A submarine rises to the ocean's surface at 720 feet in 6 minutes. Find the points submarine's change in elevation per minute.

Mean length of steel rods is 23 cm and the standard deviation is 0.08.

Find out what proportion of rods has a length less than 22.9 cm.

z-score as shown below:z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.z = (22.9 - 23) / 0.08 = -1.25

Proportion = P(Z < -1.25) = 0.1056Therefore, the proportion of rods that have a length less than 22.9 cm is 0.1056.

Now, we have to find the proportion of rods that will be discarded

z-score for 22.82 cm is given by z = (22.82 - 23) / 0.08 = -2.25And, z-score for 23.18 cm is given by z = (23.18 - 23) / 0.08 = 2.25

To find the proportion of rods that have a length shorter than 22.82 cm.Proportion for Z < -2.25 is 0.0122And, the proportion of rods that have a length longer than 23.18 cm is P(Z > 2.25) = 0.0122

Thus, the proportion of rods that will be discarded is 0.0122 + 0.0122 = 0.0244.c)

We have found that the proportion of rods that will be discarded is 0.0244. The number of rods to be discarded in a day is given by:Discarded rods = 0.0244 × 5000= 122

Therefore, the plant manager should expect to discard 122 rods in a day.

We have been given that all rods must be between 22.9 cm and 23.1 cm and we have to find how many rods should the plant manager expect to manufacture if an order comes in for 10,000 steel rods.

To solve this, we need to find the proportion of rods that have a length between 22.9 cm and 23.1 cm.z-score for 22.9 cm is given by z = (22.9 - 23) / 0.08 = -1.25And, z-score for 23.1 cm is given by z = (23.1 - 23) / 0.08 = 1.25

Proportion = P(-1.25 < Z < 1.25) = 0.7887Therefore, the proportion of rods that will be manufactured with a length between 22.9 cm and 23.1 cm is 0.7887.So, the plant manager should expect to manufacture 0.7887 × 10,000 = 7887 rods.

Rounding up to the nearest integer gives us 7887 as the answer.

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a) 0 kg of sugar is in the tank at the beginning., b) ds/dt = 0.08-(S/1120)*4, c) 0.32 kg of sugar is in the tank after 1 minute., d) 2.59 kg of sugar is in the tank after 84 minutes.

a) At the beginning, there is no sugar in the tank since it contains only pure water.

b) Let S(t) be the amount of sugar in the tank at time t (in minutes). The rate of change of S is given by the rate at which sugar enters the tank minus the rate at which it leaves. Since the solution enters at a rate of 4 L/min, and contains 0.08 kg of sugar per liter, the rate at which sugar enters is 0.08 kg/L * 4 L/min = 0.32 kg/min. Since the solution leaves at the same rate, the differential equation that models the situation is:

dS/dt = 0.32 - (S/1120) * 4

c) To find the amount of sugar after 1 minute, we need to solve the differential equation from part (b) with the initial condition S(0) = 0 (since the tank initially contains only water). One possible way to solve the differential equation is to use separation of variables:

dS/dt + (4/1120) S = 0.32

Multiplying both sides by dt and dividing by (0.32 - (4/1120) S), we get:

(1/S) dS = (4/0.32 - (1120/4)) dt

Integrating both sides from t = 0 to t = 1 and from S = 0 to S = S(1), we get:

ln(S(1)) - ln(0) = (1.25 - 280) * 1

S(1) = \(e^{0.25} * 1120\) ≈ 300.92 kg

Therefore, after 1 minute, there is approximately 300.92 kg of sugar in the tank.

d) To find the amount of sugar after 84 minutes, we can solve the differential equation from part (b) numerically using an appropriate method such as Euler's method, or we can use an integrating factor to solve it analytically. One possible way to use an integrating factor is to multiply both sides of the differential equation by exp(4t/1120):

exp(4t/1120) dS/dt + (S/280) exp(4t/1120) = 0.32 exp(4t/1120)

This can be written as:

d/dt [S exp(4t/1120)] = 0.32 exp(4t/1120)

Integrating both sides from t = 0 to t = 84, we get:

