Prove that for all a,b∈Z+, if a|b, then a≤b.
Let a and b be positive integers. Prove that if a|b and b|a, then a=b.

The sample size needed to achieve a margin of error of 1.12 with a 90% confidence level is 215.

To determine the sample size required for a 90% confidence level and a margin of error less than or equal to 1.12, we need to use the formula for sample size calculation. Given that the population standard deviation is 10, we can use the formula:

n = (Z * σ / E)²

Where:

n is the required sample size,

Z is the z-score corresponding to the desired confidence level (90% corresponds to a z-score of approximately 1.64),

σ is the population standard deviation (10), and

E is the desired margin of error (1.12).

Plugging in the values, we have:

n = (1.64 * 10 / 1.12)² = 214.18

Since the sample size must be a whole number, we round up to 215. Therefore, a sample size of 215 is needed to achieve a margin of error less than or equal to 1.12 with a 90% confidence level.

To learn more about “sample size” refer to the https://brainly.com/question/17203075

#SPJ11

Mayan notation for the given numbers 23. 135 24. 234 25. 360 26. 1,215 27. 10,500 28. 1,100,000 is written as 55.0.0.0.0.

Mayan civilization is renowned for its advanced math and astronomy. Mayans had a distinctive numbering system.

The Mayans used a counting system based on multiples of twenty, which included elements that represented zero.

This system of counting was used to measure time and space.

The following are the conversions of the given numbers to Mayan notation:

23 in Mayan notation is written as 1.3.

This is computed as 20 + 3 = 23.135 in Mayan notation is written as 7.15.

This is computed as 7 times 20 + 15 = 135.234 in Mayan notation is written as 11.14.

This is computed as 11 times 20 + 14 = 234.360 in Mayan notation is written as 18.0.

This is computed as 18 times 20 + 0 = 360.1,215 in Mayan notation is written as 3.15.15.

This is computed as 3 times 20 times 20 + 15 times 20 + 15 = 1,215.10,500 in Mayan notation is written as 34.0.0.

This is computed as 34 times 20 times 20 + 0 times 20 + 0 = 10,500.1,100,000 in Mayan notation is written as 55.0.0.0.0.

This is computed as 55 times 20 times 20 times 20 times 20 + 0 times 20 times 20 times 20 + 0 times 20 times 20 + 0 times 20 + 0 = 1,100,000.

Hence, the conversions of the given numbers to Mayan notation are given above.

To know more about Mayan notation, visit:

https://brainly.com/question/30650712

#SPJ11

The dependent variable in a correlational study of amounts of sunlight and the heights of tomato plants is the heights of tomato plants.

In a correlational study, the researcher examines the relationship between two variables to determine if there is a statistical association between them. In this case, the two variables of interest are the amounts of sunlight and the heights of tomato plants.

The dependent variable is the variable that is being measured or observed and is expected to be influenced or affected by the independent variable. It is the outcome variable of interest. In this study, the heights of tomato plants would be the dependent variable.

The independent variable, on the other hand, is the variable that is manipulated or controlled by the researcher. It is the variable that is believed to have an effect on the dependent variable. In this study, the independent variable would be the amounts of sunlight.

The researcher would collect data on the amounts of sunlight received by the tomato plants and measure the corresponding heights of the plants. By examining the relationship between these variables, the researcher can determine if there is a correlation between the amounts of sunlight and the heights of the tomato plants.

It is important to note that in this specific study, the dependent variable is not the types of tomato plants or the numbers of hours of sunlight or the angle of the sunlight. These variables may be relevant factors to consider, but in a correlational study, the focus is on examining the relationship between the two variables of interest (amounts of sunlight and heights of tomato plants) rather than investigating the influence of other variables.

Learn more about correlational here

https://brainly.com/question/28567944

#SPJ11

From the given points, only point B (5, 5√3) satisfies the equation of circle C. Therefore, the correct option is B. (5, 5√3) lies on circle C.

To determine which of the given points lies on circle C, we can use the equation of a circle centered at the origin.

The equation of a circle with center (h, k) and radius r is given by:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

In this case, since the center of circle C is at the origin (0, 0), the equation of the circle can be simplified to:

[tex]x^2 + y^2 = r^2[/tex]

Given that point Q(10, 0) lies on circle C, we can substitute the coordinates of Q into the equation:

[tex]10^2 + 0^2 = r^2[/tex]

[tex]100 = r^2[/tex]

So, the radius of circle C is r = √100 = 10.

Now, let's check which of the given points satisfy the equation of circle C.

