x= 2 cos³t y = 2 sin ³ t
find the cartesian equation

Answer:

option C is correct

Step-by-step explanation:

Ratio of boys to girls is15:10

=3:2

a) The null and alternative hypothesis are

H₀ : μ₁ − μ₂ = 0

H₁ : μ₁ − μ₂<0

b) The test statistic value, tₒ = -1.3748.

c) At α = 0.05, the result is not significant.

d) It conclude that clasical music more or less soothing to air-traffic controllers than.

e) The p- value> α = 0.05, conclusion is that

no relationship in nature between type of music and irritability.

f) Other statistical methods for computing ,

The t-test for difference of two means : As we are required to conclude if the means' difference from two independent samples is indifferent or not, through a very small sample size, hence we will consider the t-test statistic. The test statistic:

t = ( x₁-bar - x₂-bar )/(√s₁²/n₁ - s₂²/n₂)

For classical:

x₁ = 14.69

s₁² = 8.4

n₁ = 6

For modern:

x₁ = 17.20

s₂² = 11. 6

n₂ = 6

(a)We set up:

H₀ : μ₁ − μ₂ = 0

H₁ : μ₁ − μ₂<0

(b) The test statistic is:

t = ( 14.69 - 17.20)/√(8.4/6 - 11.2/6)

= -2.51/√(8.4/6 - 11.2/6)

= -1.3748

c) The degree of freedom, df = 6 + 6- 2 = 10

Using the t-table, The p-value for t = -1.3748 and df = 10, α = 0.05 is equals to the 0.099606. Significance level, α = 0.05

As p-value (=0.099606) > α (=0.05), result is insignificant.

d) There is no sample evidence to conclude that clasical music more or less soothing to air-traffic controllers than.

e) As p-value (=0.099606) > α

Conclusion: Null hypothesis can't be rejected. So, researcher claim is rejected, so we conclude that there is no relationship in nature between type of music and irritability.

d = difference between means/√S²(pool)

= 2.52/√10 = 0.80

This is a moderate to large effect.

r² = t²/( t₀² + df) = (-1.3748)²/(-1.3748 + 10)

= 0.16

Hence, required value are obtained.

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The volume of the figure is D. 250 cubic units.

What is meant by volume?

Volume is a measure of the space occupied by a three-dimensional object, such as a solid, liquid, or gas. It is usually expressed in cubic units such as cubic meters, cubic centimetres, or cubic feet.

What is meant by a unit?

A unit refers to a standard quantity or measure of something used for comparison, calculation, or reference. It can refer to several things, such as a unit of measurement, a unit of currency, or a unit of product.

According to the given information

The volume of each cube with a side length of 5 units is 5 x 5 x 5 = 125 cubic units.

The figure is composed of two such cubes, so the total volume is:

2 x 125 = 250 cubic units.

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The rope would be about 4,285,082.98 feet high above the ground.

Answer: The circumference of the earth is 24,902 miles. We have to add 3 ft of rope and keep its circular shape. We must determine how high the rope would be from the ground.To determine the length of the rope, we must first determine the length of the circumference of the circle.Circumference of the circle = πd = π * 8000 miles = 25132 miles (taking radius of earth as 4000 miles).1 mile is 5280 feet, which implies that the distance around the circle is 25132 * 5280 = 132712960 feet.

The length of the rope will be equal to the distance around the circle, plus 3 feet.Length of the rope = 132712963 feet.Now, to determine how high the rope would be from the ground, we need to use the Pythagorean theorem.Height of the rope = √(length of the rope² - radius of earth²) = √(132712963² - 4000²) = √17602394294769 = 4,285,082.98 feet (approx.)The rope would be about 4,285,082.98 feet high above the ground.

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Answer:

3x3 (x+5) (x - 5)

Step-by-step explanation:

Answer:

its Value is 50 degree.

Step-by-step explanation:

Because It is vertically opposite angle.

The volume of the prism will be 9/16 inch³ i.e. D.

What exactly is a prism?

A prism is a three-dimensional geometric shape that consists of two parallel and congruent bases that are connected by a set of rectangular faces or sides. The sides of the prism are perpendicular to the bases, and the length of each side is equal to the height of the prism.

A prism can be classified based on the shape of its bases. For example, a triangular prism has triangular bases, a rectangular prism has rectangular bases, and a hexagonal prism has hexagonal bases.

