Calculate the standard score of the given X value, X = 77.4 where µ = 79.2 and σ = 74.4 and indicate on the curve where z will be located. Round the standard score to two decimal places.

The 95% confidence interval for this estimate is (2.0, 6.5).

The 95% confidence interval for this estimate is (1.01, 1.04).

The confidence interval is (0.85, 9.8).

The confidence interval is (0.92, 1.08).

The confidence interval is (0.05, 1.05).

a) The rate of disease is 3.5 times higher in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate is (2.0, 6.5).

b) The rate of disease is 1.02 times higher in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate is (1.01, 1.04).

c) The rate of disease is 6.0 times higher in the exposed group compared to the nonexposed group. However, the 95% confidence interval for this estimate is wide and includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.85, 9.8).

d) The rate of disease is 0.97 times lower in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.92, 1.08).

e) The rate of disease is 0.15 times lower in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.05, 1.05).

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Answer:

  x ≈ 2.467

Step-by-step explanation:

You want the solution to 5^(3x -2) = 7^(x +2).

Logarithms turn an exponential problem into a linear problem. Taking logs, we have ...

  (3x -2)·log(5) = (x +2)·log(7)

  x(3·log(5) -log(7)) = 2(log(7) +log(5)) . . . . . separate variables and constants

  x = log(35²)/log(5³/7) = log(1225)/log(125/7) . . . . divide by x-coefficient

  x ≈ 2.46693

__

Additional comment

A graphing calculator can solve this nicely as the x-intercept of the function f(x) = 5^(3x-2) -7^(x+2). Newton's method iteration is easily performed to refine the solution to calculator precision.

For some function f(x), suppose that for a certain interval [a, b], we have:
∫(f(x))dx from a to b = 1
And for another interval [c, d], we have:
∫(f(x))dx from c to d = 10

In mathematics, a (real) interval is a set of real numbers that includes all the real numbers between two numbers in the set. For example, the set x of numbers satisfying 0 ≤ x ≤ 1 is the range containing 0, 1, and every number in between. Other examples of ranges are the set of numbers such as 0 < x < 1, the set of all real numbers {R}, the set of negative numbers, positive real numbers, free space, and a singular (similar sets).

Real numbers play an important role together because they are the simplest numbers whose "length" (or "measure" or "size") is easy to define. The concept of measure can be extended to more complex real numbers, giving rise to the Boral measure and eventually the Lebesgue measure.

To find the values of other integrals involving this function, you would need to either use additional information about the function or be provided with the specific integral expressions and interval limits.
For some function f(x), suppose that for a certain interval [a, b], we have:
∫(f(x))dx from a to b = 1
And for another interval [c, d], we have:
∫(f(x))dx from c to d = 10


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(a)  The probability of rolling at least one 6 is approximately 0.665, or about 66.5%.

(b)  The probability of rolling 5 different numbers is approximately 0.0772, or about 7.72%.

(a) To calculate the probability of rolling at least one 6 in 6 rolls of a fair die, we can use the complement rule: the probability of the complement (rolling no 6s) is easier to calculate, and then we subtract that from 1.

The probability of rolling no 6s in a single roll is 5/6, so the probability of rolling no 6s in 6 rolls is (5/6)^6.

Therefore,  probability of rolling at least one 6 in 6 rolls is:

1 - (5/6)^6 ≈ 0.665

So the probability of rolling at least one 6 is approximately 0.665, or about 66.5%.

(b) To calculate the probability of rolling 5 different numbers in 6 rolls of a die, we can use the formula for combinations. There are 6 possible numbers that could be rolled first, 5 possible numbers that could be rolled second (since we want 5 different numbers), and so on down to 2 possible numbers that could be rolled fifth.