A. (3, 5√3)

[tex]=(3)^2 + (5√3)^2[/tex]

= 9 + 75

= 84

≠ 100

B. (5, 5√3)

=[tex](5)^2 + (5√3)^2[/tex]

= 25 + 75

= 100

C. (4, 5√3)

=[tex](4)^2 + (5√3)^2[/tex]

= 16 + 75

= 91

≠ 100

D. (6, 4)

=[tex](6)^2 + (4)^2[/tex]

= 36 + 16

= 52

≠ 100

To know more about equation,

https://brainly.com/question/31333717

#SPJ11

If Q(10,0) lies on circle C, another point that lies on circle C is:
B. (5,5√3)

To determine which point lies on circle C, we can use the distance formula to calculate the distance between each point and the center of the circle (origin). If the distance is equal to the radius, the point lies on the circle.

Let's examine each option.

Point A: (3, 5√3)

Distance from center (0, 0) to A:

dA = √((3 - 0)² + (5√3 - 0)²)

dA = √(9 + 75)

dA = √84

Point B: (5, 5√3)

Distance from center (0, 0) to B:

dB = √((5 - 0)² + (5√3 - 0)²)

dB = √(25 + 75)

dB = √100

dB = 10

Point C: (4, 5√3)

Distance from center (0, 0) to C:

dC = √((4 - 0)² + (5√3 - 0)²)

dC = √(16 + 75)

dC = √91

Point D: (6, 4)

Distance from center (0, 0) to D:

dD = √((6 - 0)² + (4 - 0)²)

dD = √(36 + 16)

dD = √52

dD = 2√13

Comparing the distances to the radius:

Radius of circle C = distance from center to point Q = distance from (0, 0) to (10, 0) = 10

Based on the calculations, only Point B: (5, 5√3) has a distance from the center of the circle equal to the radius. Therefore, Point B lies on circle C.

Answer: B. (5, 5√3)

Learn more about circle here:

https://brainly.com/question/29614115

#SPJ11

Answer:

  x = 3

Step-by-step explanation:

You want the equation of the vertical asymptote of the function h(x)=6log₂(x−3)−5.

The vertical asymptote of the parent log function log(x) is x = 0. For the given function it will be located where the argument of the log function is zero:

  x -3 = 0

  x = 3

The equation of the vertical asymptote is x = 3.

__

Additional comment

The leading coefficient (6) and the base (2) serve only as vertical scale factors of the log function. The added value -5 shifts the curve down 5 units, so has no effect on the vertical asymptote. The horizontal translation of the function right 3 units is what moves the asymptote away from x = 0.

<95141404393>

The given function is h(x)=6log_2(x-3)-5. We know that the vertical asymptote is a vertical line that indicates a point where the function will be undefined. It occurs at x=c where the denominator of the fraction f(x) is equal to zero.

The given function is h(x)=6log_2(x-3)-5. We know that the vertical asymptote is a vertical line that indicates a point where the function will be undefined. It occurs at x=c where the denominator of the fraction f(x) is equal to zero.Therefore, we need to check if the given function is undefined at any particular value of x. If yes, then the vertical asymptote will be the value of x that makes the function undefined. Let's find the value of x where the function is undefined.

We know that the logarithmic function is undefined for negative arguments. Hence, the function h(x)=6log_2(x-3)-5 is undefined for x \le 3. Therefore, the vertical asymptote of the given function is x = 3.

Answer: x = 3

To know more about vertical asymptote visit: https://brainly.com/question/29260395

#SPJ11

The following is the R code that produces the given plot layout 2 5 1 3 with the red-colored square box, using the "layout()" function in R:> x<-c(1:50)> y<-run if(50, min = 0, max = 10)> par(mar=c(5, 4, 4, 5))> plot.

new()# Creating layout> layout(matrix(c(2, 5, 1, 3),

2, 2,

b y row = TRUE),

c(3, 2), c(2, 3))#

Creating plotting area for 2 and 5> plot(x, y, type='l', col='blue',

l w d=2, x lab = "X label",

y lab = "Y label",

main="Plotting area 2 and 5")> plot(x, y, type='o', col='green', l w d=1.5, x lab = "X label",

y lab = "Y matrix", main="Plotting area 2 and 5")#

Creating plotting area for 1> plot(x, y, type='h', col='red', l  w d=1.5, x lab = "X label", y lab = "Y label", main="Plotting area 1")# Creating plotting area for 3> plot(x, y, type='b', col='purple', l w d=1.5, x lab = "X label", y lab = "Y label", main="Plotting area 3")# Creating the red colored box> rect(0.5, 0.5, 2.5, 2.5, border="red", l w d=2, l t y=2).

To produce the above plot layout, firstly, create two vectors "x" and "y" containing 50 random numbers between 0 and 10 using the "run if()" function in R. Then, define the plotting area and margin sizes of the plot using the "par()" function .Next, create a new plot using the "plot .new()" function to clear the graphics window.

