin, the slope is calculated as

slope = y/x

This gives

y/x = 8/-4

Evaluate

y/x = -2

This gives

y = -2x

Hence, the equation is y = -2x

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The midpoint of the second class ([40-49]) is calculated by adding the lower and upper limits of the class and dividing by 2. So, (40+49)/2 = 44.5.

The relative frequency of the second class is calculated by dividing the frequency of the second class by the total frequency. So, 8/38 = 0.211.

The class width is the difference between the upper and lower limits of the class. So, [59.5-69.5] has a class width of 10.

The fourth actual class is [60-69], which has a midpoint of (60+69)/2 = 64.5.

Therefore, the answer is a. 44.5, 0.211, 10 and [59.5-69.5].

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Myra should choose the container with a volume of 343 in.³

This container is the closest to her requirement of 288 in.³ and will have less leftover sand compared to the other options.

We have,

Myra needs a container with a volume of at least 288 in.³ of sand.

Out of the three options, the only one that meets this requirement is the container with a volume of 336 in.³

This container has more than enough sand for Myra to build her sandcastle.

However, Myra will be deducted points for having too much sand left over in her container.

So, she should choose the smallest container that meets her requirement of 288 in.³.

The container with a volume of 336 in.³ is too large and will likely result in too much leftover sand.

Therefore,

Myra should choose the container with a volume of 343 in.³

This container is the closest to her requirement of 288 in.³ and will have less leftover sand compared to the other options.

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The mean of the sampling distribution of the sample means is equal to the population mean, which is 5.5 inches. The standard deviation of the sampling distribution of the sample means is equal to the population standard deviation divided by the square root of the sample size.

So, for a sample size of 11, the standard deviation of the sampling distribution is:

standard deviation = 1.4 / sqrt(11) = 0.42 inches

To find the probability that the mean length of the 11 items is less than 13.4 inches, we need to standardize this value using the formula:

z = (x - mu) / (sigma / sqrt(n))

where:

x = 13.4 (the mean length we're interested in)

mu = 5.5 (the population mean)

sigma = 1.4 (the population standard deviation)

n = 11 (the sample size)

Substituting the values, we get:

z = (13.4 - 5.5) / (1.4 / sqrt(11)) = 14.31

Using a standard normal distribution table, we can find that the probability of getting a z-score of 14.31 or more is practically zero.

Therefore, the probability that the mean length of the 11 items is less than 13.4 inches is practically 1 or 100%.

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The graph is attached.

Given is a triangle ABC, we need to find its coordinates if it is reflected over y = -x,

The rule of reflection over y = -x is,

(x, y) = (-x, -y)

So,

A = (-5, -4)

B = (1, -4)

C = (-1, -5)

After reflection,

A' = (5, 4)

B' = (-1, 4)

C' = (1, 5)

Hence the points after reflection are A' = (5, 4), B' = (-1, 4) and C' = (1, 5)

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Answer:

3.6 kilograms

Step-by-step explanation:

divide the mass value by 2.205

The measure of the side c is 41  meters

It is important to note that the different trigonometric identities are listed thus;

From the information given, we have the sides;

angle, θ = 28 degrees

The opposite angles = 19m

Hypotenuse side = c

Using the sine identity, we have;

Substitute the values

sin 28 = 19/c

cross multiply, we get;

c = 19/sin 28

find the value and substitute

c = 19/0.4695

divide the values

c = 41 meters

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To verify if the set {-696, -36, -19, 3, 7, 12, 99} is a complete system of residues modulo 7, we need to check if all residue classes modulo 7 are represented by elements in the set.

The residue classes modulo 7 are {0, 1, 2, 3, 4, 5, 6}. To check if the given set is a complete system of residues modulo 7, we need to check if each residue class is represented by at least one element in the set.
- 0: None of the elements in the set is divisible by 7, so none of them leave a residue of 0 when divided by 7.
- 1: 12 and -696 leave a residue of 1 when divided by 7.
- 2: -36 leaves a residue of 2 when divided by 7.
- 3: 99 and -19 leave a residue of 3 when divided by 7.
- 4: None of the elements in the set leave a residue of 4 when divided by 7.
- 5: 7 leaves a residue of 5 when divided by 7.
- 6: 3 leaves a residue of 6 when divided by 7.
Since every residue class modulo 7 is represented by at least one element in the set, we can conclude that the set {-696, -36, -19, 3, 7, 12, 99} is a complete system of residues modulo 7.