For the sixth roll, any of the 5 previous numbers would result in 5 different numbers, so there are 5 choices. Therefore, the total number of ways to roll 5 different numbers is:

6 × 5 × 4 × 3 × 2 × 5 = 3600

To find the probability, we divide this by the total number of possible outcomes for rolling a die 6 times, which is 6^6 = 46656. Therefore, the probability of rolling 5 different numbers in 6 rolls of a die is:

3600/46656 ≈ 0.0772

So the probability of rolling 5 different numbers is approximately 0.0772, or about 7.72%.

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The sum of the resulting limits of integration is:

a + b + 0 + Jo = a + b + Jo

Assuming the limits of integration for the first integral are 0 to Jo for y and some limits for x.

The limits for the second integral are also 0 to Jo for y and the same limits for x, we can combine the integrals as follows:

3∫∫ f(x,y) dy dx + 3∫∫ f(x,y) dy dx

= 3∫∫ f(x,y) + f(x,y) dy dx (by combining the two integrals)

= 6∫∫ f(x,y) dy dx

Now, to switch the order of integration, we need to express the limits of integration of y in terms of x. Let's assume the limits of integration for x are a to b:

6∫∫ f(x,y) dy dx = 6∫[a,b]∫[0,Jo] f(x,y) dy dx

We can integrate with respect to y first, then with respect to x, so:

6∫[a,b]∫[0,Jo] f(x,y) dy dx = 6∫[a,b] (∫[0,Jo] f(x,y) dy) dx

The limits of integration for y are constant, so we can take them out of the inner integral:

6∫[a,b] (∫[0,Jo] f(x,y) dy) dx = 6∫[a,b] f(x,y) * Jo|0 dx

The limits of integration for x are a to b, and the limits for y are 0 to Jo, so the sum of the resulting limits of integration is:

a + b + 0 + Jo = a + b + Jo

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The standard deviation of the random variable is approximately 3.4082.

To compute the mean and standard deviation of the given discrete probability distribution, we need to use the following formulas:

Mean (μ) = ∑ [xi * P(xi)]

Variance (σ^2) = ∑ [(xi - μ)^2 * P(xi)]

Standard deviation (σ) = sqrt(σ^2)

where xi represents each possible value of the random variable and P(xi) represents the probability of each value.

Using the given probability distribution, we can compute the mean as:

Mean (μ) = (-5 * 0.14) + (-4 * 0.03) + (0 * 0.17) + (3 * 0.30) + (4 * 0.23) + (5 * 0.16) = 1.39

Therefore, the mean of the random variable is 1.39.

To compute the variance, we first need to compute the squared deviation of each value from the mean. Using the formula for variance, we get:

Variance (σ^2) = (-5 - 1.39)^2 * 0.14 + (-4 - 1.39)^2 * 0.03 + (0 - 1.39)^2 * 0.17 + (3 - 1.39)^2 * 0.30 + (4 - 1.39)^2 * 0.23 + (5 - 1.39)^2 * 0.16 = 11.6109

Finally, we can compute the standard deviation by taking the square root of the variance:

Standard deviation (σ) = sqrt(11.6109) = 3.4082

Therefore, the standard deviation of the random variable is approximately 3.4082.

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Geometric mean (G(n)): It is defined as the square root of the product of all the positive factors of n. In other words, G(n) = √(f1 * f2 * f3 * ... * fn), where fi represents the positive factors of n.

Arithmetic mean (A(n)): It is defined as the sum of all the positive factors of n divided by the total number of factors. In other words, A(n) = (f1 + f2 + f3 + ... + fn) / k, where fi represents the positive factors of n and k represents the total number of factors.

To determine when G(n) is an integer, we need to find values of n for which all the factors of n can be paired such that each pair multiplies to an integer.

For example, if n has four factors (f1, f2, f3, f4), and we can pair them as (f1 * f4) and (f2 * f3), such that both products are integers, then G(n) would be an integer.

This condition can be satisfied when n is a perfect square, as each factor will have an even count and can be paired.

To find values of n for which A(n) is an integer, we need to determine when the sum of all the factors of n is divisible by the total number of factors (k).

This condition can be satisfied when n has an odd number of factors, as the sum of factors will always be an integer and can be divided evenly by k.

