sinusoidal wave in a string is described by the wave function y 5 0.150 sin (0.800x 2 50.0t) where x and y are in meters and t is in seconds. The mass per length of the string is 12.0 g/m. (a) Find the maximum transverse acceleration of an element of this string. (b) Determine the maximum transverse force on a 1.00-cm segment of the string . (c) State how the force found in part (b) compares with the tension in the string

Answer:

x= 7.5 mile/h

y= 3.5 miles / h

Step-by-step explanation:

Given x and y

so, x+y = speed down river

and x-y= speed up river

using

travel time = distance / speed for each case;

we get;

22/x+y=2  ---(1)

and

22/x-y=5.5 -----(2)

Solving equations simultaneously

eq 1 gives

2x+2y=22  ---(3)

eq 2 gives

5.5x-5.5y=22 ----(4)

Multiply eq 3 by 2.75

==> 5.5x+5.5y=60.5 --- (5)

adding eq 4 with eq 5

==> 11x =82.5

==> x= 7.5 miles/h

put this value in eq 3

==> 2(7.5)+2y=22

==> 2y=22-19.8

==> 2y=7

==> y = 3.5

Once setting up two equations based on downstream and upstream velocities, using variables x for Pratap's rowing speed and y for the current's speed, you can solve for these variables. The results show Pratap can row at a speed of 7.5 mph in still water, while the rate of the current is 3.5 mph.

Firstly, we have to understand that Pratap's speed with the current and against the current will be affected. Let's say x is the speed of Pratap rowing in still water, and y is the speed of the current. When going downstream (in the direction of the current), Pratap's speed is (x + y) since the current helps him. He covered a distance of 22 miles in 2 hours, so the rate (velocity) can be found by dividing distance by time. Hence, x + y = 22/2 = 11 miles per hour.

On the other hand, when rowing upstream (against the current), Pratap's speed is (x - y), because now the current hinders his rate. He covered the same distance in 5 1/2 hours, so x - y = 22/(5.5) = 4 miles per hour.

Now, we have two equations: x + y = 11 and x - y = 4. Solving these equations (by adding them), we find that 2x = 15 hence x = 7.5. Pratap can row in still water at a speed of 7.5 miles per hour. Substituting x=7.5 into either of the original equations we get y = 3.5. Hence, the rate of the current is 3.5 miles per hour.

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Answer:

summation of five times negative three to the power of n from n equals zero to infinity

Step-by-step explanation:

Summation Notation

It represents the sum of a finite or infinite number of terms. Let's analyze the terms of the given succession:

5-15+45-135+...

If we take 5 as a common factor, we have

5(1-3+9-27+...)

The parentheses contain the alternate sum/subtraction of powers of 3. The odd terms are positive, the even terms are negative, thus the exponent must be n starting from 0 or n-1 starting from 1

The summation is then represented by

[tex]\sum_{n=0}^{\infty}5(-3)^n[/tex]

This corresponds with the option:

summation of five times negative three to the power of n from n equals zero to infinity

Final answer:

The correct summation notation for the series 5 - 15 + 45 - 135 + ... is the summation of five times negative three to the power of n from n equals zero to infinity.

Explanation:

The given sequence is 5 - 15 + 45 - 135 + ... which can be seen as a geometric series with a pattern of alternating signs and a common ratio of -3. The first term of the sequence is 5 (when n=0), and each subsequent term is multiplied by -3. Therefore, to write this pattern using summation notation, the nth term can be represented as 5 × (-3)^n. So, the summation notation for the entirety of the series from n equals 0 to infinity is:

Σ [5 × (-3)^n], from n = 0 to infinity

This matches the option: summation of five times negative three to the power of n from n equals zero to infinity.

Answer:

Step-by-step explanation:

Variability refers to the degree in which a set of data set spreads out or dispersed or clustered together. There are four measure of variability namely: range, interquartile range (IQR) variance and standard deviation. The consequence of increasing variability is that the distance from one score to another tends to increase, and a single score tends to provide a less accurate representation of the entire distribution.

In the context of a data set, increasing variability implies that scores in the dataset are dispersing more, increasing the distance from one score to another. Therefore, a single score provides a less accurate reflection of the entire distribution. The correct answer to the question is B.

The subject of the question relates to the concept of variability in a data set, which is a measure of how much scores differ from each other in a distribution. The question asks about the consequences of increasing variability.

