Use Fermat's little theorem to find 82035 mod 17

(a)The power is 1/2, which is less than 1 the improper integral ∫(1 / √(x)) dx from a to infinity diverges.

b)The value of the integral of ∫(1 / √(x)) dx from 1 to infinity is ln(√2 + 1).

c) The general solution to the differential equation dy/dx = (2x - 2) / (x² + 3x - 2) is y = ln|x - 1||x + 2| + C, where C is a constant.

To determine if the improper integral converges or diverges, to evaluate the integral:

∫(1 / √(x)) dx from a to infinity

This integral represents the area under the curve of the function 1/√(x) from x = a to x = infinity.

To determine convergence or divergence, the p-test for improper integrals. For the p-test, the power of x in the denominator, which is 1/2.

If the power is greater than 1, the integral converges. If the power is less than or equal to 1, the integral diverges.

To confirm the result using a trigonometric substitution, let's substitute x = tan²(t):

√(x) = √(tan²(t)) = tan(t)

dx = 2tan(t)sec²(t) dt

substitute these values into the integral:

∫(1 / √(x)) dx = ∫(1 / tan(t))(2tan(t)sec²(t)) dt

= ∫2sec(t) dt

To determine the limits of integration. Since the original integral was from 1 to infinity, to find the corresponding values of t.

When x = 1, tan²(t) = 1, which implies tan(t) = ±1. the positive value because dealing with positive values of x.

tan(t) = 1 when t = π/4

The integral with the appropriate limits:

∫(1 / √(x)) dx = ∫2sec(t) dt from t = 0 to t = π/4

Evaluating the integral:

∫2sec(t) dt = 2ln|sec(t) + tan(t)| from t = 0 to t = π/4

Plugging in the limits:

2ln|sec(π/4) + tan(π/4)| - 2ln|sec(0) + tan(0)|

ln(√2 + 1) - ln(1)

ln(√2 + 1)

The given differential equation is:

dy/dx = (2x - 2) / (x^2 + 3x - 2)

To find the general solution,  by factoring the denominator:

dy/dx = (2x - 2) / [(x - 1)(x + 2)]

decompose the fraction into partial fractions:

dy/dx = A/(x - 1) + B/(x + 2)

To find the values of A and B, both sides of the equation by the denominator (x - 1)(x + 2):

2x - 2 = A(x + 2) + B(x - 1)

Expanding the right side and collecting like terms:

2x - 2 = Ax + 2A + Bx - B

Matching the coefficients of x and the constant terms on both sides, the following system of equations:

A + B = 2 (coefficient of x)

2A - B = -2 (constant term)

Solving this system of equations,  A = 1 and B = 1.

Substituting these values back into the partial fraction decomposition:

dy/dx = 1/(x - 1) + 1/(x + 2)

integrate both sides with respect to x:

∫ dy = ∫ (1/(x - 1) + 1/(x + 2)) dx

Integrating each term separately:

y = ln|x - 1| + ln|x + 2| + C

Combining the logarithmic terms using properties of logarithms:

y = ln|x - 1||x + 2| + C

This is the general solution to the given differential equation.

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The height of the radio tower to the nearest tenth of a metre is 6.9 m.

Let the height of the radio tower be h metresFrom Sun's point of view, the top of the radio tower is right-angled to the horizontal line through his eye. This implies that the length of the radio tower is opposite the angle of elevation.

Thus, the distance of Sun from the radio tower is equal to the length of the adjacent side of the right-angled triangle formed.

Thus, from the tangent ratio:tan 31° = h / 14.5 + 1.6 (since Sun's eye level is 1.6m above the ground)h = (14.5 + 1.6) tan 31° = 6.9 m (to one decimal place)From Natalia's point of view,

the radio tower makes an acute angle with the line of sight from her eye to the top of the radio tower. This implies that the length of the radio tower is adjacent to the angle of elevation.

Thus, the distance of Natalia from the radio tower is equal to the length of the hypotenuse of the right-angled triangle formed.

Thus, from the sine ratio:sin 89° = h / 14.5 - 1.6 (since Natalia's eye level is 1.6m above the ground)

h = (14.5 - 1.6) sin 89° = 12.9 m (to one decimal place)

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Answer:

37.999749573966

Step-by-step explanation:

Brainliest?

The number of iterations required is 6.

Given a starting guess of [90, yo] = [1, 2], Newton's method for optimization takes 6 iterations to find a minimum of the function f(x, y) = 12a^2 – 7x + 7 +10y – xy to a tolerance of 10^-8.Step-by-step explanation:

Newton's method is an iterative process to approximate the roots of an equation or optimization of a function.

To solve this problem, we need to apply Newton's method to the function f(x, y) = 12a^2 – 7x + 7 +10y – xy as follows:

Given a starting guess of [90, yo] = [1, 2], we can find the solution using the following formula:xn+1 = xn - f'(xn)/f''(xn)where xn is the current guess, f'(xn) is the derivative of f at xn, and f''(xn) is the second derivative of f at xn.

Here, f(x, y) = 12a^2 – 7x + 7 +10y – xy, so we have (x, y) = [-7 - y, 10 - x]f''(x, y) = [0, -1; -1, 0]

We need to apply this formula until the difference between xn and xn+1 is less than or equal to the tolerance of 10^-8.

This means that we need to keep iterating until |xn+1 - xn| <= 10^-8.

Given the assumption of quadratic convergence, we can also calculate the number of iterations required as follows:|xn+1 - xn| ≈ |xn - xn-1|^2/|xn+1 - 2xn + xn-1|

Using this formula, we can calculate the number of iterations required to reach the tolerance of 10^-8.

Here are the results of the first six iterations:x1 = [1, 2]x2 = [4.125, 2.9]x3 = [4.223382352941176, 2.979411764705882]x4 = [4.223639551849474, 2.979707510729614]x5 = [4.223639551862697, 2.979707510767132]x6 = [4.223639551862697, 2.979707510767132]

We can see that the difference between x5 and x6 is less than 10^-8, so we have found a solution to the desired tolerance.

Therefore, the number of iterations required is 6.

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Answer:

what?

Step-by-step explanation:

Answer:

3/20

Step-by-step explanation:

1/1-1/4-3/5= (money spent)

1×4×5/(1×4×5)-1×1×5/(1×4×5)-3×1×4/(1×4×5)

20/20-5/20-12/20

(20-5-12)/20=3/20

Answer:

$y - $0.85

Step-by-step explanation:

y represents how much money he had.

[tex]\frac{3}{5}+\frac{1}{4} =\frac{17}{20}[/tex]

$y - $0.85

Answer: May

Step-by-step explanation: 9 is greater than 8 don't worry about the signs

Answer:

Only the mean increased.

Step-by-step explanation:

The mean is the total divided by the number of scores, and since adding 21 will increase the total score significantly, the mean increases.

The median is still 7.

The translation for this problem is classified as follows:

The four translation rules are defined as follows:

For this problem, we have a translation 2 units left, which is an horizontal translation, and then a translation 4 units down, which is a vertical translation.

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Divide total miles by number of days:

151.2 / 24 = 6.3 miles per day

Answer:

lets divide the figure into two parts.

triangle base=2ft+2ft+6ft

triangle base=10ft

area of triangle = 1/2×base×height

area of triangle = 1/2×10ft×12ft

area of triangle =60ft²

area of square=side²

area of square=(6ft)²

area of square=36ft²

Answer:

The triangle is isosceles.

Step-by-step explanation:

This means that one angle's sine is the same as the other's cosine.

it is True ..............................................

Answer:

True

Step-by-step explanation:

2-72+3+4x=ax+b

73 + 4x = ax + b

Since there are 3 variables in this equation you need 3 equations. Since there is only 1 you cannot continue solving this equation after you simplified.

Hope this helps!

Answer:

48 degrees

Step-by-step explanation:

Supplementary angles add up to 180 degrees.

If one angle is 132 degrees, the other must be 48 degrees.

Answer:

$20

Step-by-step explanation:

Answer:

c ) 2480 cm²

Step-by-step explanation:

Surface Area of a Triangular prism =

S = bh + lb + 2ls

b = 30cm

h = 8cm

s = 17 cm

l = 35cm

The surface area = 30cm × 8 cm + 35 × 30 cm + 2(35 × 17)

= 240 cm² + 1050 cm² + 1190 cm²

= 2480 cm²

Option c is the correct option

Answer:

50/100, simplified to 1/2

Step-by-step explanation:

50% means half of 1.

1/2=.5 which is half of 1.

1=100%

.5=50%

1/2=50%

Answer:

[tex] \displaystyle \frac{1}{2} [/tex]

Step-by-step explanation:

we are given a parcentage

we want to convert it into fraction

remember that,

[tex] \displaystyle \% = \frac{1}{100} [/tex]

therefore

substitute:

[tex] \displaystyle 50 \times \frac{1}{100} [/tex]

reduce fraction:

[tex] \displaystyle \cancel{50} \times \frac{1}{ \cancel{100} \: ^{2} }[/tex]

[tex] \displaystyle 1 \times \frac{1}{2} [/tex]

simplify multiplication:

[tex] \displaystyle \frac{1}{2} [/tex]

hence,

[tex] \displaystyle 50\% = \frac{1}{2} [/tex]

Answer:

94.25 feet.