To determine when A(n) is equal to 6 or 124, we need to find values of n for which the sum of all the factors of n is equal to 6k or 124k, where k is a positive integer.

This condition can be satisfied when n is a multiple of 6 or 124, respectively.

During the process of solving these problems, interesting conjectures may arise, such as the conjecture that G(n) is an integer if and only if n is a perfect square, and that A(n) is an integer if and only if n has an odd number of factors.

These conjectures can be proved using mathematical reasoning and properties of factors and means.

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Answer:

19

Step-by-step explanation:

f'(c) = (f(b) - f(a))/(b - a)

In this case, we have a = 1 and b = 5, so we can write:

f'(c) = (f(5) - f(1))/(5 - 1)

Solving for f(5), we get:

f(5) = f(1) + f'(c)(5 - 1)

Since we know that f '(x) ≥ 1 for 1 ≤ x ≤ 5, we have:

f(5) = f(1) + f'(c)(5 - 1) ≥ 15 + 1(5 - 1) = 19

Therefore, the smallest value that f(5) can possibly be is 19.

*IG:whis.sama_ent

We would need to collect at least 268 service times to return a 95% confidence interval whose width is at most 20 seconds.

(a) We can use the formula for a confidence interval for the mean of a normal distribution with known standard deviation:

CI = X ± z*(σ/√n)

where X is the sample mean, σ is the population standard deviation (in this case, the sample standard deviation is used as an estimate of the population standard deviation since it is known), n is the sample size, and z is the critical value from the standard normal distribution for the desired level of confidence.

For a 95% confidence interval, the critical value is z = 1.96. Plugging in the values, we get:

CI = 8 ± 1.96*(7/√100) = 8 ± 1.372

Therefore, a 95% confidence interval for the mean service time is (6.63, 9.37) minutes.

(b) To find the sample size required to return a 95% confidence interval whose width is at most 20 seconds, we can use the formula for the margin of error:

ME = z*(σ/√n)

where ME is the maximum allowed margin of error (which is 1/3 minute or 0.33 minutes in this case).

Solving for n, we get:

n = (z*σ/ME)^2

For a 95% confidence interval, the critical value is z = 1.96. Plugging in the values, we get:

n = (1.96*7/0.33)^2 ≈ 267.17

Therefore, we would need to collect at least 268 service times to return a 95% confidence interval whose width is at most 20 seconds.

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Answer:

Step-by-step explanation:

If 60% of a number is 80, find 6% of that number.

The cost to surround the region with border is $283.20.

Perimeter is the summation of the length of sides of a given figure.

The perimeter of a rectangle can be determined as;

perimeter of a rectangle = 2(length + width)

From the given question, we have to determine the width of the rectangular region.

area of rectangle = length x width

width = area of rectangle/ length

         = 7.02 x 10^3/ 1.17 x 10^2

         = 6.0 x 10^1

So that;

perimeter of the rectangular region = 2(1.17 x 10^2  + 6.0 x 10^1)

                                              = 354 feet

The cost to surround the region with a border that costs $0.80 per foot is;

354 x $0.8 = $283.2

The required cost is $283.20.

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Answer: see attached file

Step-by-step explanation:

Answer:

(x, y) (1/3, 3)

Step-by-step explanation:

substitute x with 4/9y-1

we will get

18(4/9y-1) -7y = -15

solving for y we get

y = 3

substitute y with 3 for the first eqaution

x=4/9(3)-1= 1/3

we will get x = 1/3

Answer:

Asiah it’s ego the answer is D

Step-by-step explanation:

I got it right

L ms Brenda

Answer: The answer is Sample D

Step-by-step explanation:

To test the given situation, you would use a one-sample z-test. For this test, the null and alternative hypotheses are as follows: Null Hypothesis (H₀): The population mean (µ) is equal to 10.