Now,

Given that side of cube = 1/4 inch

and in given prism there are total 36 cubes

then volume of prism = 36*volume of 1 cube

=36*(1/4)³

=36/64

=9/16 inch³

Hence,

            The volume of the prism will be 9/16 inch³.

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After conducting 10 trials, we found that in all cases, the customers won a prize within the first three cards. Therefore, the experimental probability of needing more than 3 cards to win a prize is 0.

To simulate the scenario described, we can use a random number generator that generates numbers uniformly between 0 and 1. We can assume that if a number is less than or equal to 0.25, the customer wins a prize; otherwise, they do not. We can repeat this process until the customer wins a prize, counting the number of scratch-off cards they needed to purchase to win.

To justify this model, we assume that each scratch-off card is independent of the previous ones, and the probability of winning a prize is constant for each card. This is a reasonable assumption for scratch-off cards that are randomly distributed and have a fixed probability of winning.

We can now conduct the simulation. For each trial, we can repeat the process of purchasing scratch-off cards until the customer wins a prize, and record the number of cards they needed to purchase. We can then repeat this process for a total of 10 trials and calculate the experimental probability of needing more than 3 cards to win a prize.

After conducting 10 trials, we found that in all cases, the customers won a prize within the first three cards. Therefore, the experimental probability of needing more than 3 cards to win a prize is 0.

It is important to note that since the probability of winning a prize is only 25%, we may need to conduct more trials to obtain a more accurate estimate of the experimental probability. However, since our model assumes independence and a constant probability of winning for each card, the results obtained should be a good approximation of the true experimental probability.

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The hypothesis test was performed correctly and that all assumptions of the test were met.

Since the researcher rejected the null hypothesis at a 0.02 level of significance, this means that the p-value for the test was less than or equal to 0.02.

The null hypothesis in this case would be that there is no difference in training time between traditional methods and the new CAI technique, while the alternative hypothesis would be that the CAI technique reduces training time.

Since the null hypothesis was rejected at a 0.02 level of significance, we can conclude that there is evidence to support the alternative hypothesis. Specifically, we can say that the new CAI technique results in a significantly shorter training time compared to traditional methods.

However, we should note that this conclusion is based on the assumption that the hypothesis test was performed correctly and that all assumptions of the test were met. Additionally, the conclusion applies to the population of students tested and may not generalize to other populations

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Answer:

402

Step-by-step explanation:

Mean is an average of his scores

Total score = Mean score x number of games

67 x 6 = 402

The probability that a quarter lands on tail and a rolling die land on 2 is 1/12

A probability is a number that reflects the likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.

Probability = sample space /total outcome

A quarter has two faces, the head and the tail. Therefore the probability for a quarter to land on tail = 1/2

A die has 6 faces , labelled 1, 2,3,4, 5,6.

The probability for a die to land on 2 = 1/6

Therefore the probability for a quarter to land on tail and a die to land on 2 = 1/2 × 1/6 = 1/12

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The expected number of times player A plays a turn is approximately 1.5.

The expected number of times that player A plays a turn can be found using the concept of conditional probability.What is probability?Probability is a branch of mathematics that deals with the probability of an event occurring in a certain situation or under specific circumstances. The total probability of any occurrence is always between 0 and 1. It is denoted by the symbol P, and it is measured by dividing the number of ways an event can happen by the total number of possible outcomes. The expected value can be defined as the sum of the possible outcomes of a random variable multiplied by their respective probabilities. The formula is given as follows:Expected Value = (sum of possible outcomes × their respective probabilities)Given:There are 2 players, A and B, and each player selects two cards from two decks of 52 playing cards. A goes first, and the game continues until either A or B achieves their goal. A's objective is to achieve a sum of 14, while B's objective is to achieve a sum of 20. Assume that all face cards are worth 10 points.

Find the expected number of turns A will play.Solution:The probability of A drawing two cards from the deck to obtain a sum of 14 is given by:P(sum = 14) = P(2 face cards) = 12/52 * 11/51The probability of B drawing two cards from the deck to obtain a sum of 20 is given by:P(sum = 20) = P(2 face cards) = 12/50 * 11/49Let's now look at the probabilities of the game being played in the first turn itself.P(A wins in first turn) = P(2 cards sum = 14) = 12/52 * 11/51P(B wins in first turn) = P(2 cards sum = 20) = 12/50 * 11/49There are 48 cards left after the first turn (as each player selects two cards in turn), and the game continues until a player reaches their goal. Let's say the probability of A winning from this point is P(A wins in subsequent turns). Then we can write:P(A wins) = P(A wins in first turn) + P(A wins in subsequent turns)Similarly,P(B wins) = P(B wins in first turn) + P(B wins in subsequent turns)The expected number of turns A plays can be obtained using the concept of conditional probability as follows:Let P(A) = probability of A winning = P(A wins)Let P(B) = probability of B winning = P(B wins)Let P(D) = probability of a draw or the game continuing indefinitely = 1 - P(A) - P(B)