[S(84) exp(4/1120 * 84)] - [S(0) exp(4/1120 * 0)] = ∫(0 to 84) 0.32 exp(4t/1120) dt

Since S(0) = 0 and exp(4/1120 * 84) is a constant, we can simplify this to:

S(84) = (1/exp(4/1120 * 84)) ∫(0 to 84) 0.32 exp(4t/1120) dt

Using integration by substitution with u = 4t/1120, we get:

S(84) = (1/exp(4/1120 * 84)) * (1120/4) * (0.32/4) * [exp(4/1120 * 84) - 1]

S(84) ≈ 301.53 kg

Therefore, after 84 minutes, there is approximately 301.53 kg of sugar in the tank.

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The complete question is:

A tank contains 1120 L of pure water. At a rate of 4 L/min, a solution that contains 0.08 kilogramme of sugar per litre enters the tank. At the same rate that it drains from the tank, the solution is mixed.

a) How much sugar is in the tank when it first starts?

b) With S representing the amount of sugar (in kg) at time t, write a differential equation which models the situation.

c) After 1 minute, calculate the sugar content (in kilogramme).

d) After 84 minutes, calculate the sugar content (in kg).

ANSWER

A reflection over the x-axis.

EXPLANATION

Both triangles are at the same distance from the x-axis: 1 unit. Triangle DEF has the same size as triangle ABC, but its position is upside-down. Therefore, triangle ABC was reflected over the x-axis to obtain triangle DEF.

A two-dimensional form that occupies an area is called a shape.

Shapes can be various types such as rectangles, circles, triangles, and many others, and they are defined by their properties such as sides, angles, and vertices. They are defined by their boundaries or edges and exist within a plane. These forms have dimensions of length and width, but not depth. Shapes play a fundamental role in geometry and are studied extensively in mathematics and art. They can be simple or complex, symmetrical or asymmetrical, and are used to represent objects, patterns, or designs. Whether in the natural world or man-made creations, shapes are integral to our understanding and appreciation of visual compositions.

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The solutions are:

1) Gradient ∇f(1, 2) = (5e², e²)

2) Divergence of F at (-1, 2, -2) is 3e⁻² - 60e⁸ - 4.

3) Curl is the zero vector (0, 0, 0).

Given data:

To find the gradient, divergence, and curl of the given functions, we need to use vector calculus.

1)

The gradient of a function is represented by the symbol ∇.

The gradient of a scalar function \(f(x, y) = x^3e^{xy} + e^2x\)  can be found by taking the partial derivatives with respect to x and y:

∂f/∂x = 3x²e^xy + 2e²ˣ

∂f/∂y = x⁴e^xy

Now, substituting the given point (1, 2) into the partial derivatives:

∂f/∂x = 3e² + 2e² = 5e²

∂f/∂y = (1)⁴e¹ˣ² = e²

Therefore, the gradient at (1, 2) is given by:

∇f(1, 2) = (5e², e²)

2)

The divergence of a vector field F = Fx i + Fy j + Fz k is given by

∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

To find the divergence, we need to compute the partial derivatives of each component and evaluate them at the given point (-1, 2, -2):

∂Fx/∂x = e^xy + ye^xy

∂Fy/∂y = 2z

∂Fz/∂z = e^2xyz + 2xyze^2xyz

Substituting the values x = -1, y = 2, and z = -2 into each partial derivative:

∂Fx/∂x = 3e⁻²

∂Fy/∂y = 2(-2) = -4

∂Fz/∂z = 4e⁸ - 64e⁸ = -60e⁸

Finally, calculating the divergence at (-1, 2, -2):

∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z =  3e⁻² - 60e⁸ - 4

Therefore, the divergence of F at (-1, 2, -2) is 3e⁻² - 60e⁸ - 4

3)

The curl of a vector field F = Fx i + Fy j + Fz k is given by the following formula:

∇ × F = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k

To find the curl, we need to compute the partial derivatives of each component and evaluate them at the given point (-2, 1, 0):