Mathematically, it can be written as: H₀: µ = 10, Alternative Hypothesis (H₁): The population mean (µ) is significantly less than 10. Mathematically, it can be written as:
H₁: µ < 10

You are given the sample size (n=16), the sample mean (X=8.4), and the population standard deviation (σ=3.2). To test the hypotheses at a 0.05 level of significance, you would calculate the z-score using the formula:

z = (X - µ) / (σ / √n)

Once you find the z-score, compare it to the critical value from the standard normal distribution table. If the z-score is less than the critical value, reject the null hypothesis, indicating that the population mean is significantly less than 10.

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Since f(n) is defined using a primitive recursive function (exponentiation) and follows a recursive structure, we can conclude that f(n) is primitive recursive.

To show that the function f(n) is primitive recursive, we need to demonstrate that it can be defined using basic primitive recursive functions (zero, successor, and projection functions) and can be composed or recursed using only primitive recursive function schemes.

Given the definition of f(n), we can write it as:
- f(0) = 0
- f(1) = 1
- f(2) = 2²
- f(3) = (3³)³
- ...

We can observe that f(n) is defined as a stack of exponentiation operations with the base and the exponent both being n. We can use the following recursive formula to define f(n):

- f(0) = 0


We know that exponentiation is primitive recursive, as it can be defined using multiplication, which is also primitive recursive. We can define exponentiation recursively as:

- exp(a, 0) = 1
- exp(a, b) = a * exp(a, b-1) for b > 0

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A function that would match the graph shown is f(x) = -5(x + 4)(x - 1)

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

The graph

The zeros of the graph are

x = -4; multiplicity 2

x = 1; multiplicity 1

So, we have

f(x) = a(x + 4)^2(x - 1)

The function intersects with the y-axis at y = 80

So, we have

a(0 + 4)^2(0 - 1) = 80

Evaluate

x = -5

So, we have

f(x) = -5(x + 4)(x - 1)

Hence, the equation is f(x) = -5(x + 4)(x - 1)

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The test statistic is 1.177, and the p-value is approximately 0.120, which is greater than the significance level of 0.05, indicating that there is not enough evidence to conclude that the proportion of defective circuit boards is greater than 3%.

To test the hypothesis that more than 3% of circuit boards are defective, we can use a one-tailed test with the following null and alternative hypotheses:

[tex]H_0[/tex]: p ≤ 0.03 (proportion of defective boards is less than or equal to 3%)

[tex]H_a[/tex]: p > 0.03 (proportion of defective boards is greater than 3%)

where p is the true proportion of defective boards in the population.

To calculate the test statistic, we can use the following formula:

z = (p-cap - p0) / √(p0(1-p0)/n)

where p is the sample proportion of defective boards, p0 is the hypothesized proportion (0.03), and n is the sample size.

In this case, we have p-cap = 0.04, p0 = 0.03, and n = 100, so the test statistic is:

z = (0.04 - 0.03) / √(0.03(1-0.03)/100) = 1.177

To determine the p-value associated with this test statistic, we can use a standard normal distribution table or a calculator to find the probability of observing a z-value of 1.177 or greater under the null hypothesis. This probability is approximately 0.120, which is the area to the right of z = 1.177 on the standard normal distribution curve.

Since this p-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis.

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The probability of obtaining a sum less than or equal to 4 is:  1/12

An experiment consists of tossing two ordinary dice and adding their numbers together. To determine the probability of obtaining a sum of 8, we need to first count the number of ways we can get a sum of 8. We can do this by listing all the possible combinations of dice rolls that add up to 8:

2+6, 3+5, 4+4, 5+3, 6+2

So there are 5 ways to get a sum of 8.

Next, we need to determine the total number of possible outcomes for this experiment. Each die has 6 sides, so there are 6 x 6 = 36 possible outcomes.

Therefore, the probability of obtaining a sum of 8 is:

Number of ways to get a sum of 8 / Total number of possible outcomes = 5/36

Now let's determine the probability of obtaining a sum less than or equal to 4. We can use the same method as before:

1+1, 1+2, 2+1

So there are 3 ways to get a sum less than or equal to 4.