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The probability that the distance between the two randomly selected points on a line of length 18, which are on opposite sides of the midpoint, is greater than 7 is 1/3.

Let the midpoint of the line be M. Since x is uniformly distributed over [0,9) and y is uniformly distributed over (9,18], the probability of selecting any point in [0,9) is 1/2 and the probability of selecting any point in (9,18] is also 1/2.

Let A be the event that the distance between x and M is less than or equal to 7, and let B be the event that the distance between y and M is less than or equal to 7.

Therefore, the probability of A is the ratio of the length of [0,9) and the length of the entire line, which is 9/18 or 1/2. Similarly, the probability of B is also 1/2.

Now, the probability that the distance between the two points is greater than 7 is the complement of the probability that either A or B occurs, which is 1 - P(A or B).

Using the formula for the probability of the union of two events, we have P(A or B) = P(A) + P(B) - P(A and B).

Since A and B are independent events, P(A and B) = P(A) * P(B) = 1/4.

Therefore, P(A or B) = 1/2 + 1/2 - 1/4 = 3/4.

Finally, the probability that the distance between the two points is greater than 7 is 1 - 3/4 = 1/4 or 0.25, which is equivalent to 1/3 when expressed as a fraction in the simplified form.

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The best estimate of the remaining life of Lourdes Corporation's bonds is 14.82 years.

Lourdes Corporation would have to set a coupon rate of 10.07% to issue new bonds at par.

To find the remaining life of the bond, we need to use the formula:

Remaining Life = (ln(Par Value/Market Price)) / (2 * ln(1 + Coupon Rate/2))

Substituting the given values, we get:

Remaining Life = (ln(1000/1327.36)) / (2 * ln(1 + 0.13/2)) = 14.82 years

B. To find the required coupon rate for new bonds, we can use the formula:

Coupon Rate = (ln(Par Value/Call Price) + Yield Rate) / (2 * ln(1 + Call Price/Market Price))

Substituting the given values, we get:

Yield Rate = 0.13 (given)

Coupon Rate = (ln(1000/1075) + 0.13) / (2 * ln(1 + 1075/1327.36)) = 0.1007 or 10.07%.

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Answer:

4.2

Step-by-step explanation:

6.3 - 2.1 = 4.2

A. y = 4x-7, the inverse function is, x = (y + 7)/4

B. y = 7x, the inverse function is, x = y/7

C. y = 7x-4, the inverse function is, x = (y + 4)/7

D. y = x-4, the inverse function is, x = y + 4

E. y = x+4, the inverse function is, x = y - 4

F. y = x-4/7, the inverse function is, x = 7y + 4

The inverse of each of the functions is calculated as follows;

y = 4x - 7

x = 4y - 7

4y = x + 7

y = (x + 7)/4  

⇒ x = (y + 7)/4

Second function;

y = 7x

x = 7y

y = x/7

⇒ x = y/7

Third function;

y = 7x - 4

x = 7y - 4

7y = x + 4

y = (x + 4)/7

⇒ x = (y + 4)/7

Fourth function;

y = x - 4

x = y - 4

y = x + 4

⇒ x = y + 4

Fifth function;

y = x + 4

x = y + 4

y = x - 4

⇒ x = y - 4

Sixth function;

y = (x - 4)/7

x = (y - 4)/7

7x = y - 4

y = 7x + 4

⇒ x = 7y + 4

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Answer:

Exact answer (x = -127/3) or Rounded answer (x = -42.3)

Step-by-step explanation:

First, we will need to find the measure of the third angle in the triangle, which we can call angle y:

The sum of all the angles in a triangle is always 180, so we can find the measure of angle y by subtracting the sum of the two angles we know from 180:

[tex]y+96+31=180\\y+127=180\\y=53[/tex]

Angle y and the angle measuring -3x° are supplementary angles, which means the sum of these two angles is 180°.

We know that they're supplementary because of the straight line that separates them, because straight lines create straight angles which are 180°

Thus, we can find the value of x by making the sum of the -3x° angle and the 53° angle equal to 180° and solve for x:

[tex]-3x+53=180\\-3x=127\\x=-43.333333=-43.3\\x=-127/3[/tex]

-127/3 is the exact answer, while -43.3 is the rounded answer.  Feel free to use any of the two.

The middle value and is not affected by outliers, while the mean represents the average and can be influenced by outliers.

a. To find the median cost of electricity, we need to arrange the bills in order from lowest to highest:

36, 44,54, 65, 80

The median is the middle value, which in this case is 54.

b. To find the mean cost of electricity, we need to add up all the bills and divide by the total number of bills:

(54 + 36 + 80 + 65 + 44) / 5 = 55.80

So the mean cost of electricity is 55.80.

that the median and mean can give different perspectives on the data. The median represents the middle value and is not affected by outliers, while the mean represents the average and can be influenced by outliers.

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To determine the magnitude of the force F exerted at the lever of the assembly, we'll need to consider the moment at point O, the distances involved, and the angle at which the force is applied.

The moment at point O is given as 150 kN·m. Let's consider the lever arm distance, which is the horizontal distance between point O and the line of action of the force F. This can be found by looking at the given measurements: 4m (distance from O to P) + 3m (distance from P to the line of action of force F) = 7m.

Now, we'll take the angle of the force into account. The force F is applied at a 60° angle. To find the horizontal component of the force, we can use the cosine of the angle:
Horizontal component of F = F * cos(60°)

The moment at point O is the product of the horizontal component of force F and the lever arm distance (7m):
Moment = (F * cos(60°)) * 7m

Given that the moment at point O is 150 kN·m, we can now solve for the magnitude of the force F:
150 kN·m = (F * cos(60°)) * 7m

To solve for F:
F = (150 kN·m) / (cos(60°) * 7m)

Calculating the value, we find the magnitude of the force F to be approximately 43.3 kN.

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Using Lagrange multipliers, The maximum value of f(x, y, z) subject to the constraint x² + y² + z² = 16 is 17√2, which is approximately 24.04. Option C is the correct answer.

To solve this problem, we will use Lagrange multipliers. We want to maximize the function f(x, y, z) = 3x + 4y + 3z subject to the constraint g(x, y, z) = x² + y² + z² = 16. We can write the Lagrangian as:

L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 16)

= 3x + 4y + 3z - λ(x² + y² + z² - 16)

We need to find the values of x, y, z, and λ that satisfy the following equations:

∂L/∂x = 3 - 2λx = 0

∂L/∂y = 4 - 2λy = 0

∂L/∂z = 3 - 2λz = 0

∂L/∂λ = x² + y² + z² - 16 = 0

From the first three equations, we can solve for x, y, and z in terms of λ:

x = 3/2λ

y = 2/λ

z = 3/2λ

Substituting these values into the equation x² + y² + z² = 16, we get:

(3/2λ)² + (2/λ)² + (3/2λ)² = 16

Solving for λ, we get:

λ = ±2

Substituting these values into the equations for x, y, and z, we get the following critical points:

(2√2, √2, 2√2)

(-2√2, -√2, -2√2)

We need to evaluate the function f(x, y, z) = 3x + 4y + 3z at these critical points to determine which one gives the maximum value. We get:

f(2√2, √2, 2√2) = 3(2√2) + 4(√2) + 3(2√2) = 17√2

f(-2√2, -√2, -2√2) = 3(-2√2) + 4(-√2) + 3(-2√2) = -17√2

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The data values that complete the the table of values are 15, 16 and 17

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

The histogram

From the histogram, we can see that

Range of the dataset is 10 to 20

This means that we can complete the table of values with any data value within this range

Examples of the data values to use are 15, 16 and 17 (we can use other values too, and the values do not need to be sequential)

Hence, the data values that complete the the table of values are 15, 16 and 17

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Angle A is measured as 39°.

Inverse trigonometric functions have been what they sound like.

The opposite direction functions of trigonometry are somewhat the inverse functions of the basic trigonometric functions. The basic trigonometric function sin = x can be replaced with sin-1 x =. In this case, x is able to be expressed as a whole number, a decimal number, a fraction, as well as an exponent.

Now, we have AB (Hypotenuse)= 54 m BC (opposite side)= 38 m in triangle ABC.

To find the angle A's measurement

By employing inverse trigonometric keys.

We are aware of the following:

The sin inverse formula is as follows:

[tex]\theta = Sin^-^1(\frac{opposite side}{hypontenuse} )[/tex]

[tex]\theta= Sin^-^1(\frac{54}{38} )[/tex]

[tex]\theta= Sin^-^1(\frac{27}{19} )[/tex]  ≈1.570796326794897−0.888179846706129

θ = 39°

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The probability that Jim gets free beer is 0.7854.

The terms we need to consider are the square frame, the round board, and the probability p of Jim getting free beer.

To calculate the probability p, we need to find the ratio of the area of the round board to the area of the square frame.

Step 1: Calculate the area of the square frame.
Since the frame is 2x2, its area is A_square = side * side = 2 * 2 = 4 square units.

Step 2: Calculate the area of the round board.
The radius of the round board is 1, so its area is A_round = π * radius² = π * 1² = π square units.

Step 3: Calculate the probability p.
The probability p that Jim gets free beer is the ratio of the round board's area to the square frame's area, which is:
p = A_round / A_square = π / 4

So the probability that Jim gets free beer is π / 4, or approximately 0.7854.

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The solution to this system of equations is dependent.

Dependent matrix is a matrix where one or more rows can be expressed as a linear combination of the other rows. This means that the rows are not linearly independent, and there is redundancy in the information they provide.

A dependent matrix has less than full rank, which means that the rank of the matrix is less than the number of rows or columns. In a dependent matrix, one or more variables can be expressed in terms of the other variables, and the system of equations represented by the matrix has infinitely many solutions.

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The evaluation of the dot plots on the top indicates that the mean is 7.5

The table to create a different dot plot with the same mean as the dot plot on top is therefore;

Value  [tex]{}[/tex]          Frequency

4     [tex]{}[/tex]                2

6 [tex]{}[/tex]                    3

8 [tex]{}[/tex]                    3

10 [tex]{}[/tex]                  4

A dot plot is a data visualization method which consists of datapoints located above a number line, such that the number of dots at a datapoint represents the data value.

The mean of the dot plot can be found as follows;

Mean = (3 + 2 × 5 + 4 × 7 + 3 × 9 + 2 × 11)/(1 + 2 + 4 + 3  + 2) = 7.5

Therefore, the sum of the values = (3 + 2 × 5 + 4 × 7 + 3 × 9 + 2 × 11) = 90

The number of dots = (1 + 2 + 4 + 3  + 2) = 12

The required dot plot should therefore, have 12 dots

A possible combination of 12 dots that have a mean of 12 is therefore;

(2 × 4 + 3 × 6 + 3 × 8 + 4 × 10)/(2 + 3 + 3 + 4)

Therefore, one possible dot plot consists of 2 dots at 4, 3 dots at 6, 3 dots at 8, and 4 dots at 10 can be presented as follows;

[tex]{}[/tex]                                             o

[tex]{}[/tex]                         o        o        o

               o[tex]{}[/tex]       o        o        o

 [tex]{}[/tex][tex]{}[/tex]              o       o        o        o

-|----|--------|--------|--------|--------|--------|-

[tex]{}[/tex] 1    2        4        6        8       10      

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Therefore, the value of the given double integral is approximately 1634.449.

The double integral:

∫ from y=0 to y=9 [ ∫ from x=In y to x=9 of [tex](7y/x) e^y dx ][/tex] dy

Using integration by parts, we can evaluate the inner integral as:

∫ from x=In y to x=9 of [tex](7y/x) e^y dx = [7y/e^x][/tex] evaluated from x=In y to x=9

= [tex]7y(e^{-9} - e^{(-lny)}) = 7y(1/y - 1/e^9) = 7 - 7e^{(9-y)[/tex]

Substituting this back into the original double integral and evaluating the integral with respect to x, we get:

∫ from y=0 to y=9 [tex][ 7y - 7y e^{(9-y)} ] dy[/tex]

Using integration by parts again, we can evaluate this integral as:

[tex][ 7y^2/2 + 7y e^{(9-y)} - 49 e^{(9-y) ][/tex] evaluated from y=0 to y=9

= [tex]3309/2 - 343 e^{-9[/tex]

So, the value of the given double integral is approximately 1634.449.

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Answer:

the answer is indeed $140.40

a) The probability that X is within three standard deviations of the mean is approximately 1.

b) the probability that the drone will land out of the targeted area exactly 4 times is 0.00052.

c) The probability that the drone will land out of the targeted area at most 4 times is 0.1029

d) The expected value of X is 9.6.

e) The meaning of the expected value in the context of the story is average landing performance of the drone based on the given probability of success.

f) The variance of X is 0.7319.

g) The probability that it missed the target at most 2 times is 3.121.

h) The probability that it missed the target at least 2 times is 0.7319.

I) The probability that X is within three standard deviations of the mean is 1.3856.

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials with a constant probability of success.

The characteristics of a binomial random variable include the number of trials (n), the probability of success (p), the number of successes (x), and the mean and variance of the distribution.

Here we have

Binomial distribution We are testing the landing performance of a new automated drone. The drone lands on the targeted area 80% of the time. We test the drone 12 times.

a. X is a binomial random variable because we have a fixed number of independent trials and each landing has only two possible outcomes (landing on the targeted area or landing outside of it) with a constant probability of success (0.8).

The characteristics of the binomial distribution are:

The number of trials is fixed (n=12)

Each trial has only two possible outcomes (success or failure)

The probability of success (p) is constant for each trial

The trials are independent of each other

b. P(X = 4) = (12 choose 4) × (0.8)⁴ × (0.2)⁸ = 0.00052

c. P(X< = 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

= 0.0687 + 0.2060 + 0.3020 + 0.2670 + 0.1854 = 0.1029

d. E(X) = np = 120.8 = 9.6

e. The expected value of X represents the average number of successful landings (in the targeted area) we would expect to see in a sample of 12 landings.

In the context of the story, it tells us the average landing performance of the drone based on the given probability of success.

f. Var(X) = np(1-p) = 120.80.2 = 1.92

g. P(X<=2 | X<=4) = P(X<=2)/P(X<=4)

= (P(X=0) + P(X=1) + P(X=2))/(P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4))

= 0.3217/0.1029 = 3.121

h. P(X>=2 | X<=4) = 1 - P(X<2 | X<=4) = 1 - P(X<=1 | X<=4) = 1 - (P(X=0) + P(X=1))/(P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)) = 1 - 0.2747/0.1029 = 0.7319

i. The standard deviation of a binomial distribution is √(np(1-p)). So, the standard deviation of X is √(120.80.2) = 1.3856. Three standard deviations above and below the mean would be 3*1.3856 = 4.1568.

Therefore,

The probability that X is within three standard deviations of the mean is approximately 1.

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The probability of rolling a 2 on a single die is 1/6, and the probability of rolling a 4 is also 1/6. To find the probability of both events occurring in sequence, you multiply their individual probabilities: (1/6) * (1/6) = 1/36, which is approximately 2.78%, rounded to the nearest whole number is 3%.

The probability of rolling a 2 on the first throw is 1/6 (since there are six equally likely outcomes when rolling a die). The probability of rolling a 4 on the second throw is also 1/6. To find the probability of rolling both a 2 and a 4, we multiply these probabilities: (1/6) x (1/6) = 1/36.

To convert this to a percentage and round to the nearest whole number, we multiply by 100 and round: 1/36 x 100 = 2.78, which rounds to 3%.

Therefore, the answer is C. 3%.

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In the rhombus, m<D is 16^o and m<C is  164^o.

A rhombus is a quadrilateral which has equal length of sides, but stands on one of its edges. One of its major properties is that the measure of opposite internal angles are congruent.

The sum of the internal angles of a rhombus gives 360^o.

So that in the given diagram, we can deduce that;

y + y + (4y + 100) + (4y + 100) = 360^o

2y + 8y + 200 = 360

10y = 360 - 200

     = 160

y = 160/ 10

  = 16

y = 16^o

So that;

(4y + 100) = 4*16 + 100

                = 64 + 100

                = 164^o

Therefore, m<D is 16^o and m<C is  164^o.

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The average customer spends 0.2 hours, or 12 minutes, in the bank during peak hours.

Little's Law states that the average number of customers in a stable system (i.e., one where the number of arrivals and departures is balanced) is equal to the average arrival rate multiplied by the average time that a customer spends in the system:

L = λW

where L is the average number of customers in the system, λ is the average arrival rate, and W is the average time that a customer spends in the system.

In this case, we are given that the average arrival rate during peak hours is λ = 40 customers per hour, and the average number of customers in the bank is L = 8 customers. We are asked to find the average time that a customer spends in the bank and the probability is unknown.

Plugging in the values, we get:

8 = 40W

Solving for W, we get:

W = 8/40

W = 0.2 hours

Therefore, the average customer spends 0.2 hours, or 12 minutes, in the bank during peak hours.

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