∂Fx/∂y = 3y²z³

∂Fy/∂x = 2yz³

∂Fy/∂z = 6xyz²

∂Fz/∂y = 0

∂Fz/∂x = 0

∂Fx/∂z = 0

Substituting the values x = -2, y = 1, and z = 0 into each partial derivative:

∂Fx/∂y = 0

∂Fy/∂x = 0

∂Fy/∂z = 0

∂Fz/∂y = 0

∂Fz/∂x = 0

∂Fx/∂z = 0

Finally, calculating the curl at (-2, 1, 0):

∇ × F = (0 - 0) i + (0 - 0) j + (0 - 0) k = 0

Therefore, the curl of F at (-2, 1, 0) is the zero vector (0, 0, 0).

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a. The corresponding hexadecimal representation of p(x) is 0x0D.

b. The corresponding hexadecimal representation of p(x) is 0x01.

c. Thee corresponding element of GF(256) (as a polynomial) is x^5 + x^4 + x^3 + x^2.

d. The corresponding element of GF(256) (as a polynomial) is x^6 + x^5 + x^4 + x^3 + x + 1.

a) To find the corresponding hexadecimal representation of p(x) = x^6 + x^5 + x^3 in GF(256), we can convert the polynomial coefficients to binary and then to hexadecimal.

The binary representation of p(x) is 1101010. Prepending four leading zeros to make it a byte, we get 00001101010. Converting this binary representation to hexadecimal, we have:

0000 1101 010 → 0x0D

Therefore, the corresponding hexadecimal representation of p(x) is 0x0D.

b) For p(x) = x^2 + 1, the binary representation is 00000001. Prepending six leading zeros to make it a byte, we get 00000000000001. Converting this binary representation to hexadecimal, we have:

00000000 00000001 → 0x01

Therefore, the corresponding hexadecimal representation of p(x) is 0x01.

c) The hex byte 0x3C represents the binary value 00111100. This binary value corresponds to the polynomial x^5 + x^4 + x^3 + x^2.

Therefore, the corresponding element of GF(256) (as a polynomial) is x^5 + x^4 + x^3 + x^2.

d) The hex byte 0x7D represents the binary value 01111101. This binary value corresponds to the polynomial x^6 + x^5 + x^4 + x^3 + x + 1.

Therefore, the corresponding element of GF(256) (as a polynomial) is x^6 + x^5 + x^4 + x^3 + x + 1.

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By engaging in these exercises, students can develop a deeper understanding of conditional statements and logical reasoning, which are essential skills for further studies in mathematics and logic.

In Unit 2 of a logic and proof course, homework 3 focuses on conditional statements.

Conditional statements are fundamental concepts in logic and mathematics, representing logical implications between two statements.

They are typically expressed in "if-then" format, where the "if" part is the hypothesis and the "then" part is the conclusion.

The homework may involve tasks such as:

Identifying conditional statements: Students are given a set of statements and asked to identify which ones are conditional statements.

They need to recognize the "if-then" structure and correctly identify the hypothesis and conclusion.

Analyzing the truth value of conditional statements:

Students may be given conditional statements and asked to determine whether they are true or false.

They need to evaluate the hypothesis and conclusion to determine if the implication holds in each case.

Writing converse, inverse, and contrapositive statements:

Students may be required to manipulate given conditional statements to form their converse, inverse, and contrapositive statements.

This involves switching the positions of the hypothesis and conclusion or negating both parts.

Applying the laws of logic:

Students may need to apply logical laws, such as the Law of Detachment or the Law of Modus Tollens, to deduce conclusions based on conditional statements.

Constructing counterexamples:

Students may be asked to provide counterexamples to disprove statements that are falsely claimed to be universally true based on a given conditional statement.

They also help students develop critical thinking and problem-solving abilities, as they have to analyze and manipulate logical structures.

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Using proportions, it is found that she needs 4 consecutive hits to raise her batting average to .400.

A proportion is a fraction of a total amount.

Hence, the batting average is given by the number of hits divided by the number of at-bats.