Step-by-step explanation:

Answer:

A would be the answer I would choose

Answer:

It increases by 2.5

Step-by-step explanation:

yeah

There is no polynomial attached ; considering the hypothetival

Answer:

x³ = leading term.

2 = leading Coefficient

1 = constant term.

Step-by-step explanation:

Considering an hypothetical situation ;

For any polynomial such as 2x³ + 3x + 1

The polynomial is represented as : ax³ + bx + c

Where a = leading Coefficient ;

x³ = leading term

c = constant term

Therefore ;

The leading term in this hypothetical question is :

x³ = leading term.

2 = leading Coefficient

1 = constant term.

The probability that a rider must wait between 1 minute and 1.4 minutes at the elevator can be determined by calculating the proportion of the uniform distribution that falls within this time interval.

The given information states that the waiting time at the elevator follows a uniform distribution between 30 and 200 seconds. To find the probability of waiting between 1 minute and 1.4 minutes, we need to convert these time values to seconds.

1 minute is equal to 60 seconds, and 1.4 minutes is equal to 84 seconds. Therefore, we are interested in finding the probability that the waiting time falls between 60 seconds and 84 seconds.

Since the waiting time follows a uniform distribution, the probability of waiting within a specific interval is equal to the length of that interval divided by the total length of the distribution.

The total length of the distribution is 200 seconds - 30 seconds = 170 seconds.

The length of the interval between 60 seconds and 84 seconds is 84 seconds - 60 seconds = 24 seconds.

Thus, the probability that a rider must wait between 1 minute and 1.4 minutes is 24 seconds / 170 seconds, which is approximately 0.1412 or 14.12%.

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I think it’s the third one. Hope that helps!

Answer:

the answers are B or C x>4/19

The answer is b=18. To solve, multiply 6 by 3 to get 18. This is called doing the inverse operation. Since the equation is a division equation, we would have to multiply in order to find the missing variable.

we need a sample size of at least 248 to have a probability of 0.8664 that the sample mean is within 0.8 of the population mean.

Given a population with standard deviation 8, we have to calculate the sample size required so that the probability is 0.8664 that the sample mean is within 0.8 of the population mean.To solve the problem, we have to use the formula as follows:$$n = \frac{z^2\sigma^2}{d^2}$$

Where, n = sample sizeσ = population standard deviation d = precision level z = z-score

So, z can be found using the standard normal table. In this case, we need to find the z-score that corresponds to the probability of 0.8664 plus half of the remaining probability of 1 - 0.8664, which is equal to 0.0668.Using the standard normal table, we find the z-score that corresponds to the 0.9334 probability, which is 1.48 (approximately).Now, we can substitute all the values into the formula and solve for n.$$n = \frac{z^2\sigma^2}{d^2}$$$$n = \frac{(1.48)^2 \cdot 8^2}{(0.8)^2}$$$$n = 247.15$$

Therefore, we need a sample size of at least 248 to have a probability of 0.8664 that the sample mean is within 0.8 of the population mean.

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The needed sample size is given as follows:

n = 250.

We first must use the z-score formula, as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which:

The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, and can be positive(above the mean) or negative(below the mean).

The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure represented by X in the distribution.

By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

The p-value of the z-score in this problem is given as follows, considering the symmetry of the normal distribution:

0.5 + 0.8864/2 = 0.9432.

Hence the z-score is given as follows:

z = 1.58.

Then the sample size is obtained as follows:

[tex]1.58 = \frac{0.8}{\frac{8}{\sqrt{n}}}[/tex]

[tex]\sqrt{n} = 15.8[/tex]

n = 15.8²

n = 250.

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Answer: 45

Step-by-step explanation:

The confidence coefficient for a 98% confidence interval is 2.33, indicating the number of standard deviations away from the mean.

To construct a confidence interval, we use a critical value that corresponds to the desired level of confidence. In this case, the confidence level is 98%, which means there is a 98% chance that the true population parameter falls within the confidence interval.

The critical value for a 98% confidence interval can be found using the standard normal distribution. Since the sample size is relatively small (13 measurements), we typically use the t-distribution instead. However, when the sample size is large (typically considered to be greater than 30), the t-distribution closely approximates the standard normal distribution.

For a 98% confidence level, the critical value is 2.33. This value represents the number of standard deviations away from the mean that includes 98% of the distribution.

Therefore, the correct answer is (D) 2.33 as the confidence coefficient for a 98% confidence interval.

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