If variability increases, it means that the scores in the dataset are spreading out more. In other words, the distance from one score to another tends to increase. Consequently, a single score tends to provide a less accurate representation of the entire distribution because the scores are more spread out and there is a higher level of diversity in the scores. So, the correct answer is B.

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Answer: option C is the correct answer

Step-by-step explanation:

Let x represent the cost of one night at the hotel.

Let y represent the cost of renting the car for one day.

Connor went to New York for vacation. He spent 3 nights at a hotel and rented a car for 4 days. If Connor paid $675, it means that

3x + 4y = 675 - - - - - - - - - - - 1

Jillian stayed at the same hotel, but spent 4 nights and rented a car for 5 days from the same company. If Jillian paid $875, It means that

4x + 5y = 875 - - - - - - - - - - - -2

Multiplying equation 1 by 4 and equation 2 by 3, it becomes

12x + 16y = 2700

12x + 15y = 2625

Subtracting, it becomes

y = 75

Substituting y = 75 into equation 1, it becomes

3x + 4 × 75 = 675

3x + 300 = 675

3x = 675 - 300 = 375

x = 375/3 = $125

The recommended number of classes is 5, and the class interval is 10. The first class's lower limit would be 50. To create a relative frequency distribution, divide each class frequency by the total

First, we have to calculate the range of the data by subtracting the smallest number from the highest number (98-51 = 47). A typical choice for the number of classes might be between five and 20. The

square root rule

suggests a round number near the square root of the number of observations, which is approx 4.47 in this case; you can round to 5. Now we divide the range by the number of classes (47/5 approximately 9). So, the class interval would be 10 (we round up as it does not make sense to have a fraction of an oil change). If the smallest number is 51, the lower limit for the first class would be 50. Hence, the

frequency distribution

would break down like this: 50-59, 60-69, 70-79, 80-89, 90-99. To create a relative frequency distribution, divide each class frequency by the total number of observations and multiply by 100 to convert to percentages. As for the shape, you would need to plot the data to see, but it may have a normal distribution as service levels often do.

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Answer:

The Spanish ship goes to Port Said and the French ship carries tea. However, tea can be carried by the Brazilian ship, too.

Step-by-step explanation:

French 5:00 tea blue Genoa

Greek 6:00 coffee red Hamburg

Brazilian 8:00 cocoa black Manila

English 9:00 rice white Marseille

Spanish 7:00 corn green Port Said

Answer:

Step-by-step explanation:

Let x represent the number of nights for which you rent a suite.

Resorts-R-Us charges $ 125 a night to rent a suite. If you choose this plan, the total cost of renting the suite for x nights would be

125 × x = 125x

If you purchase their one year membership fee for $350, you only pay $75 a night. This means that the total cost of renting a suite for x nights would be

75x + 350

For the plan involving membership to be a better deal,

75x + 350 < 125x

350 < 125x - 75x

350 < 50x

50x > 350

x > 350/50

x > 7

After 7 nights, the membership option would be a better deal. If you intend to rent the suite for lesser than 7 nights in a year, then the first option is a better deal.

11°

The solution is in the attachment

The angle of subtends the hill will make with horizontal is 17.04 degrees.

The trigonometric functions found in the four quadrants, as well as their graphs, domains, and differentiation and integration, will all be understood.

The trigonometric function is very good and useful in real-life problems.

Given below is the image of the situation,

In triangle ABC →

Tan25° = (125+240)/x

x = 782.745 feet

Now in triangle ADC →

Tan(y) = 240/x

Tan y = 240/782.745

y = 17.04°

Hence "The angle of subtends the hill will make with horizontal is 17.04 degrees".

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Final answer:

After calculating the z-scores for a high temperature of 55 degrees in both January and July, it was found that a temperature of 55 degrees is more unusual in July than in January due to the higher absolute value of its z-score.

Explanation:

To determine in which month it is more unusual to have a day with a high temperature of 55 degrees, we compare the standardized scores (z-scores) of 55 degrees for January and July. The z-score is calculated by subtracting the mean from the observation and then dividing the result by the standard deviation. For January, with a mean of 36 and standard deviation of 10, the z-score is (55 - 36) / 10 = 1.9. For July, with a mean of 74 and standard deviation of 8, the z-score is (55 - 74) / 8 = -2.375.

Comparing the absolute values of the z-scores, the z-score for July is higher in absolute value, indicating that a temperature of 55 degrees is more unusual in July than it is in January.