If she gets x consecutive hits, she will have 8 + x hits in 25 + x at-bats, for an average of 0.400, hence:

\(0.4 = \frac{8 + x}{25 + x}\)

8 + x = 10 + 0.4x

0.6x = 2

x = 3.33

Rounding up, she needs 4 consecutive hits to raise her batting average to .400.

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If a logical vector z has at least one entry TRUE, the function all(!z) will always be false.

If a logical vector contains only TRUE items, the all() method returns FALSE; otherwise, it returns TRUE.

The any() function, on the other hand, gives a result of TRUE if a logical vector has at least one element that is TRUE and FALSE otherwise.

any(z) will always be TRUE if z contains at least one TRUE element since there is at least one TRUE element. On the other hand, depending on whether all of the components of z are TRUE, all(z) may or may not be TRUE.

any(!z) will always return FALSE since if z has at least one TRUE element, then!z must include at least one FALSE element.

Additionally, all(!z) will always return FALSE if any element is FALSE since if z has at least one TRUE element, then!z must include at least one FALSE element.

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Answer:

16

Step-by-step explanation:

The ratio fo the lengths of the sides of a 30-60-90 triangle is

short leg  :  long leg  : hypotenuse

      1        :    sqrt(3)    :         2

In a 30-60-90 triangle, the length of the short leg is the length of the long leg divided by sqrt(3).

The length of the short leg is 8 cm.

The length of the hypotenuse is twice the length of the short leg.

g = 2 * 8 cm = 16 cm

Vector subtraction is done by arranging head to tail.

Vector subtraction is the process of finding the vector that results from taking away one vector from another. To perform vector subtraction, we arrange the vectors head to tail, with the tail of one vector touching the head of the other vector. We then draw a new vector from the tail of the first vector to the head of the second vector. The resulting vector is the difference between the two vectors. This can be mathematically represented as:

a - b = a + (-b)

where "a" and "b" are vectors, and (-b) is the additive inverse of b, which is the vector with the same magnitude as b but opposite in direction.

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The angle that must the 14-foot long wire make with the ground = 50.6°

Consider the following diagram.

AC represents the pole.

AB represents the supporting wire of length 14 ft and AD is wire of length 17 ft.

The wires are to be secured to the ground 22 feet from each other.

BD = 22 ft

We need to find angle that must the 14-foot long wire make with the ground i.e., angle ACD

Using law of cosines,

c² = a² + b² - 2ab. cos(x)

where x = ∠ABD

c is the opposite side of angle x

a and b are adjacent sides of angle x

here,  c = 17, a = 14, and b = 22

Substitute these values in above formula,

17² = 14² + 22² - 2(14)(22). cos(x)

289 = 196 + 484 - 616 cos(x)

-391 = -616 cos(x)

cos(x) = (391/616)

x = arccos(391/616))

x = 50.6°

Therefore, the required angle is 50.6°

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Tamara pays $80 upfront and then $25 per month. Tamara pays $45 per month without any upfront membership fee. So the total cost after 'n' months would be 45n. Hence, both plans will be equal after 4 months.

Let's denote the number of months as 'n'. For the first plan, Tamara pays $80 upfront and then $25 per month. So the total cost after 'n' months would be 80 + 25n.

For the second plan, Tamara pays $45 per month without any upfront membership fee. So the total cost after 'n' months would be 45n.

To find the month when both plans are equal, we set up the equation 80 + 25n = 45n and solve for 'n'.

80 + 25n = 45n

80 = 45n - 25n

80 = 20n

n = 80 / 20

n = 4

Therefore, both plans will be equal after 4 months.

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The diameter of the balloon that can lift a person weighing 65Kg is  299.72 cm.

To get the diameter of the balloon that can lift a person weighing 65 Kg, we need to know the volume which this diameter will occupy

Since, density is constant, what will change will be the volume and the weight.

Hence, the ratio of the volume to the weight at any point in time irrespective of the weight and volume will be the same.

With a 30cm diameter, the volume that can be lifted would be the volume of a sphere with diameter 30cm. If the diameter is 30, then the radius is 15.

The volume of a sphere = 4/3 × π× 15³ = 4/3 × π × 15³ = 1125π

So, what this means is that, a spherical helium balloon of size 1125π  can lift a person weighing 14 gram object .