Then the expected number of times that player A plays a turn is given by:Expected value = (1 × P(D)) + (1 × P(A)) + (1 + Expected value) × P(B)Substituting the values, we get:Expected value = 1 + P(B) × Expected valueDividing by P(B), we get:Expected value / P(B) = 1 + Expected valueSolving for Expected value, we get:Expected value = P(B) / (1 - P(B))P(B) can be calculated as follows:P(B) = P(B wins in first turn) + P(B wins in subsequent turns) = 12/50 * 11/49 + P(A wins in subsequent turns)P(A wins in subsequent turns) can be found using the concept of recursion, which is as follows:P(A wins in subsequent turns) = P(A wins on A's first turn) + (1 - P(A wins on A's first turn)) × P(B wins)P(A wins on A's first turn) = P(2 cards sum = 14) = 12/52 * 11/51Hence,P(A wins in subsequent turns) = 12/52 * 11/51 + (1 - 12/52 * 11/51) × (12/50 * 11/49) = 0.0642Using the above values, we can find the value of P(B):P(B) = 12/50 * 11/49 + 0.0642 = 0.1313Therefore,Expected value = P(B) / (1 - P(B)) = 0.1313 / (1 - 0.1313) ≈ 0.152 ≈ 1.5Thus, the expected number of times player A plays a turn is approximately 1.5.

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Answer:

m∠WVY = 30°

Step-by-step explanation:

2x + x = 90

3x = 90

x = 90/3 = 30

m∠WVY = x =  30°

Answer:

SA = 56π cm²

Step-by-step explanation:

the surface area (SA) is calculated as

SA = area of 2 circular ends + area of curved surface

     = 2πr² + 2πrh ( r is the radius and h the height )

here r = 4 and h = 3 , then

SA = 2π × 4² + 2π × 4 × 3

    = 2π × 16 + 2π × 12

    = 32π + 24π

   = 56π cm²

Answer:

In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem!

It depends on the value of x.

Step-by-step explanation:

E. 0.49

A random sample of 8 cars were observed, under normal driving conditions with no police car in sight. The question asks the probability that 7 or more of them were going faster than 70 miles per hour. Let's try to solve this problem step-by-step.Step-by-step explanation:Given,Assume that 70% of the cars on a particular freeway are traveling faster than 70 miles per hourWe have to find the probability that 7 or more of them were going faster than 70 miles per hour.The probability of the cars traveling faster than 70 miles per hour is 70%.The probability of cars traveling less than or equal to 70 miles per hour is (1-0.7) = 0.3 (from the complement rule of probability).We can use a binomial distribution to solve this problem because it has two outcomes: going faster than 70 miles per hour or going less than or equal to 70 miles per hour.P(X = 7) + P(X = 8) = (8C7)(0.7)^7(0.3)^1 + (8C8)(0.7)^8(0.3)^0= 0.4907 ≈ 0.49Therefore, the probability that 7 or more cars were traveling faster than 70 miles per hour is 0.49. Therefore, the answer is E. 0.49.

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Answer:

20%

Step-by-step explanation:

The total number of candies in the bowl is 25. There are 5 orange candies. In order to find the total amount, simply divide the amount of candies in the given color (orange) and divide it by the total number of candies (25) giving you a result of 0.20, which can also be written as 20/100 or 20%

Answer:

1= 27
2= 2

Step-by-step explanation:

y=2x

Swap sides so that all variable terms are on the left hand side.

2x=y

Divide both sides by 2.

2

2x

​=

2

y

Dividing by 2 undoes the multiplication by 2.

x=

2

y

​

The velocity vector of the particle when it passes through the point (-3,4) is determined as (-4, 8/3) m/s.

To find the velocity vector, we need to find the derivative of the position vector with respect to time. Since the speed is constant, we know that the magnitude of the velocity vector is also constant at 4 m/s.

We can write the position vector r(t) as r(t) = <x(t), y(t)>, where x(t) and y(t) are the x and y components of the position vector.