Answer:

[tex]C. \: 9 \times 7[/tex]

Answer:

m<7=65

m<4 = 115

m<6 = 115

m<1=65

m<16 = 60

m<18 = 60

m<21 = 120

m<10 = 55

m<11 = 125

m<12 = 55

Step-by-step explanation:

(1) m<7 = 65° (corresponding angles are equal)

(2) m<4 = 180 - 65 = 115 (angles on a straight line)

(3) m<6 = 115 (angles on a straight line)

(4) m<1 = 180-115 = 65

(5) m<16 = 180 - 120 = 60°

(6) m<18 = m<14 = 60°(corresponding angles are equal)

(7) m<21 = 180 - 60 = 120

(8) m<10 = m<12 = 180-(65+60)=55(sum of angles in a triangle)

m<10 = 55°

(9) m<11 = 180 - 55 = 125°

(10)m<12 = 55°(sum of angles in a triangle)

Answer:

Step-by-step explanation:

The value of a is 80

Step-by-step explanation:

The distance of a point [tex](x_0,y_0)[/tex] from the y-axis can be written as

[tex]d_y = |x_0|[/tex]

because the x-coordinate of the y-axis is zero.

Similarly, the distance of a point [tex](x_0,y_0)[/tex] from the x-axis can be written as

[tex]d_x=|y_0|[/tex]

Since the y-coordinate of the x-axis is zero.

In this problem:

- The distance of the point A (−30, −45) from the y-axis can be written as

[tex]d_A = |-30|=30[/tex]

- The distance of point B (a,a) from the x-axis can be written as

[tex]d_B = |a| = a[/tex]

Since [tex]a>0[/tex].

We are told that 2/3 of the distance from the y-axis to point A (−30, −45) is equal to 1/4 of the distance from the x-axis to point B(a, a), which means

[tex]\frac{2}{3}d_A = \frac{1}{4}d_B[/tex]

Therefore,

[tex]\frac{2}{3}(30)=\frac{1}{4}a[/tex]

And solving for a,

[tex]20=\frac{1}{4}a\\a=80[/tex]

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Answer:

40

Step-by-step explanation:

Answer:

5x + 13

Step-by-step explanation:

(3x + 9 ) + (2x + 4)

It's quite simple, we just have to add the parts that have x to each other and those that don't have x to each other

3x + 2x + 9 +4 =

then the result is

5x + 13

Answer:

226.08 mm³

Step-by-step explanation:

The figure below shows a capsule mad of;

To determine the amount of vitamin that can be enclosed in the capsule, we determine the volume of the capsule;

Volume of a hemisphere = 2/3 πr³

Therefore; Taking π to be 3.14

Then;

Volume of the hemisphere = 2/3 × 3.14 × 3³

                                             = 56.52 mm³

But, since there are two equal hemispheres;

Then;

Volume of hemispheres = 56.52 mm³ × 2

                                        = 113.04 mm³

Volume of the cylinder is given by;

Volume = πr²h

Height = (10 mm - (2× 3mm)

           = 4 mm

Thus, taking π to be 3.14

Then;

Volume of the cylinder = 3.14 × 3² × 4

                                      = 113.04 mm³

Thus, Volume of the capsule = 113.04 mm³ + 113.04 mm³

                                                = 226.08 mm³

Answer:

4r

Step-by-step explanation:

If you take away the variable, add the 3 and metaphorical 1

3 + 1 = 4

Adding like terms (3+1) and then putting the variable back in makes 4r

Hope this helps, mark brainliest

Answer:

4r

Step-by-step explanation:

I hope this helps!

Answer:b also d

Step-by-step explanation:

Adrian's recipe has a greater ratio of sauce to dough, and the ratios can be compared by comparing 15 to 18 because 30 is in both dough columns.

In Adrian's recipe, the ratio of sauce to dough is 15:30, which simplifies to 1:2.

In Juliet's recipe, the ratio of sauce to dough is 18:30, which also simplifies to 1:2.

Since both ratios simplify to the same value of 1:2, it means that Adrian's and Juliet's recipes have equal ratios of sauce to dough.

Therefore, the statement "Adrian's and Juliet's recipes have equal ratios of sauce to dough" is true.

The reason we can compare the ratios by looking at 15 and 18 is because both of these values represent the amount of sauce, and 30 represents the amount of dough in both recipes.

By comparing the sauce portions directly, we can determine that the ratios are indeed equal.

It's important to note that the absolute values of the sauce and dough quantities are less relevant in this context than their proportions, which are expressed in the ratios.