The probability of obtaining a sum less than or equal to 4 is:

Number of ways to get a sum less than or equal to 4 / Total number of possible outcomes = 3/36 = 1/12

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The angles must be of 90°, using that, we will find that:

x = 47

y = 3

If the two lines AB and CD are perpendicular, then all the formed angles must be 90° angles.

Then we need to have:

2x - 4 = 90

34y - 12 = 90

Solving these linear equatons we will get:

2x = 90 + 4

2x = 94

x = 94/2 = 47

And the other linear equation gives:

34y - 12 = 90

34y = 90 + 12

34y = 102

y = 102/34

y = 3

These are the two values.

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The area of each isosceles triangle is [tex]120cm^{2}[/tex]. Some of these triangles are used to make a larger triangle.

To solve the problem, we need to know more information about how the smaller triangles are arranged to form the larger triangle. However, we can make some observations based on the given information.

Since the isosceles triangle has a base of 16cm and a height of 15cm, we can use the formula for the area of a triangle:

Area [tex]= (1/2)[/tex]x base x height

Area[tex]= (1/2)[/tex] x [tex]16cm[/tex] x [tex]15cm[/tex]

Area [tex]= 120cm^{2}[/tex]

So the area of each isosceles triangle is [tex]120cm^{2}[/tex].

If we know the number of isosceles triangles used to make the larger triangle and how they are arranged, we could find the dimensions and area of the larger triangle using geometric properties and formulas.

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Por lo tanto, el radio de la circunferencia cuyo perímetro es de 18 metros mide aproximadamente 2.8648 metros.

Hola, entiendo que quieres saber cuánto mide el radio de una circunferencia cuyo perímetro es de 18 metros. Para

resolver este problema, utilizaremos la fórmula del perímetro de una circunferencia, que es P = 2πr, donde P es el

perímetro y r es el radio.

Paso 1: Identificar el perímetro (P) y la fórmula del perímetro de una circunferencia.

P = 18 metros

Fórmula: P = 2πr

Paso 2: Despejar la variable r (radio) de la fórmula.

Para hacer esto, dividiremos ambos lados de la ecuación por 2π.

r = P / 2π

Paso 3: Sustituir el valor de P en la ecuación despejada y calcular el valor de r.

r = 18 / (2 × π)

r ≈ 18 / 6.2832 (aproximadamente, porque 2 × π ≈ 6.2832)

r ≈ 2.8648

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To calculate the probability of having a positive profit after 6 months, we need to consider two scenarios: the stock price being higher than 42 and the stock price being lower than 42.

If the stock price is higher than 42, then the profit will be the absolute difference between the stock price and 42. Let's call this difference "x". In this case, the profit will be x, since the call option will be in the money and the put option will be out of the money.

If the stock price is lower than 42, then the profit will be the absolute difference between 42 and the stock price. Let's call this difference "y". In this case, the profit will be y, since the put option will be in the money and the call option will be out of the money.

To calculate the probability of having a positive profit, we need to find the probability of the stock price being higher than 42, multiplied by the expected profit in that scenario, plus the probability of the stock price being lower than 42, multiplied by the expected profit in that scenario.

Let's assume that the stock price follows a normal distribution with a mean of 42 and a standard deviation of σ. The probability of the stock price being higher than 42 can be calculated as follows:

P(X > 42) = 1 - P(X < 42) = 1 - Φ((42 - 42)/σ) = 1 - Φ(0) = 0.5

Where Φ is the standard normal cumulative distribution function.

The expected profit in this scenario is x, which can be calculated as follows:

E(x) = ∫[42, +∞] x * f(x) dx

Where f(x) is the probability density function of the normal distribution.