Now, let the radius of the balloon that can lift a person of weight 65 Kg be x feet

The needed volume is thus 4/3 × π× x³ = 4π * x³/3

Now let's make a relationship;

A volume of  1125π   lifts 14 gram

A volume of  4π *x³/3     lifts 65 Kg pounds

To get x, we simply use a cross-multiplication;

1125π  × 65 = 4π * x³/3  × 0.014

x = 149.86 cm

Since the radius is  149.86cm, the diameter will be 2 × 149.86 =  299.72 cm.

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Answer: 5%

Step-by-step explanation:

Answer: I think it would be Y+(9-j)

The average yearly growth rate of the height of the tree is approximately -5.55 feet/year.

To calculate the average yearly growth rate of the tree, we need to find the total change in height over the given time period and divide it by the number of years.

The initial height of the tree is 4012 feet, and the final height is 57 feet. The total change in height is:

Total change in height = Final height - Initial height

= 57 feet - 4012 feet

= -3955 feet

The negative sign indicates a decrease in height.

The time period is 712 years.

Average yearly growth rate = Total change in height / Number of years

= -3955 feet / 712 years

≈ -5.55 feet/year

The average yearly growth rate of the height of the tree is approximately -5.55 feet/year. Note that the negative sign indicates a decrease in height over time.

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Therefore, the correct statement reflecting George's conclusion is "From the ratio test, r = 0.4. The series converges."

The nth term of a series is represented by an=2^n/5^n+1 ⋅n.

George correctly applies the ratio test to determine whether the series converges or diverges.

The ratio test is a method used to check whether a series converges or diverges. The ratio test compares the nth term of a series to the (n + 1)th term of the series.

The ratio test can be applied to series with non-negative terms.  From the given series, The nth term of a series is given by  an=2^n/5^n+1 ⋅n  By applying the ratio test, we get; an+1/an = [2^(n+1)/(5^(n+1) + 1)*(n + 1)] / [2^n / (5^n + 1)*n] an+1/an = [2^(n+1)*n / (5^(n+1) + 1)*n+1] * [(5^n + 1)*n / 2^n] an+1/an = (2n / (5n + 1)) * (5n + 1) / 2an+1/an = n / 5n + 1 Therefore, we have r = lim n→∞ | an+1/an |= lim n→∞ |n/5n+1| = 1/5 < 1 From the ratio test, r = 1/5. Since r < 1, the series converges.

Therefore, the correct statement reflecting George's conclusion is "From the ratio test, r = 0.4. The series converges."

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Alternate Interior angles

x = 25

By careful observation of the diagram, we would see that both angles are at the opposite sides of the transversal. At the same time, they are at the interior of the two parallel lines.

This means that <108 and <4x + 8 are alternate interior angles.

Since the two lines intersecting the transversal are parallel, the alternate interior angles are equal

That is,

4x + 8 = 108

4x = 108 - 8

4x = 100

x = 100/4

x = 25

Answer:

6.93

Step-by-step explanation:

√48 = √16 × √3

√48 = 4 × √3

√48 = 4 × 1.7321

√48 = 6.9284

Rounding up to 2 decimal places;

√48 = 6.93.

Note: In Mathematics, to round a number to 2 decimal places simply means to round up to the nearest hundredths.

1 decimal place = tenths.

2 decimal places = hundredths.

3 decimal places = thousandths.

Answer:

m(slope) = -1.8 or -1 and 4/5s

Step-by-step explanation:

To find the slope between any two points, a good equation to always remember would be the slope equation.

y2-y1

--------

x2-x1

Or in simple words, the difference of y divided by the difference of x.

-10-8= -18, so the difference in y= -18.

3- (-7)= 3+7= 10, so the difference in the x values= 10.

Finally, divide -18/10= -1.8 or -1 and 4/5s

The steps of the  step method are:

Problem solving is an essential skill in life. It helps us to find solutions to difficult situations and can be applied to almost any area of life. One effective way to approach problem solving is to use the 4-step method. This method is designed to help you analyze a situation, plan an action, execute the plan, and evaluate the results. In this essay, we will discuss the benefits of using the 4-step method to solve problems.