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Answer:

No

Step-by-step explanation:

The majority fairness criteria states that a person with majority of votes should be the winner and it is not violated by the plurarity method. Rather pluarity is always satisfied with majority fairness criterion.

Answer: Extraneous variable

Step-by-step explanation:

In an experiment , Independent variable can be manipulated by the experimenter to see the change in the dependent variable or response variable. But there are some variable known as extraneous variable that is attached to independent variable and make experimenter confuse.  

Extraneous variables is defined as :

∴ A variable that is systematically linked with the factor you believe is causing the overall effect in your research is called the extraneous variable variable. The presence of such a variable can prevent you from knowing what is really causing the effect.

Answer:

4.8 hours

Step-by-step explanation:

Distance = speed × time

If Tim drove 4 hours at 60 miles per hour

Than he drove a total of 4×60 miles

He drove 240 miles

Time = distance ÷ speed

If he drove 240 miles at 50 miles per hour

Than he took 240÷50 hours

He took 4.8 hours (4 hours and 48 minutes)

Answer:

         [tex]\large\boxed{\large\boxed{3\times 10^8meters}}[/tex]

Explanation:

One of the applications of scientific notation is to make estimations rounding the whole part of the numbers, i.e. the digits before the decimal point, to the nearest integer, and adding the exponents of the powers of base 10.

Here, you must estimate this product of two numbers written is scientific notation:

          [tex](3.21\times 10^3)(1.13\times 10^5)[/tex]

Then, for an estimation you round 3.21 to 3 and 1.13 to 1, then multiply 3 × 1 = 3. That will be the coefficient of your new power of 10.

The power or exponent will be the sum of the powers of the numbers that are being multiplied, i.e. 3 + 5 = 8.

And the result is [tex]3\times 10^8[/tex]

The unit is meters, so you write your answer as: [tex]3\times 10^8meters[/tex]

Answer:

b. 3*10^8

Step-by-step explanation:

Answer:

Step-by-step explanation:

If I'm understanding this correctly, the rate of decay function is

[tex]f(x)=1000(.5)^{\frac{x}{30}}[/tex]

and we want to solve for the amount of element left after x = 40 years.  That would make our equation

[tex]f(40)=1000(.5)^{1.33333333333}[/tex]

Multiply the repeating decimal by .5 to get

f(40) = 1000(.3968502631)

and f(40) = 396.85

So no, it's not safe for human habitation.

f(40) ≈ 793.7 kilograms. The area will not be safe for human habitation by 2025.

For the decay, to find f(40), we substitute x=40 into the function f(x)=1000(0.5)x/30.

f(40)=1000(0.5)40/30

f(40)=1000(0.5)4/3

f(40) ≈ 1000(0.7937)

f(40) ≈ 793.7 kilograms

Since the question asks if the area will be safe for human habitation by 2025, we need to check if f(40) is less than or equal to 100 kilograms.

But the value of f(40) is about 793.7 kilograms, which is greater than 100 kilograms.

Therefore, the area will not be safe for human habitation by 2025.

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Final answer:

After Sam eats 1/2 and Jack eats 1/4 of the remaining chocolate bar, 3/8 of the original chocolate bar is left uneaten.

Explanation:

Sam eats 1/2 of a chocolate bar. Jack then eats 1/4 of the remaining chocolate bar. To determine how much of the chocolate bar is left, we need to perform a couple of simple calculations. After Sam eats his portion, 1/2 of the bar is left. Jack eats 1/4 of what remains after Sam, which is 1/4 of 1/2, or 1/2 * 1/4 = 1/8 of the original bar.

This means that Jack's portion is 1/8 of the entire chocolate bar. Now, the total amount eaten by Sam and Jack together is 1/2 (Sam's portion) + 1/8 (Jack's portion). To add these fractions, they need to have a common denominator, which in this case is 8. This makes Sam's portion 4/8 when expressed with the common denominator. Now, add 4/8 (Sam's adjusted portion) + 1/8 (Jack's portion) = 5/8.

Therefore, the total amount eaten by both is 5/8 of the bar, leaving 3/8 of the original chocolate bar uneaten.

Answer:

10.2 feet.

Step-by-step explanation:

We have been given that Derek places  a standard 12-inch ruler next to the goal post and measures the shadow of the ruler and the backboard. If the ruler has a shadow of 10 inches. We are asked t find the height of the backboard, if the backboard has a shadow of 8.5 feet.