To know more about matrix visit:

https://brainly.com/question/28180105

#SPJ11

Answer:

17 inches, 4 inches

Step-by-step explanation:

Let the width = x.

Then the length is 2x + 9.

area = length × width

area = (2x + 9)x

area = 2x² + 9x

area = 68

2x² + 9x = 68

2x² + 9x - 68 = 0

2 × 68 = 136

136 = 2³ × 17

8 × 17 = 136

17 - 8 = 9

2x² + 17x - 8x - 68 = 0

x(2x + 17) - 4(2x + 17) = 0

(x - 4)(2x + 17) = 0

x = 4 or x = -17/2

2x + 9 = 8 + 9 = 17

The length is 17 inches, and the width is 4 inches.

Answer:

The dimensions are 17 inches (length) by 4 inches (width).  

Step-by-step explanation:

W = Width

L = Length

Since the problem says that the length, L, equals 9 more inches than 2 times its width, the equation would be:

L = 9+2*W

This would be the same as:

L = 2W + 9

The formula for the area of a rectangle is:

L*W = Area

In the problem, we are given that the area equals 68 inches, so after plugging in the variables for the equation, we get:

(2W+9) * (W) = 68

Then we distribute:

2W^2 + 9W = 68

Then we set it equal to zero:

2W^2 + 9W - 68 = 0

Then we factor it out:

(2W+17) (W-4) = 0

We set each part equal to zero:

2W +17 = 0

2W = -17

W = -17/2

W-4 = 0

W = 4

Since we know that the lengths can only be positives, we disregard the negative solution. Therefore, W, the width, is equal to 4.

We then plug it into the equation to solve for length.

L = 2(4) + 9

L = 17

Then we plug in the lengths and widths to the solution. (FYI: it is typically written as length x width.)

We get:

The dimensions are 17 inches by 4 inches.

The distance d is approximately equal to 84.17.

Given, the sin 32° = 0.53, cos 32° = 0.85 and tan 32° = 0.62.

Find the distance marked d in the diagram. We can use the trigonometric ratios to find the value of d.

In right-angled triangle ABC, we have;

tan θ = AB/BC   (1)

We can rewrite equation (1) as:

BC = AB/tan θ   (2)

Also, cos θ = AC/BC    (3)

We can rewrite equation (3) as:

BC = AC/cos θ      (4)

Equating equations (2) and (4), we have:

AB/tan θ = AC/cos θ

AB/0.62 = AC/0.85

AB = 0.62 × AC/0.85

AB = 0.729 × AC  (5)

Again, in right-angled triangle ACD, we have;

sin θ = d/AC

=> AC = d/sin θ     (6)

Substituting the value of AC from equation (6) into equation (5), we have:

AB = 0.729 × d/sin θ

AB = 0.729 × d/sin 32°

AB = 1.39 × d        (7)

Therefore, d = AB/1.39

= 117/1.39

≈ 84.17

Hence, the distance d is approximately equal to 84.17.

To know more about trigonometric ratios visit:

https://brainly.com/question/23130410

#SPJ11

Answer:

  -8748

Step-by-step explanation:

You want the 8th term of the geometric sequence that begins 4, -12, 36, ....

A geometric sequence is characterized by a common ratio r. When the first term is a1, the n-th term is ...

  an = a1×r^(n-1)

Here, the first term is 4, and the common ratio is -12/4 = -3. That means the n-th term is ...

  an = 4×(-3)^(n-1)

and the 8th term is ...

  a8 = 4×(-3)^(7-1) = -8748

The 8th term is -8748.

__

Additional comment

The attachment shows the 8th term and the first 8 terms of the sequence.

<95141404393>

the 8th term of the geometric sequence  4 , − 12 , 36 , . . . 4,−12,36,... is -4374 by using formula of a:an = a1 * rn-1Wherean = nth terma1 = first term r = common ratio

Given the first three terms of a geometric sequence: 4, -12, 36, ...To find the 8th term of the geometric sequence, we need to first find the common ratio of the sequence which can be found using the formula:Common ratio, r = Term 2 / Term 1= -12 / 4= -3The nth term of a geometric sequence is given by the formula:an = a1 * rn-1Wherean = nth terma1 = first term r = common ratio To find the 8th term, we use the formula:a8 = a1 * r8-1= 4 * (-3)7= -4374Therefore, the 8th term of the geometric sequence is -4374.

To know more about geometric sequence Visit:

https://brainly.com/question/27852674

#SPJ11

The equation that describes the height (h) of a projectile as a function of time (t) can be given by the equation:

[tex]h(t) = -16t^2 + v_0t + h_0[/tex]

Where:

h(t) is the height of the projectile at time t,

v₀ is the initial velocity (speed) of the projectile,

h₀ is the initial height of the projectile.