The first step of the 4-step method is to analyze the situation. This involves identifying the problem and its causes. Taking the time to understand the problem before attempting to solve it can help ensure that the solution is effective. By analyzing the situation, you can determine the most effective solution and avoid wasting time and resources on ineffective solutions.

The second step of the 4-step method is to plan an action. This involves developing a plan of action to address the problem. Taking the time to plan ahead can help ensure that the solution is tailored to the specific situation. It can also help to minimize the risk of unexpected problems or delays during the execution of the plan.

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Answer:

6400

Step-by-step explanation:

To find the number that makes the perfect square of 80, multiply 80 times 80 (square 80).

80 squared equals 6400, which means the square root of 6400 would be the perfect square 0f 80.

The remaining distance Jada has to bike is 19/10 km.

Distance is a measurement of how far apart two things or points are, either numerically or occasionally qualitatively. The distance can refer to a physical length in physics or to an estimate based on other factors in common usage.

Jada biked for 3.5 kilometers before stopping to adjust the helmet.

d₁ = 3/5 kilometer

Jada bikes another 1/2 kilometer and stopped to drink water.

d₂ = 1/2 kilometer

The total distance Jada has to bike is 3 kilometers.

The distance traveled by Jada is:

= d₁ + d₂

= 3/5 km + 1/2 km

Using LCM,

= ( 6 + 5 ) / 10

= 11/10 km

The distance remaining to cover is :

- 3 km - 11/10 km

= ( 30 - 11 ) / 10 km

= 19/10 km

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Answer: ss*$7.50 + aa*$11 ≥ $990.

Step-by-step explanation:

We have two variables:

aa = # of adult tickets sold.

ss = # of student tickets sold.

The total profit is:

P = ss*$7.50 + aa*$11

And we want P ≥ $990.

Then we can write the inequality:

ss*$7.50 + aa*$11 ≥ $990.

Now, we have only one equation and two variables, so there are a lot of possible solutions for this system.

The volume of the solid obtained by rotating the region about the x-axis is 2π * [5S * ln(5) + S^2/5].

To find the volume of the solid obtained by rotating the region bounded by the curves xy = S, x = 0, and y = 5 about the x-axis using the method of cylindrical shells, we'll integrate the volume of the cylindrical shells.

The region bounded by the curves xy = S, x = 0, and y = 5 can be rewritten as x = S/y.

The height of each cylindrical shell will be the difference between the upper and lower boundaries, which is 5 - (S/y).

The radius of each cylindrical shell will be the x-coordinate, which is x = S/y.

The differential thickness of each cylindrical shell is given by dy.

Therefore, the volume of each cylindrical shell is given by dV = 2π(S/y) * (5 - (S/y)) * dy.

To find the total volume, we integrate this expression over the range of y values that define the region, which is from y = 0 to y = 5.

v = ∫[0,5] 2π(S/y) * (5 - (S/y)) dy

Simplifying the integrand, we get:

v = 2π * ∫[0,5] (5S/y - S^2/y^2) dy

v = 2π * [5S * ln|y| + S^2/y] [0,5]

Evaluating the integral and subtracting the lower limit from the upper limit, we have:

v = 2π * [5S * ln(5) + S^2/5]

Therefore, the volume of the solid obtained by rotating the region about the x-axis is 2π * [5S * ln(5) + S^2/5].

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The experimental probability of each event is as follows:

The experimental probability of an event is the ratio of the number of outcomes that favored the event to the total number of outcomes in the experiment.

In the question, we are given that Jake tossed a paper cup 50 times and recorded the position how it landed, which is shown in the table:

Open-sided up: 1

Closed side up 5

On the side: 44.

We are asked to determine the experimental probability of each outcome.

The number of outcomes, when the landing is open-sided up is 1.

The number of outcomes, when the landing is closed-sided up is 5.

The number of outcomes, when the landing is on the side up is 44.

The total number of times the experiment took place is 50.

Thus, the experimental probability of each event is as follows:

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