We will use proportions to solve our given problem as ratio between sides ruler will be equal to ratio of sides of background.

[tex]\frac{\text{Actual height of ruler}}{\text{Shadow of ruler}}=\frac{\text{Actual height of backboard}}{\text{Shadow of backboard}}[/tex]

[tex]\frac{12}{10}=\frac{\text{Actual height of backboard}}{8.5}[/tex]

[tex]\frac{12}{10}*8.5=\frac{\text{Actual height of backboard}}{8.5}*8.5[/tex]

[tex]1.2*8.5=\text{Actual height of backboard}[/tex]

[tex]10.2=\text{Actual height of backboard}[/tex]

Therefore, the actual height of the back-board is 10.2 feet.

Answer:

Step-by-step explanation:

Since the backboard and the ruler are both vertical and the sun is at the same position in the sky, the triangle made by the backboard and its shadow is similar to the triangle made by the ruler and its shadow.

The ratio of the corresponding sides of the triangles are equal and the height of the backboard can be determined by solving the following proportion.

Therefore, the top of the backboard is 7.64 feet high.

Answer:

Each interior angle = 135 degrees,

Step-by-step explanation:

The exterior angles of all convex polygons add up to 360 degrees.

So for a regular octagon each exterior angle = 360 / 8

= 45 degrees.

Therefore each interior angle = 180 - 45 = 135 degrees,

Final answer:

To find the intersection points of the paths of two particles, one needs to set their position functions equal and solve for the variable t. If no solution exists, then the intersection points do not exist, and the answer is DNE.

Explanation:

To find the points where the paths of two particles intersect, we set their respective position functions r1(t) and r2(t) equal to each other and solve for t. The position functions given are:
r1(t) = (t, t^2, t^3)
r2(t) = (1 + 4t, 1 + 16t, 1 + 52t).

Their paths intersect when:
x1 = x2
y1 = y2
z1 = z2.

By solving these equations, we find the common t values that make both position vectors equal. If there is no common t that satisfies all three conditions, then the particles never collide, and the answer is DNE (Does Not Exist).

Answer:

The average weight of the newborn otter is  [tex]\frac{71}{40}\ kg.[/tex]

Step-by-step explanation:

Given:

A baby otter was born 3/4 of a month early at first it's weight was 7/8 kg which is 9/10 kg less than the average weight of newborn otter in the aquarium.

Now, to find the average weight of the newborn otter.

Let the average weight of the newborn otter be [tex]x.[/tex]

It's weight was = [tex]\frac{7}{8} \ kg.[/tex]

It's weight is less than the average weight of newborn by = [tex]\frac{9}{10} \ kg.[/tex]

According to question:

[tex]x-\frac{9}{10}=\frac{7}{8}[/tex]

Adding both sides by [tex]\frac{9}{10}[/tex] we get:

[tex]x=\frac{9}{10} +\frac{7}{8}[/tex]

[tex]x=\frac{36+35}{40}[/tex]

[tex]x=\frac{71}{40} \ kg.[/tex]

Therefore, the average weight of the newborn otter is [tex]\frac{71}{40}\ kg.[/tex]

Answer:

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Step-by-step explanation:

Given:

In Right Angle Triangle PQR at ∠Q = 90° such that

RQ = 3     ....Adjacent side of angle R

PQ = 8     ....Opposite side of angle R

To Find:

m∠R = ?

Solution:

In Right Angle Triangle PQR, Tan Identity,

[tex]\tan R= \dfrac{\textrm{side opposite to angle R}}{\textrm{side adjacent to angle R}}[/tex]

Substituting the values we get

[tex]\tan R= \dfrac{PQ}{QR}=\dfrac{8}{3}=2.6666\\\\\angle R=\tan^{-1}(2.666)=69.439=69.4\°[/tex]

Therefore,

[tex]m\angle R = 69.4\°[/tex]

Using the Pythagorean theorem, the distance from the bottom of the ladder to the wall is found to be 5 ft, after solving the quadratic equation that arises from the conditions given.

The question asks for the distance from the bottom of the ladder to the wall when a ladder is leaning against it. We can use the Pythagorean theorem to solve this problem as it forms a right-angled triangle. Let's denote the distance from the bottom of the ladder to the wall as x, then the distance from the top of the ladder to the bottom of the wall would be x + 7 ft. Since the ladder's length, which represents the hypotenuse, is 13 ft, we can set up the equation:

x2 + (x + 7)2 = 132

Expanding this, we get:

x2 + x2 + 14x + 49 = 169

Combining like terms:

2x2 + 14x - 120 = 0

Dividing everything by 2 to simplify:

x2 + 7x - 60 = 0

Factoring the quadratic equation:

(x + 12)(x - 5) = 0

The possible values for x are -12 and 5. Since distance cannot be negative, the distance from the bottom of the ladder to the wall is 5 ft.

