Problem 5. Apply the Rabin-Miller primality test to the following cases. In each case state if n probably prime or if it is composite. If possible also give factorization. (1) For n= 1729 using a = 671 (note that n-1 = 26 · 27). (2) For n= 104513 with a = 3 (you may use that n-1 = 26 · 1633)

To solve the equation \(6sin^2(x)\) + 6sin(x) + 1 = 0, we can use algebraic methods and the unit circle to determine the values of x that satisfy the equation.

1. Start by rearranging the equation to a quadratic form: \(6sin^2(x)\) + 6sin(x) + 1 = 0.

2. Notice that the equation resembles a quadratic equation in terms of sin(x). Let's substitute sin(x) with a variable, such as u: \(6u^2\) + 6u + 1 = 0.

3. Solve this quadratic equation for u. You can use the quadratic formula or factorization methods to find the values of u. The solutions are u = (-3 ± √3) / 6.

4. Since sin(x) = u, substitute back the values of u into sin(x) to obtain the values for sin(x): sin(x) = (-3 ± √3) / 6.

5. To find the values of x, we can use the inverse sine function. Take the inverse sine of both sides: x = arcsin[(-3 ± √3) / 6].

6. The arcsin function has a range of [-π/2, π/2], so the values of x lie within that range. Use a calculator to find the approximate values of x based on the values obtained in step 5.

7. Plot the obtained x-values on the graph to show the solutions of the equation \(6sin^2(x)\) + 6sin(x) + 1 = 0. The graph will illustrate the points where the curve intersects the x-axis.

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Answer:

75%

Step-by-step explanation:

80 x 0.75 = 60

80 + 60 = 140

Answer:

Approximately 11.1666666667 rows

Step-by-step explanation:

134/12= 11.1666666667

Answer:

  12 rows

Step-by-step explanation:

  (134 students) / (12 students/row) = 11 2/12 rows

The 6th-graders will need 12 rows of seats. The last row will not be filled.

Answer:

March, June, and November

Step-by-step explanation:

See attached image for the utility bills by month.  Draw a line along the $40 and $50 lines on the y axis and select the months that fall within these two lines.

Answer:

It should be 16

Answer:

18.78

Step-by-step explanation:

\(a^{2} + b^{2} = c^{2} \\8^{2} + 17^2 = c^2\\64 + 289 = 353\\\sqrt{353} = 18.78\)

Answer:

8

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

slope is rise over run 4/1 so the slope is 4 and intercept is 3.

Step-by-step explanation:

Expression = x + 2x + 7 + 3x + 1

Simplication

x + 2x + 7 + 3x + 1

6x + 8

The value of investment P at any time t is given by the function P(t) = 1500e^(2t). This equation shows that the value of investment P grows exponentially with time, with a rate of growth proportional to its instantaneous value.

The given differential equation, dP/dt=rP, implies that the instantaneous rate of change of the value of investment P is proportional to its value at any given time. Here, r is the proportionality constant, which is equal to 2. If P(0) = 1500, it means that the initial value of investment P was 1500 units.
To find the value of P at any time t, we need to solve the differential equation. Integrating both sides, we get:
ln(P) = rt + C
where C is the constant of integration. To determine the value of C, we can use the initial condition P(0) = 1500. Substituting t = 0 and P = 1500 in the above equation, we get:
ln(1500) = r(0) + C
C = ln(1500)
Thus, the equation for the value of investment P at any time t is given by:
ln(P) = 2t + ln(1500)
P(t) = e^(2t+ln(1500))
Simplifying the above equation, we get:
P(t) = 1500e^(2t)
Therefore, the value of investment P at any time t is given by the function P(t) = 1500e^(2t). This equation shows that the value of investment P grows exponentially with time, with a rate of growth proportional to its instantaneous value.

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(REMEMBER: Final label is ft.)

Hallway = (1/2 in. (width) x (2 in. (length) + 3 in. (length)) x 6 ft. (actual size)

Hallway = (1/2 in. (width) x 5 in. (length)) x 6 ft. (actual size)

Hallway = 5/2 in. (area) x 6 ft. (actual size)

Hallway = 15 ft^2 (final answer)

Kitchen = (3 in. (width) x 3 in. (length)) x 6 ft. (actual size)

Kitchen = 9 in. (area) x 6 ft. (actual size)

Kitchen = 54 ft^2 (final answer)

The slope of the line that passes through point (0,-1) and (-1,-4) is 3.

Slope is simply expressed as change in y over the change in x.

Slope m = ( y₂ - y₁ )/( x₂ - x₁ )

Given the data in the question;

Point 1( 0,-1 )

Point 2( -1,-4 )

To determine the slope of the line, plug the coordinates into the slope formula above.

Slope m = ( y₂ - y₁ )/( x₂ - x₁ )

Slope m = ( -4 - (-1) )/( -1 - 0 )

Slope m = ( -4 + 1 )/( -1 )

Slope m = ( -3 )/( -1 )

Slope m = 3

Therefore, the slope of the line is 3.

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Answer:

48in^2 or just 48 dont forget to include the inches and squared

Step-by-step explanation:

Answer:

..........

Step-by-step explanation:

Answer:

jj

Step-by-step explanation:

Answer:

(i) 88°

(ii) 23°

Step-by-step explanation:

See the picture below.

The numbers added in red are the angle measures. The numbers added in black are the order in which each angle measure was written.

ABCD is a rhombus, so all sides are congruent.

Triangle BCD is isosceles with sides DC and BC congruent, so <DBC is congruent to <BDC. From the sum of the measures of the angles of a triangle, we get m<DCB. Opposite angles of a rhombus are congruent, so m<A = m<DCB. From the sum of the measures of the angles of a triangle and from isosceles triangle ADB we get m<ADB and m<ABD. From m<DBC and supplementary angles, we get m<CBE. From isosceles triangle BCE, we get m<BCE.

Answer:

85 m²

Step-by-step explanation:

*Multiply length times width (7m x 15m)

*Find the area of the missing space by multiplying the length times the width (4m x 5m)

*Subtract the missing space area from the first calculation

\( \large \bf \implies85 \: {m}^{2} \)

= Length × width

= (15 × 7) = 105 m²

= Length × width

= (5 × 4) = 20 m²

= Area of Rectangle ABCF - Area of Rectangle DEGH

= 105 m² - 20 m² = 85 m²

Answer:

y=-17

Step-by-step explanation:

1-2(9)=1-18=-17

Answer: y = -17

Step-by-step explanation:

Using PEMDAS, you need to do the multiplication first. 2 times x is 18, because the value of x is 9. You will then get 1 - 18. This is -17, so y = -17. I hope that this helped! :)

The resulting equivalent expression is -2k + 6

Equivalent expression.

Equivalent expression means an expressions that work the same even though they look different. If we said the two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).

Given,

Here we have the expression -2k - (-5) + 1

Now, we have to combine the like terms and then we have to solve it in order to find the equivalent expression.

First, we have to write the given expression,

=> -2k - (-5) + 1

Here we know that (-) x (-) = (+), therefore, the given expression is rewritten as follows:

=> -2k + 5 + 1

Here the only like term is the constant 5 and 1, so, we have to combine these constant, the  we get the expression like the following;

=> -2k + (5 + 1)

No, we have to add the constant in order to get the equivalent expression,

=> - 2k + 6

Therefore, the equivalent expression i s -2k + 6.

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Answer:

610000

Step-by-step explanation:

Srry i misread that i thought it said round to the nearest thousandth f

After rounding the number to the nearest thousand we get 609635.783

Here,

The given number to round nearest thousand is 609635.782852

What is rounding off of a number?

Rounding off means a number is made simpler by keeping its value intact but closer to the next number.

Now,

In rounding off a number nearest thousand we can write a number up to 3 decimal places

So after rounding 609635.782852 to the nearest thousands we get,

609635.783

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The mean and MAD of the two sets are solved

Given data ,

Set A: {1,2,3,4,10}

Mean of Set A: (1 + 2 + 3 + 4 + 10) / 5 = 4

To find the mean absolute deviation (MAD), we first find the absolute difference of each value from the mean:

|1 - 4| = 3

|2 - 4| = 2

|3 - 4| = 1

|4 - 4| = 0

|10 - 4| = 6

Next, we find the average of these absolute differences:

MAD of Set A = (3 + 2 + 1 + 0 + 6) / 5 = 2.4

And ,

Set B: {1, 2, 3, 4, 5}

Mean of Set B: (1 + 2 + 3 + 4 + 5) / 5 = 3

To find the mean absolute deviation (MAD), we first find the absolute difference of each value from the mean:

|1 - 3| = 2

|2 - 3| = 1

|3 - 3| = 0

|4 - 3| = 1

|5 - 3| = 2

Next, we find the average of these absolute differences:

MAD of Set B = ( 2 + 1 + 0 + 1 + 2 ) / 5 = 1.2

Hence , mean absolute deviation of Set A is less than the mean absolute deviation of Set B and mean of Set A is greater than the mean of Set B

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By using the concept of compact set, it can be proved that

|f(x)−f(y)|≤A∥x−y∥+ϵ,∀x,y∈K.

What is compact set?

A set K is said to be compact if every open cover of K has a finite subcover.

Let K⊆Rn is compact  f:K→R is continuous, and ϵ>0

Let there exist \(x_n, y_n\) ∈ K such that |f(\(x_n\))−f(\(y_n\))| > n∥\(x_n\)−\(y_n\)∥+ϵ,

Since K is compact there is a subsequence \(x_{nk}\) and \(y_{nk}\) of \(x_n, y_n\) respectively such that \(x_{nk}\) converges to x and \(y_{nk}\) converges to y.

So,  |f(\(x_{nk}\))−f(\(y_{nk}\))| > \(n_k\)∥\(x_{nk}\)−\(y_{nk}\)∥+ϵ,

Since f is continuous,

We can write

|f(x)−f(y)| > \(n_k\)∥x - y∥+ϵ,

This is true for infinite many \(n_k\)

So ||x - y|| = 0

|f(x) - f(y)| > ϵ, a contradiction since f is continuous

So,  there is a number A>0 such that

|f(x)−f(y)|≤A∥x−y∥+ϵ,∀x,y∈K.

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Answer:

120

Step-by-step explanation:

Answer:

x+6y=20

x=8, y=2

Andy did 8 problems

Step-by-step explanation:

The number of significant digits the product should have = 2

The given numbers are:

\(4.5160 \times 10^{-3}\\ 2.09 \times 10^7\\ 5.8 \times 10^3\)

We first multiply them.

\(4.5160 \times 10^{-3}\times 2.09 \times 10^7\times 5.8 \times 10^3 = 54.743\times10^7\)

In measurement problems, we usually write it as \(5.474\times10^7\)

Significant digits are the numbers which decide the precision and accuracy in measurement problems.

The fewest number of significant digits in each factor of multiplication is considered as the significant digits of the product.

So number of significant digits in \(4.5160\times10^{-3}\) = 5

Number significant digits in \(2.09\times10^7\) = 3

Number of significant digits in \(5.8\times10^3\) = 2

So the product should be rounded into 2 significant digits.

So the product,  \(5.474\times10^7\)= \(5.5\times10^7\) with 2 significant figures.

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The solution of the function g(x) is (-1/7)x² + (4/7)x + (6/7).

We know that g(f(x)) = 4x² + 4x + 3. This means that the input of g is actually the output of f.