In this case, the tennis ball machine serves the ball vertically into the air from a height of 2 feet, with an initial speed of 110 feet. So, the equation for the projectile's height versus time would be:

[tex]h(t) = -16t^2 + 110t + 2[/tex]

Therefore, the correct equation for the given scenario is [tex]h(t) = -16t^2 + 110t + 2[/tex].

To know more about Equation visit-

brainly.com/question/14686792

#SPJ11

Theorem 3.1.9(d) states that if f(x) is a continuous function at c and g(x) is a continuous function at f(c), then the composition g(f(x)) is continuous at c.

To prove this theorem using the Sequential Characterization of Continuity, we need to show that for any sequence {x_n} that converges to c, the sequence {g(f(x_n))} converges to g(f(c)).

Let {x_n} be a sequence that converges to c. Since f(x) is continuous at c, by the Sequential Characterization of Continuity, we know that f(x_n) converges to f(c).

Similarly, since g(x) is continuous at f(c), by the Sequential Characterization of Continuity, we know that g(f(x_n)) converges to g(f(c)).

Therefore, we have shown that for any sequence {x_n} converging to c, the sequence {g(f(x_n))} converges to g(f(c)). This satisfies the conditions of the Sequential Characterization of Continuity, which proves that the composition g(f(x)) is continuous at c.

Hence, Theorem 3.1.9(d) holds true based on the Sequential Characterization of Continuity.

To know more about sequence visit-

brainly.com/question/31390966

#SPJ11

The correct options for analyzing the study using the methods for conducting a hypothesis test regarding two independent proportions are:

B. n1p11−p1≥10 and n2p21−p2≥10

E. The samples are independent.

Explanation:

A. The assumption of normal distribution is not required for conducting a hypothesis test regarding two independent proportions. Therefore, option A is incorrect.

C. The sample size being less than 5% of the population size for each sample is not a requirement for analyzing the study using the methods for conducting a hypothesis test regarding two independent proportions. Therefore, option C is incorrect.

D. The sample size being more than 5% of the population size for each sample is also not a requirement for analyzing the study using the methods for conducting a hypothesis test regarding two independent proportions. Therefore, option D is incorrect.

E. The independence of the samples is a crucial assumption for conducting a hypothesis test regarding two independent proportions. If the samples are not independent, then different methods need to be used. Therefore, option E is correct.

F. The samples being dependent is not consistent with the assumption for conducting a hypothesis test regarding two independent proportions. If the samples are dependent, different methods need to be used. Therefore, option F is incorrect.

The correct options are B and E.

To know more about hypothesis visit-

brainly.com/question/30885920

#SPJ11

The correct answer of the given equation of the interval is θ = 14.4°, 57.6°, 102.4°, 165.6°, 197.6°, 282.4°, 297.6°, 342.4°

.The given equation is cot⁡(θ) - 4cos(θ) - 5 = 0. We are supposed to solve the equation for solutions over the interval [0,360]. We'll use the substitution

u = cos(θ). Then cot⁡(θ) = cos(θ)/sin(θ) = u/√(1 - u²).

We have

cot(θ) - 4cos(θ) - 5 = 0u/√(1 - u²) - 4u - 5 = 0u - 4u√(1 - u²) - 5√(1 - u²) = 0(4u)² + (5√(1 - u²))² = (5√(1 - u²))²(16u² + 25(1 - u²)) = 25(1 - u²)25u² + 25 = 25u²u² = 0.

Then u = 0. For u² = 1/5, we obtain

5θ = ±72°, ±288°.

Then

θ = 14.4°, 57.6°, 102.4°, 165.6°, 197.6°, 282.4°, 297.6°, 342.4°.

Therefore, the solutions of the given equation in the interval

[0,360] are θ = 14.4°, 57.6°, 102.4°, 165.6°, 197.6°, 282.4°, 297.6°, 342.4°.

Hence, the correct answer is

θ = 14.4°, 57.6°, 102.4°, 165.6°, 197.6°, 282.4°, 297.6°, 342.4°.

To know more about equation visit:

https://brainly.com/question/29657983

#SPJ11

The discount rate for the shirt is approximately 62.5%. This means that the shirt was discounted by 62.5% off the original price of $40, resulting in the sale price of $15.

To calculate the discount rate for the shirt, we need to find the difference between the original price and the sale price, and then express that difference as a percentage of the original price.

The original price of the shirt was $40, and it was on sale for $15. To find the discount amount, we subtract the sale price from the original price:

Discount amount = Original price - Sale price

= $40 - $15

= $25

To determine the discount rate as a percentage, we divide the discount amount by the original price and then multiply by 100:

Discount rate = (Discount amount / Original price) × 100

= ($25 / $40) × 100

≈ 62.5%

For more such more questions on Discount rate:

https://brainly.com/question/29067322

#SPJ8

The integral of 4x²/(x²+9) is equal to 2 ln |x² + 9| - 18/(x²) + C, where C is the constant of integration.