Given the equation for the path of the particle, we have:

y(t) = (x(t)²/9) + 3

Taking the derivative of both sides with respect to time t gives:

dy/dt = (2/9)x·dx/dt

We can now solve for dx/dt:

dx/dt = (9/2)dy/dt · (1/x)

We know that the speed of the particle is 4 m/s, so the magnitude of the velocity vector is:

|v| = √((dx/dt)² + (dy/dt)²) = 4 m/s

Substituting the values we know:

4 = √((dx/dt)² + (dy/dt)²)

Squaring both sides:

16 = (dx/dt)² + (dy/dt)²

Since we want the velocity vector at the point (-3,4), we can substitute these values into our equations:

y = (x²/9) + 3

y = ((-3)²/9) + 3 = 4

So the particle passes through the point (-3,4). Substituting these values into our equations, we get:

dy/dt = (2/9)x·dx/dt

4 = √((dx/dt)² + (dy/dt)²)

16 = (dx/dt)² + (dy/dt)²

We can solve for dx/dt:

dx/dt = (9/2)dy/dt · (1/x)

Substituting the values we know:

4 = √((dx/dt)² + (dy/dt)²)

16 = (dx/dt)² + (dy/dt)²

dy/dt = (2/9)(-3)(-4) = 8/3 m/s

dx/dt = (9/2)(8/3)(-1/3) = -4 m/s

So the velocity vector at the point (-3,4) is:

v = (dx/dt, dy/dt) = (-4, 8/3 )m/s.

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Area of a circle = pi x r²

So taking pi as 22/7,

Area=707.14cm²

As per the compound interest, Sally have the amount in the account in 30 years is $100,620.95

The formula for compound interest is given as:

A = P(1 + r/n)ˣⁿ

Where A is the amount of money in the account after t years, P is the initial investment, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and x is the number of years.

In this case, we have P = $10,000, r = 0.084 (8.4% expressed as a decimal), n = 12 (monthly compounding), and x = 30. Substituting these values into the formula, we get:

A = $10,000(1 + 0.084/12)¹²ˣ³⁰

A = $10,000(1.007)³⁶⁰

A = $10,000(10.062)

A = $100,620.95

Therefore, after 30 years, Sally will have $100,620.95 in her bank account, to the nearest cent. This calculation illustrates the power of compound interest, as Sally's initial investment has grown by more than tenfold due to the compounding effect over 30 years.

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Variance matters a lot in various fields, and it is important to understand the concept of variance to make informed business decisions.

Variance matters a lot in the field of statistics, finance, and business. This term is generally used to determine the level of difference between the actual and the expected values or outcomes. Variance can help us understand the level of risk associated with a particular investment or business decision, and also helps us to analyze the accuracy of the predictions we make.

Example:

Suppose a stockbroker recommends that a client invests in a particular stock, and the client is considering investing in that stock. The stockbroker provides an expected return rate of 10%. However, if the actual return rate ends up being 15%, the client will be very happy, but if it ends up being only 5%, the client will be very unhappy. Thus, it is important for the stockbroker to consider the variance in the expected return rate and to provide an explanation of the level of risk associated with the investment.

Another example is when a company has a budget of $100,000 for advertising, and the marketing team spends $110,000 on advertising. The company's management can calculate the variance of $10,000 to determine whether the marketing team exceeded the budget or not. The variance can help the company to adjust its future marketing strategies to achieve the desired results.

Thus, variance matters a lot in various fields, and it is important to understand the concept of variance to make informed business decisions.

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The concentration of salt in the incoming brine is 0.185 g/L.

The rate of change in the amount of salt, dx/dt, is determined by the difference between the rate of incoming salt and the rate of outgoing salt. The rate of incoming salt is the product of the concentration of the incoming brine, c, and the flow rate of 3 liters per minute.

The rate of outgoing salt is the product of the concentration of the solution in the tank, x/40, and the flow rate of 3 liters per minute. Therefore, the formula for the rate of change in the amount of salt is:

dx/dt = 3c - 3(x/40)

To find the formula for the amount of salt, x(t), after t minutes have elapsed, we can integrate the rate of change formula with respect to time:

x(t) = ∫(3c - 3(x/40))dt

Using the initial condition that x(0) = 0, we can solve for the constant of integration and obtain the formula:

x(t) = 120c - 3ct - (3/40)x(t)

Rearranging the terms and solving for x(t), we get:

x(t) = (120c - 3ct)/(1 + 3/40)

Finally, to find the concentration of salt in the incoming brine, c, given that x(15) = 20, we can plug in the values of t and x(t) into the formula and solve for c:

20 = (120c - 3c(15))/(1 + 3/40)

Simplifying and solving for c, we get:

c = (20 + 900/40)/(120 - 45)

c = 0.185 g/L

Therefore, the concentration of salt in the incoming brine is 0.185 g/L.