Since the normal distribution is symmetric around the mean, we can assume that the expected profit in the lower scenario is the same as in the upper scenario, but with a negative sign:

E(y) = -E(x)

Therefore, the expected total profit is:

E(x+y) = E(x) + E(y) = 0

Since the expected total profit is zero, the probability of having a positive profit is the same as the probability of having a negative profit. Therefore, the answer is:
B At least 0.35 but less than 0.40

To answer your question, follow these steps:

Step 1: Understand the problem
You have bought a 6-month straddle that pays the absolute difference between the stock price after 6 months and 42. You need to calculate the probability of having a positive profit after 6 months.

Step 2: Identify the profit condition
For a positive profit, the payout should be greater than the cost of the straddle. Since we do not have the cost of the straddle, we cannot determine the exact probability of having a positive profit after 6 months.

However, we can infer that a higher probability of the stock price deviating significantly from 42 after 6 months will increase the likelihood of a positive profit. Unfortunately, without more information on the stock price distribution or the cost of the straddle, we cannot provide a definite answer within the given answer choices (A, B, C, D, or E).

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The sample size must be at least 12.

The formula for the standard error of the mean is:

[tex]SE = \sigma / \sqrt(n)[/tex]

where SE is the standard error, σ is the population standard deviation, and n is the sample size.

In this case, we are given that the population mean is 100 and the population variance is 25. Therefore, the population standard deviation is:

[tex]\sigma = \sqrt(\sigma^2) = \sqrt(25) = 5[/tex]

We want the standard error of the mean to be 1.5, so we can set up the following equation:

[tex]1.5 = 5 / \sqrt(n)[/tex]

Solving for n, we get:

[tex]\sqrt(n) = 5 / 1.5[/tex]

[tex]\sqrt(n) = 3.33[/tex]

[tex]n = (3.33)^2[/tex]

n = 11.0889

Since we need a whole number of samples, we can round up to the next integer and say that the sample size must be at least 12.

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The four possible times that Car B could take to complete one lap are 2 seconds, 3 seconds, 5 seconds, and 15 seconds.

150 is represented as a factor of 2, 3, 5 and 5. As 150/t is considered as an integer, t should be a divisor of 150. Hence, we can consider the potential values for t which are:

- t=2 seconds (because 150/2 equals to 75, making it an integer)

- t=3 seconds (as 150/3 equates to 50, also forming an integer)

- t=5 seconds (since 150 divided by 5 produces 30 which is another integer)

- t=15 seconds (seeing that when dividing 150 with 15 gives us 10 which is likewise an integer).

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The conjugate of the expression square root (4x² + 3x) - 2x is √(4x² + 3x) + 2x.

The conjugate of a binomial is found by taking the inverse operation of the sign in between the terms.

Here given a binomial.

√(4x² + 3x) - 2x

Here, √(4x² + 3x) is one term and 2x is the other term.

The operation in between is minus sign.

Inverse operation of minus is plus sign.

So the conjugate is √(4x² + 3x) + 2x.

Hence the conjugate of the given expression is √(4x² + 3x) + 2x.

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After calculating, we get that the probability that a randomly selected moped will have a maximum speed greater than 51.3 km/h is approximately 0.0051 or 0.51%.Hi, I'm happy to help with your question involving probability and maximum speed.

To get the probability that a randomly selected moped will have a maximum speed greater than 51.3 km/h, given that the maximum speed follows a normal distribution with a mean of 46.8 km/h and a standard deviation of 1.75 km/h, follow these steps:
Step:1. Calculate the z-score for 51.3 km/h:
  z = (x - mean) / standard deviation
  z = (51.3 - 46.8) / 1.75
  z ≈ 2.57
Step:2. Look up the probability of the z-score in a standard normal distribution table or use a calculator that can compute this probability. The table or calculator will give you the probability that a moped has a speed less than or equal to 51.3 km/h.
Step:3. Since we want to find the probability of a moped having a speed greater than 51.3 km/h, subtract the obtained probability from 1:
  P(x > 51.3) = 1 - P(x ≤ 51.3)
After calculating, we find that the probability that a randomly selected moped will have a maximum speed greater than 51.3 km/h is approximately 0.0051 or 0.51%.