To find g(x), we need to undo the composition of functions and get g(x) in terms of x. We can do this by working backwards. Let's start by finding f(g(x)):

f(g(x)) = 2g(x) + 1

Now, we know that g(f(x)) = 4x² + 4x + 3, which means that:

g(f(x)) = ag(x)² + bg(x) + c

Substituting f(x) = 2x + 1, we get:

g(2x + 1) = a(2g(x) + 1)² + b(2g(x) + 1) + c

Expanding the right-hand side, we get:

g(2x + 1) = 4ag(x)² + (4a + 2b)g(x) + (a + b + c)

We can now compare this equation with the one we got for g(f(x)):

g(f(x)) = ag(x)² + bg(x) + c

Comparing the coefficients of g(x) in both equations, we get:

4a + 2b = b => 4a = -b

Comparing the constant terms in both equations, we get:

a + b + c = 3

We now have two equations with three unknowns (a, b, and c). However, we can use the equation 4a = -b to eliminate one of the unknowns. Substituting -4a for b in the equation a + b + c = 3, we get:

a - 4a + c = 3 => -3a + c = 3 => c = 3 + 3a

We now have expressions for b and c in terms of a. Substituting these expressions into the equation 4a = -b, we get:

4a = -(-4a - 3a - 3) => 4a = 7a + 3 => a = -1/7

Substituting this value of a into our expressions for b and c, we get:

b = -4a = 4/7 c = 3 + 3a = 6/7

Therefore, g(x) = ax² + bx + c = (-1/7)x² + (4/7)x + (6/7).

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Answer:

You can use y − y1 = m(x − x1) and get 2-y= 1/6(9-3) so y = 1

Step-by-step explanation:

Answer:

y = 3x + 60

Step-by-step explanation:

Step-by-step explanation:

Given

f(x) = |x + 2|

g(x) = - (x)²

Now

(g•f) (3) = g ( 3 + 2 )

= g(5)

= - (5)²

= -25

Hope it will help :)

The probability of h that can be found using the discrete model but not the normal model is 0.03.

A discrete probability distribution or DPD (also known as a discrete probability model) records all possible values of a discrete random variable and gives their probabilities.

The normal probability model involves when the distribution of the continuous outcome conforms reasonably well to a normal or Gaussian distribution, which resembles a bell-shaped curve.

H represents the height of women

Determine P ( H < 60 ) using either a discrete or normal model

using discrete model

P ( H < 60 ) = ∑ relative frequencies of the women with height < 60 inches

P ( H < 60 ) = 0.01 + 0.02 = 0.03

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In linear equation , 13 April  is day was the flower 16. 5 centimeters tall.

What are a definition and an example of a linear equation?

Rate of change = 17.4 - 15/16 - 8

                        = 2.4/3

                         = 0.3 cm/day

Difference between 16.5 cm and 15 cm = 16.5 - 15  = 1.5 cm

Number of days required to grow from 15 cm to 16.5 cm = 1.5/0.3 = 5 days

Date on which the flower was 16.5 cm tall = 8th April + 5 = 13 April

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The correct interpretation is: We can be 95% confident that the interval 1030 to 1050 contains the average score for the entire population of Logitano students.

The statement is referring to a confidence interval estimate for the average score of the entire population of Logitano students who took the GMAT Preparation course during the past two years. The interval 1030 to 1050 is constructed based on a sample of students and provides a plausible range for the true population average.

The interpretation "We can be 95% confident that the interval 1030 to 1050 contains the average score for the entire population of Logitano students" means that if we were to repeat the process of taking samples and constructing confidence intervals multiple times, about 95% of those intervals would contain the true average score of all Logitano students who took the course.

In other words, the statement implies that we have a high level of confidence that the true population average score falls within the interval 1030 to 1050. This level of confidence is derived from the statistical methods used to estimate the population parameter (average score) based on the sample data. The 95% confidence level indicates that there is a 95% probability that the true population average lies within the stated interval.

It's important to note that this interpretation is based on the assumptions and conditions of the statistical analysis performed, such as the random sampling of students, the validity of the estimation method, and the assumption of normality in the population distribution.

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