The integral of `4x²/(x² + 9)` can be found by performing a substitution. The substitution u = x² + 9 can be used to convert the integral into a more manageable form. Therefore, `du/dx = 2x` or `x dx = (1/2) du`.Substituting `u = x² + 9` in the integral:∫(4x² / (x² + 9)) dxLet `u = x² + 9`, then `du = 2x dx` or `(1/2) du = x dx`.Substituting this into the integral:∫(4x² / (x² + 9)) dx= ∫(4x² / u) (1/2) du= 2 ∫(x² / u) du= 2 ∫(x² / (x² + 9)) dx= 2 [ln |x² + 9| - 9/x² + C]

Putting back the value of `u`:= 2 ln |x² + 9| - 18/(x²) + C The integral of `4x² / (x² + 9)` is equal to `2 ln |x² + 9| - 18/(x²) + C`. Therefore, the integral of 4x²/(x²+9) is equal to 2 ln |x² + 9| - 18/(x²) + C, where C is the constant of integration.

To know more about integral visit:-

https://brainly.com/question/31059545

#SPJ11

The relative frequencies of the eight and two classes are 21.05% and 5.26%, respectively.

Given that, a frequency table of grades has five classes (A, B, C, D, F) with frequencies of 4, 10, 14, respectively.

Using percentages, we have to find the relative frequencies of the five 8, and 2 classes.

Complete the following table: Class Frequency Percentage Relative Frequency

A 4

B 10

C 14

D  Eight  

F  Two Total

We can get the total by adding all the frequencies.

The total is: Total = 4 + 10 + 14 + 8 + 2 = 38We can find the percentage by using the formula given below:

Percentage = (Frequency / Total) × 100Substituting the values in the above formula, we get the following table:

Class Frequency Percentage    Relative Frequency

A              4                10.53%                       10.53%

B              10        26.32%                      26.32%

C              14       36.84%

__                  --                   --                                        --

Total 38 100.00% 100.00%

Hence, the relative frequencies of the eight and two classes are 21.05% and 5.26%, respectively.

Know more about relative frequencies here:

https://brainly.com/question/3857836

#SPJ11

To find f'(x), we differentiate the function f(x) = 2e^(2x²  ) using the chain rule. The derivative is f'(x) = 4xe^(2x²).

(a) To find f'(x), we differentiate the function f(x) = 2e(2x²  ) using the chain rule. The derivative is f'(x) = 4xe(2x²).

(b) To find the value of x where the slope of the tangent line is equal to 4, we set f'(x) = 4 and solve for x. So, 4xe(2x²) = 4.

Simplifying, we get xe(2x²) = 1. Unfortunately, this equation cannot be solved algebraically, and we need to use numerical methods or approximation techniques to find the value of x.

(c) To find the value(s) of x where the tangent line intersects the x-axis at the point (2,0), we set f(x) = 0 and solve for x. So, 2e(2x²) = 0. However, there is no value of x that satisfies this equation since e(2x²) is always positive and cannot be zero.

Therefore, there is no value of x for which the tangent line intersects the x-axis at the point (2,0).

Learn more about function

brainly.com/question/3072159

#SPJ11

To prove that if I is an eigenvalue of matrix A, then q(1) is an eigenvalue of q(A), we will show that the eigenvectors corresponding to eigenvalue I of A are also eigenvectors of q(A) with eigenvalue q(1). Then, using part (a), we will calculate the eigenvalues of q(A) for the given matrix which are 19, 35, and 58 .

(a) Let v be an eigenvector of A corresponding to eigenvalue I. We have Av = Iv = v. Now consider q(A)v = (A² - 2A - I + 21)v. Applying A to both sides, we get A(q(A)v) = A(A² - 2A - I + 21)v. Simplifying, we have A(q(A)v) = (A³ - 2A² - A + 21A)v = (I - 2A - A + 21I)v = (20I - 3A)v = q(1)v. Thus, q(1) is an eigenvalue of q(A) corresponding to the eigenvector v.
(b) For the given matrix A, we need to find the eigenvalues of q(A). First, we find the eigenvalues of A, which are λ₁ = 0, λ₂ = -2, and λ₃ = -3. Then, using part (a), we substitute these eigenvalues into q(1) to obtain the eigenvalues of q(A): q(1) = (1 - 2 - 1 + 21) = 19. Therefore, the eigenvalues of q(A) for the given matrix A are λ₁ = 19, λ₂ = 35, and λ₃ = 58.
Hence, the eigenvalues of q(A) for the given matrix A are 19, 35, and 58.