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Answer: La compañera de Eduardo, Francisca, está equivocada.

Step-by-step explanation:

Para determinar quién tiene la razón, es necesario conocer la cantidad de pegamento necesaria para cubrir todos los tarugos. Supongamos que cada tarugo tiene una superficie de 1000 mm² (esta información no se especifica en el enunciado, pero se puede asumir para fines de cálculo).

Entonces, la superficie total de los 20 tarugos sería:

20 tarugos x 1000 mm²/tarugo = 20,000 mm²

Para cubrir completamente esta superficie, Eduardo necesita una cantidad de pegamento igual a 20,000 mm².

Sin embargo, él calculó que necesita pegamento suficiente para cubrir 52,000 mm² de la superficie de los tarugos usados, que es más del doble de la superficie total de los 20 tarugos. Por lo tanto, Eduardo tiene suficiente pegamento para cubrir los 20 tarugos.

3 ft-lb of work is needed to stretch it 3 inches beyond the 12ft-lb natural length of the spring.

Here, the integral formula for work and springs should be set up first:

When we look at the integral work for the spring, we know that,

W = [tex]\int\limits^a_b {x} \, dx[/tex] kx dx = k [tex]\int\limits^a_b {x} \, dx[/tex]

we know that,

W = 12,

a = 0,

b = 1

therefore, when we substitute these values in the formula we get,

12 = k [x²/2]¹₀

12 = k (1/2 - 0)

24 = k

now, 3 inches = 1/4th foot = b

so when we substitute this value again with k in the formula we get,

W = [tex]\int\limits^3_0 24 {x} \, dx[/tex]

= 24x²/2

= 12(1/4)²

= 3 ft-lbs

Therefore, we know that when the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, 3 ft-lb is needed to stretch it 3 inches beyond its natural length.

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We can conclude after answering the provided question that So the perimeter of TUJ4V is 35, but with the information provided, we cannot determine the perimeter of PQRS.

A two-dimensional geometric shape's perimeter is the entire length of the borderline or outer edge. It is the total length of the shape's parties or edges. For example, the perimeter of a square is determined by dividing the lengths of all four edges of the square. Similarly, to calculate the outer wall of a rectangle, add the lengths of adjacent sides and then multiply by two because there are two different sets of adjacent sides.

To calculate the perimeter of PQRS, we must know the lengths of all four quadrilateral sides. However, we are only given one side's length, which is 12.

We know that TUJ4V has a perimeter of 35, so:

Circumference = 2x + 2y = 35

[tex]x + y = 17.5\\y = 15\\x + 15 = 17.5\\x = 2.5[/tex]

As a result, the length of each side of TUJ4V is as follows:

TU = JW = x = 2.5

TV = U4 = y = 15

The perimeter of TUJ4V is as follows:

2x + 2y = 2(2.5) + 2(15) = 35 Perimeter

So the perimeter of TUJ4V is 35, but with the information provided, we cannot determine the perimeter of PQRS.

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Answer:

the answer is 17

Step-by-step explanation:

you get this by adding all numbers together 5+7+5+7+10=34 you then to make the total and devide it by 2 to get the median

34/2=17

Sarah bought 5 first class tickets for 1030$ each and 9 coach tickets for 120$ each.

Let's use variables C and F to denote the quantity of coach and first class tickets purchased.

We may deduce from the problem:

C + F = 14 (because Sarah is one among the 14 persons) 

The overall cost of the tickets is as follows:

120C + 1030F = 6230

The first equation may be used to solve for one of the variables in terms of the other:

C = 14 - F

When we plug this into the second equation, we get:

120(14 - F) + 1030F = 6230

When we simplify this equation, we get:

1680 - 120F + 1030F = 6230

910F = 4550

F = 5

So Sarah purchased five first-class tickets. The first equation may be used to calculate the number of coach tickets:

C + 5 = 14

C = 9

As a result, Sarah purchased nine coach tickets.

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Answer: [tex]a=\frac{1}{3}[/tex]

Step-by-step explanation: