8. Two friends, Suban and Jas, start from the same point but ride their bikes down two different paths. The paths diverge at an angle of 38º. Suban rides down his path at 12 km/h, and Jas rides down his path at 14 km/h. Determine how far apart the two are after riding at their respective speeds for 90 min. (App/4) Problem: Show all steps of your work for full credit. 9. Gursant and Leo are both standing on the north side of a monument that is 6.0 m tall. Leo is standing 3.5 m closer to the monument than Gursant. Leo measures the angle from the ground to the top of the monument to be 41º. Determine the angle that Gursant would measure from the ground to the top of the monument, to the nearest degree. (App/4)

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 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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Yes, I can help you with these Laplace transform problems.

1. To find the Laplace transform of f(t)=e-2t sin (5t), we can use the formula:

L{e-at sin(bt)} = b / (s+a)2 + b2

Applying this formula, we get:

L{e-2t sin (5t)} = 5 / (s+2)2 + 52

Therefore, the Laplace transform of f(t)=e-2t sin (5t) is:

L{f(t)} = 5 / (s+2)2 + 25

2. To find the Laplace transform of f(t)=tsin (3t), we can use integration by parts, followed by applying the Laplace transform:

L{f(t)} = L{t} L{sin (3t)} - L{dt/ds} L{sin (3t)}

Using the Laplace transform of t and sin(3t), we get:

L{t} = 1 / s2

L{sin(3t)} = 3 / (s2 + 32)

Differentiating sin(3t) with respect to t gives:

d/dt sin(3t) = 3 cos(3t)

Taking the Laplace transform of both sides gives:

L{d/dt sin(3t)} = s L{cos(3t)} - cos(0)

Since L{cos(3t)} = s / (s2 + 32), we can simplify to:

L{d/dt sin(3t)} = 3s / (s2 + 32)

Therefore, the Laplace transform of f(t)=tsin (3t) is:

L{f(t)} = (2s3) / (s2 + 32)2

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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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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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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:

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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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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Since the graph was obtained by transforming the graph of the square root function, an equation for the function the graph represent is: [tex]g(x) = -\sqrt{9(x-1)} +2[/tex]

In Mathematics and Geometry, a square root function is a type of function that typically has this form f(x) = √x, which basically represent the parent square root function i.e f(x) = √x.

In Mathematics and Geometry, a horizontal translation to the right is modeled by this mathematical equation g(x) = f(x - N) while a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) + N.

Where:

Therefore, the required square root function can be obtained by applying a set of transformations to the parent square root function as follows;

f(x) = √x

g(x) = -√9(x - 2) + 2

[tex]g(x) = -\sqrt{9(x-1)} +2[/tex]

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The probability that the answer is correct given that the

a) Let's use Bayes' theorem to calculate the probability that the answer is correct given that the person you asked said "East". Let H be the event that the person is honest, L be the event that the person is a liar, E be the event that Julia resides in the East and W be the event that Julia resides in the West. Then we have:

P(E|H) = 2/3 (the probability that an honest person gives the correct answer)

P(E|L) = 1/2 (the probability that a liar gives the correct answer)

P(H) = 1/4 (the probability that the person is honest)

P(L) = 3/4 (the probability that the person is a liar)

By the law of total probability, we have:

P(E) = P(E|H)P(H) + P(E|L)P(L) = (2/3)(1/4) + (1/2)(3/4) = 5/12

Then, using Bayes' theorem, we have:

P(H|E) = P(E|H)P(H)/P(E) = (2/3)(1/4)/(5/12) = 2/5

So the probability that the answer is correct given that the person said "East" is 2/5.

b) The probability that the same person gives the same answer twice in a row is:

P(E∩E) = P(E)P(E|H)P(H) + P(E)P(E|L)P(L) = (5/12)(2/3)(1/4) + (5/12)(1/2)(3/4) = 5/24

Using Bayes' theorem again, we have:

P(H|EE) = P(EE|H)P(H)/P(EE) = (2/3)^2(1/4)/(5/24) = 8/15

So the probability that the answer is correct given that the person said "East" twice in a row is 8/15.

c) The probability that the same person gives the same answer three times in a row is:

P(E∩E∩E) = P(E)P(E|H)^2P(H) + P(E)P(E|L)^2P(L) = (5/12)(2/3)^2(1/4) + (5/12)(1/2)^2(3/4) = 5/32

Using Bayes' theorem again, we have:

P(H|EEE) = P(EEE|H)P(H)/P(EEE) = (2/3)^3(1/4)/(5/32) = 4/5

So the probability that the answer is correct given that the person said "East" three times in a row is 4/5.

d) The probability that the same person gives the same answer four times in a row is:

P(E∩E∩E∩E) = P(E)P(E|H)^3P(H) + P(E)P(E|L)^3P(L) = (5/12)(2/3)^3(1/4) + (5/12)(1/2)^3(3/4) = 5/48

Using Bayes' theorem again, we have:

P(H|EEEE) = P(EEEE|H)P(H)/P(EEEE) = (2/3)^4(1/4)/(5/48) = 16/25

So the probability that the answer is correct given that the

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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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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 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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Answer:

The answer is 10 and -10

You get this answer because in the middle in a number line is 0 and if it 10 units from 0 then the other side of 0 will be -10 (negative ten) units from 0

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 difference between the two possibilities is based on theory and mathematics. The experimental probability is based  on the results of several tests or experiments, but the theortical result is  calculated by comparing the positive results with all the results.

Theoretical probability of an event  occurring based on theory and reasoning. It is determined by dividing  number of favourable results by total  result. On the other hand, the experimental depend on the results of  various trials or tests.

The difference between theoretical probability and testing probability is that theory is based on knowledge and mathematics. Theoretical probability is what it should be. The test will appear as a result. For example, if I flip a coin, 50 times, the theoretical number of heads of the coin is 25. Coin flip probability = 0.5

Number of flips = 50

Theoretical number of heads = 0.5 × 50

= 25

If I actually flip a coin 50 times, 25 heads may or may not come up. If we have 21 heads, the test probability is 21 out of 50 heads, or 0.42. So the theoretical probability of getting heads in this example = 0.5

The experimental probability of landing heads = 0.42. Hence, both probabilities are not the same.

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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 value of X in the diagram provided is

We can use the trigonometric function of sine to find the angle θ, where θ is the angle between the opposite side and the hypotenuse.

sin(θ) = opposite / hypotenuse

sin(θ) = 12 / 13

To find θ, we can take the inverse sine of both sides:

θ = sin⁻¹(12/13)

θ = sin⁻¹(0.9231)

θ = 67.38°

Note that we use calculator to find the θ

Therefore, the angle in the right-angled triangle with opposite side 12, adjacent side 5, and hypotenuse 13 is approximately 67.38 degrees.

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Answer:

[tex]3^{21}[/tex]

Step-by-step explanation:

using the rules of exponents

[tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]

[tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]

given

(3² × [tex]3^{5}[/tex] )³

= ([tex]3^{(2+5)}[/tex] )³

= ([tex]3^{7}[/tex] )³

= [tex]3^{7(3)}[/tex]

= [tex]3^{21}[/tex]

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 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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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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Answer:

the answer is indeed $140.40

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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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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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)

For the polygons in the attached figure,

a) Scale factor = 2.5

b) Scale factor = 2.5

c) Scale factor = 1.5

d) Scale factor = 2

We knoa that a scale factor is nothing but the ratio between the scale of a original object and a transformed object.

Here, the polygons are similar.

We know that the corresponding sides of similar figure are in proportion.

a) 10/6 = 2.5

15/6 = 2.5

18/7.2 = 2.5

So, the scale factor would be 2.5

b)

25/10 = 2.5

15/6 = 2.5

S0, the scale factor = 2.5

c)

12/8 = 1.5

6/4 = 1.5

9/6 = 1.5

So, the scale factor = 1.5

d)

12/6 = 2

16/8 = 2

20/10 = 2

so, the scale factor is 2

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Answer:

  -0.1, 1.3

Step-by-step explanation:

You want the solutions to the quadratic equation 5x² -2x -1 = 4x.

The equation can be put in standard form by subtracting 4x:

  5x² -6x -1 = 0

  5(x² -6/5x +(6/10)²) -1 -5(6/10)² = 0 . . . . . complete the square

  5(x -0.6)² = -2.8 . . . . . . . . . . . . . subtract 2.8

  x = 0.6 ± √0.56 = -0.1 or 1.3 . . . . . . . divide by 5 and take square root

Solutions to the equation are x = -0.1 and x = 1.3.

__

Additional comment

The square is completed by making the trinomial in parentheses have the form x² -2ax +a², where 'a' is half the coefficient of the x-term. When we add a² inside parentheses, we need to subtract an equivalent quantity outside parentheses.

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The solution of the inequality is q < -8.

We have,

38 < 30 - q

Now, solving the inequality

Subtract 30 from both of inequality as

38 - 30 < 30 - q - 30

8 < -q

Now, to make the variable q is positive then the sign of inequality change.

-8 > q

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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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