Answer:

A. three planes intersecting at a point

Step-by-step explanation:

Whenever three planes intersect at a point, then we have an ordered triplet (x,y,z) that is a solution to the associated system of equation.

This is a unique solution and it is finite.

However three planes intersecting at a line gives infinitely many solutions.

Three or two parallel planes may have no solution or or infinitely many solutions when they coincide.

Answer:

(a) 0.4096

(b) 0.64

(c) 0.7942

Step-by-step explanation:

The probability that the player wins is,

[tex]P(W)=0.80[/tex]

Then the probability that the player losses is,

[tex]P(L)=1-P(W)=1-0.80=0.20[/tex]

The player is playing the video game with 4 different opponents.

It is provided that when the player is defeated by an opponent the game ends.

All the possible ways the player can win is: {L, WL, WWL, WWWL and WWWW)

(a)

The results from all the 4 opponents are independent, i.e. the result of a game played with one opponent is unaffected by the result of the game played with another opponent.

The probability that the player defeats all four opponents in a game is,

P (Player defeats all 4 opponents) = [tex]P(W)\times P(W)\times P(W)\times P(W)=[P(W)]^{4} =(0.80)^{4}=0.4096[/tex]

Thus, the probability that the player defeats all four opponents in a game is 0.4096.

(b)

The probability that the player defeats at least two opponents in a game is,

P (Player defeats at least 2) = 1 - P (Player losses the 1st game) - P (Player losses the 2nd game) = [tex]1-P(L)-P(WL)[/tex]

                                    [tex]=1-(0.20)-(0.80\times0.20)\\=1-0.20-0.16\\=0.64[/tex]

Thus, the probability that the player defeats at least two opponents in a game is 0.64.

(c)

Let X = number of times the player defeats all 4 opponents.

The probability that the player defeats all four opponents in a game is,

P(WWWW) = 0.4096.

Then the random variable [tex]X\sim Bin(n=3, p=0.4096)[/tex]

The probability distribution of binomial is:

[tex]P(X=x)={n\choose x}p^{x} (1-p)^{n-x}[/tex]

The probability that the player defeats all the 4 opponents at least once is,

P (X ≥ 1) = 1 - P (X < 1)

             = 1 - P (X = 0)

             [tex]=1-[{3\choose 0}(0.4096)^{0} (1-0.4096)^{3-0}]\\=1-[1\times1\times (0.5904)^{3}\\=1-0.2058\\=0.7942[/tex]

Thus, the probability that the player defeats all the 4 opponents at least once is 0.7942.

Final answer:

The probability of defeating all four opponents in one game is 40.96%. The probability of defeating at least two opponents is 87.04%. Playing the game three times, the probability of defeating all four opponents at least once is 88.47%.

Explanation:

To calculate the probability of different outcomes when playing a video game against four opponents, we'll use basic probability rules and assumptions given in the question.

A. Probability of Defeating All Four Opponents

The probability of defeating each opponent is 80% or 0.8. Since the fights are independent, to find the probability of defeating all four, we multiply the probabilities together:
0.8 × 0.8 × 0.8 × 0.8 = 0.4096 or 40.96% chance of defeating all four opponents.

B. Probability of Defeating At Least Two Opponents

To find the probability of defeating at least two opponents, we must consider all possible combinations of winning 2, 3, or 4 opponents' fights, and add those probabilities together. Let W represent a win and L represent a loss:

We already calculated P(WWWW) as 0.4096. The other probabilities can be calculated using similar multiplication of respective individual probabilities (0.8 for W, 0.2 for L). After calculation, we sum these to find the cumulative probability, which is 0.8704 or 87.04%.

C. Probability of Defeating All Four Opponents At Least Once Over Three Games:

The probability of not defeating all four opponents in one game is 1 - P(WWWW) = 0.5904. The events in each game are independent, so the probability of not defeating all four opponents in all three games is (0.5904)^3. Therefore, by subtracting this result from 1, we get the probability of defeating all opponents at least once: 1 - (0.5904 × 0.5904 × 0.5904) = 0.8847 or 88.47%.