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264 is the value of x³ - 1/x³ in linear equation.

What is a linear equation in mathematics?

A linear equation is an algebraic equation of the form y=mx b. which contains only a constant and  first-order (linear) term, where m is the slope and b is the y-intercept.

                      Sometimes the above is called a "linear equation in two variables" where y and x are the variables. A linear equation can have more than one variable. If a linear equation has two variables,  it is called a bivariate linear equation, etc.

x - 1/x = 7

x³ - 1/x³ = ?

(x - 1/x )³ = 7³

x³  - 1/x³ - 3 * x * 1/x (x - 1/x) = 243

x³  - 1/x³ - 3 * 7 = 243

x³  - 1/x³ - 21 = 243

x³  - 1/x³ = 243 + 21

x³  - 1/x³ =  264

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Answer:

We can evaluate this integral using trigonometric substitution. Let x = 8 sin(θ). Then dx = 8 cos(θ) dθ. Substituting gives us:

```

∫√1−64x2dx = ∫√1−64(8sin(θ))^2(8cos(θ))dθ = ∫√1−64sin^2(θ)cos(θ)dθ

```

We can now use the identity sin^2(θ) + cos^2(θ) = 1 to simplify this integral:

```

∫√1−64sin^2(θ)cos(θ)dθ = ∫√cos^2(θ)cos(θ)dθ = ∫8cos^3(θ)dθ

```

We can now integrate using the power rule:

```

∫8cos^3(θ)dθ = 8cos^4(θ)/4 + C = 2cos^4(θ) + C

```

To reverse the substitution, we need to solve for θ in terms of x. We have:

```

x = 8sin(θ)

```

```

sin(θ) = x/8

```

```

θ = sin^-1(x/8)

```

Substituting gives us:

```

2cos^4(θ) + C = 2cos^4(sin^-1(x/8)) + C

```

```

= 2(1 - sin^2(sin^-1(x/8)))^2 + C

```

```

= 2(1 - (x/8)^2)^2 + C

```

```

= 2(1 - x^2/64)^2 + C

```

Therefore, the integral is equal to:

```

∫√1−64x2dx = 2(1 - x^2/64)^2 + C

```

Step-by-step explanation:

The integral ∫√(1-64x²)dx is equal to (1/128)(asin(8x) + 8x√(1-64x²) + C), where C is the constant of integration.

To solve this integral, we use trigonometric substitution. Let x = (1/8)sin(θ), so dx = (1/8)cos(θ)dθ. The integral becomes ∫√(1-64((1/8)sin(θ))²)(1/8)cos(θ)dθ = ∫(1/8)cos²(θ)dθ.

Now, apply the power-reduction formula: cos²(θ) = (1+cos(2θ))/2. The integral becomes ∫(1/16)(1+cos(2θ))dθ. Integrate with respect to θ: (1/16)(θ+(1/2)sin(2θ)) + C. Convert back to x using θ = asin(8x): (1/128)(asin(8x) + 8x√(1-64x²)) + C.

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The autonomous differential equation dp/dt = p(a + bp) has a solution that is increasing if -a/b < P < 0 (option c). This is because the rate of change of P (dp/dt) is positive when -a/b < P < 0, leading to an increasing solution.

The given differential equation is autonomous, which means it does not explicitly depend on time 't'. We can find the equilibrium solutions by setting dp/dt = 0. So, we have p(a+bp) = 0, which gives p = 0 and p = -a/b as equilibrium solutions.

Now, we can analyze the behavior of the solution by considering the sign of dp/dt for different values of p.

For p < -a/b, we have a+bp < 0, which implies dp/dt < 0. So, the solution is decreasing in this region.

For -a/b < p < 0, we have a+bp > 0, which implies dp/dt > 0. So, the solution is increasing in this region.

For p > 0, we have a+bp > 0, which implies dp/dt > 0. So, the solution is increasing in this region.

Therefore, the correct answer is (c) increasing if -a/b < p < 0.

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