Learn more about eigenvalues here
https://brainly.com/question/30357013

#SPJ11

1a) x0 = 28

1b) x0 ≈ 154.465

1c) x0 ≈ -70.714

1d) x0 ≈ 154.465

2) μ ≈ 146.958

3a) Approximately 68%

3b) Approximately 95%

3c) Approximately 99.7%

4a) IQR = 111

4b) IQR/s ≈ 1.947

4c) No, IQR/s is not approximately equal to 1.3. It implies a relatively large spread or variability in the data.

5a) P(x > 240) ≈ 0.494

5b) P(230 ≤ x < 240) ≈ 0.112

5c) P(x > 264) ≈ 0.104

6a) μ = 87.5

6b) σ ≈ 8.12

6c) z-score ≈ 1.47

6d) Approximate probability: P(x ≥ 100) ≈ 0.071

To know more about probability visit-

brainly.com/question/31581379

#SPJ11

The indefinite integral of cos14 x sin x dx= -cos 14x cos x + 14 sin x + C, Where C is the constant of integration.

We can solve the given problem by using the substitution method.

Using the formula:

∫u'v = uv - ∫uv' dx

Consider,

cos 14x as u' and sin x dx as v

Now we find v' as derivative of

v.∫ cos14 x sin x dx = ∫ u'v

Now,

v' = sin x

Therefore, v = -cos x

Now we have:

∫ cos 14x sin x dx

= ∫ u'v∫ cos 14x sin x dx

= -cos 14x cos x + ∫ cos x * 14 * sin x dx

Now we apply the formula again. We get:

∫ cos14 x sin x dx = -cos 14x cos x + 14 ∫ cos x sin x dx

Now we apply the same substitution again. We get:

∫ cos14 x sin x dx = -cos 14x cos x + 14 ∫ cos x sin x dx

u' = cos xv = sin x dxv' = cos x

Therefore, the solution is:

∫ cos14 x sin x dx

= -cos 14x cos x + 14 ∫ cos x sin x dx

= -cos 14x cos x + 14 sin x + C, Where C is the constant of integration.

To know more about indefinite integral, visit:

https://brainly.com/question/32465992

#SPJ11

Answer:

All of her work is correct

Step-by-step explanation:

If you go the opposite way, you can do 5 * 5 which is 25.

25 * 2 is 50

Adding the square root sign you get _/50, which is what she started with

meaning that she did the right work

a)Let ε be a small, positive number. We can find a δ such that if x is within δ of 0, then g(x) is within ε of g(0).We have:|g(x) - g(0)| =[tex]|3√x - 3√0||g(x) - g(0)| = |3√x - 0| = 3√x[/tex]

This means that we need to find δ such that if x is within δ of 0, then 3√x < ε. Therefore, if we choose δ to be ε^3, then 3√x < ε, as required.

b) Let g(x) = 3√x.Let δ be a positive number, and let x and c be real numbers such that |x - c| < δ. We need to show that |g(x) - g(c)| < ε. Since g(x) = 3√x, we have g(x)^3 = x. Similarly, g(c) = 3√c, so g(c)^3 = c. Then|[tex]g(x) - g(c)| = |3√x - 3√c||g(x) - g(c)| = |3√(g(x)^3) - 3√(g(c)^3)[/tex].

Using the inequality[tex]|a + b| ≤ |a| + |b|[/tex], we can simplify the denominator to get[tex]|g(x) - g(c)| = |3√(g(x)^3 - g(c)^3) / (3√(g(x)^2) + 3√(g(x)g(c)) + 3√(g(c)^2))|≤ |3√(g(x)^3 - g(c)^3)| / (3√(g(x)^2) + 3√(g(x)g(c)) + 3√(g(c)^2))[/tex]Using the inequality a + b ≤ 2√(a^2 + b^2), we can further simplify the denominator to get= [tex]2δ / (3√c + 3√δ)(√c + √δ)^2 < ε[/tex]if we choose δ to be [tex]ε^3 / (9c^2 + 3cε^3 + ε^6)[/tex]. This completes the proof that g is continuous at c.

To know more about continuity visit:

https://brainly.com/question/31523914

#SPJ11

Step-by-step explanation:

imagine the triangle is rotated and twisted so that the vertex with the 60° angle is the bottom left vertex and therefore the center of the trigonometric circle around the triangle.

so, y is the radius of that circle.

4 = sin(60)×y

x = cos(60)×y

sin(60) = sqrt(3)/2

cos(60) = 1/2

y = 4/sin(60) = 4 / sqrt(3)/2 = 8/sqrt(3)

x = cos(60)× 8/sqrt(3) = 1/2 × 8/sqrt(3) = 4/sqrt(3)

The given vector field F is not conservative.

To determine if a vector field F is conservative, we need to check if it satisfies the condition of being the gradient of a potential function. In other words, we need to find a function f such that ∇f = F.

In this case, the given vector field F = <(y + 4z + 5) sin(x), −cos(x), −4 cos(x)>. To check if it is conservative, we compute the partial derivatives of its components with respect to each variable.

The partial derivative of the first component with respect to x is (y + 4z + 5) cos(x), the partial derivative of the second component with respect to y is 0, and the partial derivative of the third component with respect to z is 0.

Since the partial derivatives do not match the components of F, we cannot find a function f such that ∇f = F. Therefore, the vector field F is not conservative.

In conclusion, the vector field F is not conservative, and there is no potential function f for F.

To learn more about vector field visit:

brainly.com/question/32574755

#SPJ11

If A = [a₁ a₂ a₃ a₄ a₅ a₆], where each aᵢ is a column vector in R², and z = [z₁ z₂ z₃ z₄ z₅ z₆] is a vector in R⁶, then T(z) = Az can be written as:T(z) = z₁a₁ + z₂a₂ + z₃a₃ + z₄a₄ + z₅a₅ + z₆a₆.

Let A be a 2x6 matrix. If we define the linear transformation T: R⁶ → R² as T(z) = Az, then the number of columns in matrix A must be equal to the dimension of the domain of T, which is 6. The number of rows in matrix A must be equal to the dimension of the range of T, which is 2. Therefore, A must be a 2x6 matrix.If we plug in a vector z from the domain of T, which is R⁶, into T(z), then we get a vector in the range of T, which is R². The entries of the output vector are obtained by taking linear combinations of the columns of matrix A, where the coefficients are the entries of z.

In other words, the i th entry of the output vector is obtained by multiplying the ith row of matrix A with the vector z, and then adding up the products. So, if A = [a₁ a₂ a₃ a₄ a₅ a₆], where each aᵢ is a column vector in R², and z = [z₁ z₂ z₃ z₄ z₅ z₆] is a vector in R⁶, then T(z) = Az can be written as:T(z) = z₁a₁ + z₂a₂ + z₃a₃ + z₄a₄ + z₅a₅ + z₆a₆.

To know more about column vector visit:-

https://brainly.com/question/31816488

#SPJ11

Answer:

(5, 0) is the set of coordinates that satisfies the equations 3x - 2y = 15 and 4x - y = 20

Step-by-step explanation:

The two equations form a system of equations and solving the system will allow us to determine the set of coordinates that satisfies the equations:

Method:  Elimination:

We can eliminate the ys first by multiplying the second equation by -2:

-2(4x - y = 20)

-8x + 2y = -40

Now we can add the two equations to solve for x:

    3x - 2y = 15

+

    -8x + 2y = -40

---------------------------

-5x = -25

x = 5

Now we can plug in 5 for x in any of the two equations to find y.  Let's use the first equation:

3(5) - 2y = 15

15 - 2y = 15

-2y = 0

y = 0

Thus (5, 0) is the set of coordinates that satisfies the equations.

Check the validity of the answer:

We can check that our answer is correct by plugging in 5 for x and 0 for y in both equations and seeing if we get 15 on both sides for the first equation and 20 on both sides for the second equation:

Checking that x = 5 and y = 0 satisfy the first equation:

3(5) - 2(0) = 15

15 - 0 = 15

15 = 15

Checking that x = 5 and y = 0 satisfy the second equation:

4(5) - (0) = 20

20 - 0 = 20

20 = 20

Thus, our answer is correct.

When a set of coordinates satisfies a system of equations, it also means that the set of coordinates is the point where the two equations intersect.  I attached a picture from Desmos that shows how the coordinates (5, 0) is the point where 3x - 2y = 15 and 4x - y = 20 intersect

the value of x from step 2 into the second equation:4(5) - y = 20y = 4(5) - 20y = 0Therefore, the set of coordinates that satisfies the given equations is (5,0).

The given equations are:3x - 2y = 154x - y = 20We are supposed to find out the set of coordinates that satisfies the given equations. In order to do that, we can use the method of substitution. The steps are given below:Rearrange the second equation in order to isolate y:4x - y = 20y = 4x - 20Substitute the value of y from step 1 into the first equation:3x - 2(4x - 20) = 153x - 8x + 40 = 15-5x = -25x = 5Substitute the value of x from step 2 into the second equation:4(5) - y = 20y = 4(5) - 20y = 0Therefore, the set of coordinates that satisfies the given equations is (5,0).

To know more about coordinates Visit:

https://brainly.com/question/22261383

#SPJ11

The equations that would require the subtraction property of equality to solve are: c. 4x - 3 = 17 and d. b - 13 = 26.

The subtraction property of equality states that if you subtract the same quantity from both sides of an equation, the equality is preserved. This property allows us to isolate the variable and solve for its value.

Based on this property, the equations in which you would use the subtraction property of equality to solve are:

c. 4x - 3 = 17

d. b - 13 = 26

To know more about equations,

https://brainly.com/question/15447331

#SPJ11

The solution to the system of linear  congruences x ≡ 7 (mod 9), x ≡ 4 (mod 12), and x ≡ 16 (mod 21) is {11096 + 2268k: k is an integer}.

We have to find all solutions, if any, to the system of congruences x ≡ 7 (mod 9), x ≡ 4 (mod 12), and x ≡ 16 (mod 21).

Using the Chinese Remainder Theorem, we can find a solution to the system of congruences.

Let m1, m2, and m3 be the moduli of the given congruences, and let M1, M2, and M3 be the moduli of the system of linear congruences.

Then, M1 = m2m3 = 12 × 21 = 252, M2 = m1m3 = 9 × 21 = 189, and M3 = m1m2 = 9 × 12 = 108.

The greatest common divisor of M1, M2, and M3 is gcd(M1, M2, M3) = 9.

Hence, we will apply the Chinese Remainder Theorem by solving the following system of linear congruences

System of linear congruences is X1 = 28, and hence the solution to the original system of congruences is

x = a1M1X1 + a2M2X2 + a3M3X3,

where X2 ≡ 1 (mod 9), X2 ≡ 0 (mod 28), X3 ≡ 1 (mod 12), and X3 ≡ 5 (mod 7).

The solution is x = 28 × 4 × 1 + 189 × 7 × 0 + 108 × 16 × 5 = 2456 + 8640 = 11096, and hence the set of solutions is {11096 + 2268k: k is an integer}.

Know more about the linear  congruences

https://brainly.com/question/24583039

#SPJ11

The sum of the 6-term Geometric series is approximately 1,527,890.

The sum of a geometric series, we can use the formula:

S = a * (1 - r^n) / (1 - r)

Where:

S is the sum of the series,

a is the first term,

r is the common ratio,

and n is the number of terms.

In this case, the first term (a) is 21 and the last term is 1,240,029. We need to determine the common ratio (r) and the number of terms (n) to calculate the sum (S).

The common ratio (r) can be found by dividing the last term by the first term:

r = (last term) / (first term)

r = 1,240,029 / 21

r ≈ 59,048.5238

Now, we can find the number of terms (n) using the formula:

(last term) = (first term) * (common ratio)^(n-1)

1,240,029 = 21 * 59,048.5238^(n-1)

To solve for n, we can take the logarithm of both sides:

log(1,240,029) = log(21 * 59,048.5238^(n-1))

log(1,240,029) = log(21) + (n-1) * log(59,048.5238)

By rearranging the equation, we can solve for (n-1):

(n-1) = (log(1,240,029) - log(21)) / log(59,048.5238)

(n-1) ≈ 3.4576

Therefore, n ≈ 4.4576 (rounded to the nearest tenth).

Now we can substitute the values into the sum formula:

S = 21 * (1 - (59,048.5238)^4.4576) / (1 - 59,048.5238)

S ≈ 1,527,890

Therefore, the sum of the 6-term geometric series is approximately 1,527,890.

For more questions Geometric .

https://brainly.com/question/30303755

#SPJ8

84 pounds of chocolate of 1st kind should be mixed with 216 pounds of chocolate of 2nd kind by using the method of alligation.

To answer this question, we can use the method of alligation. We will use the following table to get the solution to the problem:

The ratio of cocoa butter to caramel in the first chocolate is 1:7, that means the proportion of cocoa butter is 1/8 and that of caramel is 7/8.The ratio of cocoa butter to caramel in the second chocolate is 1:9, that means the proportion of cocoa butter is 1/10 and that of caramel is 9/10.

Mixing 1/8 part chocolate with 1/10 part chocolate, we get 1/9 part of the mixture as cocoa butter and 8/90 + 9/90 = 17/90 parts as caramel.

Therefore, we need 17/90 part of the mixture as caramel. The total amount of chocolate is 300 pounds.

Let the quantity of chocolate of 1st kind to be mixed be x.

Then, the quantity of chocolate of 2nd kind to be mixed = (300 – x).

We have to find the quantity of 1st kind of chocolate needed to make 1:9 parts mixture.

x/ (7/8) = (300 – x) / (9/10 * 8/10)

Solving this equation, we get x = 84 pounds.

Hence, 84 pounds of chocolate of 1st kind should be mixed with 216 pounds of chocolate of 2nd kind.

To know more about alligation visit:

https://brainly.com/question/30711902

#SPJ11