Find the area of the triangle with vertices: Q(2,-1,1), R(3,-2,-2), S(5,1,-1).

The statement " for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1" is proved.

If η is the Euler totient function defined by η(n)=n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk) then for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

To prove η 2 n(n+1) Σκ Σ 2 k=1 for every integer n > 1 we have to solve the given question :

1) We know that η(n) = n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk).and

let S = Σκ Σ 2 k=1

2) For n = 2 we have η(2) = 2 * (1 - 1/2) = 1

Hence, S = Σκ Σ 2 k=1 = 1*2=2

Now, η(4) = 4 * (1 - 1/2)(1 - 1/2) = 2 and η(6) = 6 * (1 - 1/2)(1 - 1/3) = 2

Therefore, η 2 n(n+1) Σκ Σ 2 k=1

Hence, S = Σκ Σ 2 k=1 = 2* (2 + 1) * 2 = 12.

3) For n=3, we haveη(3) = 3 * (1 - 1/3) = 2S = Σκ Σ 2 k=1 = 1 * 2 + 2 * 3 = 8

Also, η(6) = 6 * (1-1/2)(1-1/3) = 2

Hence, η 2 n(n+1) Σκ Σ 2 k=1

Thus, S = Σκ Σ 2 k=1 = 2* (3 + 1) * 2 = 16

Therefore, for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

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Answer:

114.51

Step-by-step explanation:

I'm not to sure what you meant by 'each' so I solved it like there was only one cylinder. hope this helped

The Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.

The differential equation is y'' - 12y' + 36y = 10 e 6x. We need to use the method of parameter variation to find the particular solution to the given differential equation. Let's begin by resolving the homogeneous differential equation. The homogenous piece of the differential condition isy'' - 12y' + 36y = 0The trademark condition is r² - 12r + 36 = 0 which can be figured as (r - 6)² = 0So, the arrangement of the homogenous piece of the differential condition is given byy_h(x) = c1 e^(6x) + c2 x e^(6x)where c1 and c2 are inconsistent constants. Presently, let us find the specific arrangement of the given differential condition utilizing the strategy for variety of boundaries. Specific arrangement of the given differential condition isy_p(x) = - y1(x) ∫(y2(x) f(x)/W(x)) dx + y2(x) ∫(y1(x) f(x)/W(x)) dxwhere, y1 and y2 are the arrangements of the homogeneous condition, W is the Wronskian of the homogeneous condition and f(x) is the non-homogeneous term of the differential condition. Hence, y_p(x) = -e(6x) (x e(6x) / e(12x)) dx + x e(6x) (e(6x) (10 e(6x)) / e(12x)) dx = -e(6x) (10x) dx + x e(6x) (10) dx = -5 That's what we know, W(x) = | y1 y2 | | y1' y2' | = e^(12x)Therefore, W(71.72) = e^(12*71.72) = 6.06 × 10²⁸Hence, the Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.

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Answer:

5..

Step-by-step explanation:

The centroid of the region bounded from above by the curve y = 4x - x² and below by the curve y = x is (2/3, 4/3).

The region is bounded from above by the curve y = 4x - x² and below by the curve y = x. We need to find the points of intersection between these two curves. Setting the equations equal to each other,

4x - x² = x

Rearranging,

x² - 3x = 0

Factoring,

x(x - 3) = 0

So, x = 0 or x = 3.

The region is bounded from x = 0 to x = 3. To find the y-values within this region, we evaluate the equations y = 4x - x² and y = x at these x-values.

For x = 0,

y = 4(0) - (0)² = 0

For x = 3,

y = 4(3) - (3)² = 12 - 9 = 3

Thus, the y-values within the region are y = 0 to y = 3. Now, we calculate the area of the region by integrating the difference of the upper and lower curves,

A = ∫[0,3] [(4x - x²) - x] dx

A = ∫[0,3] (3x - x²) dx

A = [3x²/2 - x³/3] evaluated from x = 0 to x = 3

A = [27/2 - 9/3] - [0 - 0]

A = [27/2 - 3] - 0

A = 21/2

Now, for the centroid,

x = (1/A) * ∫[0,3] x * [(4x - x²) - x] dx

Simplifying,

x = (1/A) * ∫[0,3] (3x² - x³) dx

x = (1/A) * [x³ - x⁴/4] evaluated from x = 0 to x = 3

x = (1/A) * [(3)³ - (3)⁴/4] - [0 - 0]

x = (1/A) * [(27) - (81)/4] - 0

x = (1/A) * [(108 - 81)/4]

x = (1/A) * (27/4)

x = 27/(4A)

x = 27/(4 * 21/2)

x = 2/3, and,

x = (1/A) * ∫[0,3] [(4x - x²) - x]² dx

Simplifying,

y = (1/A) * ∫[0,3] (16x² - 8x³ + x⁴) dx

y = (1/A) * [(16x³/3 - 8x⁴/4 + x⁵/5)] evaluated from x = 0 to x = 3

y = (1/A) * [(16(3)³/3 - 8(3)⁴/4 + (3)⁵/5)] - [0 - 0]

y = (1/A) * [(16 * 27/3 - 8 * 81/4 + 243/5)]

y = (1/A) * [(144/3 - 648/4 + 243/5)]

y = (1/A) * [(480 - 972 + 243)/60]

y = (1/A) * (480 - 972 + 243)/60

y = -83/(20A)

Since A = 21/2, we can substitute it in,

y = -83/(20 * 21/2)

y = -83/(210/2)

y = -83/(105)

y = -4/5

Therefore, the centroid of the region is (2/3, 4/3).

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A⁣⁣⁣⁣nswer i⁣⁣⁣s i⁣⁣⁣n a p⁣⁣⁣hoto. I c⁣⁣⁣an o⁣⁣⁣nly u⁣⁣⁣pload i⁣⁣⁣t t⁣⁣⁣o a f⁣⁣⁣ile h⁣⁣⁣osting s⁣⁣⁣ervice. l⁣⁣⁣ink b⁣⁣⁣elow!

bit.[tex]^{}[/tex]ly/3a8Nt8n

Answer:

me

Step-by-step explanation:

becssu imthe best guy

Answer:

12 oz

Step-by-step explanation:

2.56 ÷ 8 = 0.32 per oz

3.66 ÷ 12= 0.305 per oz

Answer:

Step-by-step explanation:

So in ratios you can mostly all of the time scale your answer. So by determining how much increase there is in the baby's thigh bone each week you can pretty much answer these questions.

keep in mind: Proportional means having the same ratio. A scale factor is the ratio of the model measurement to the actual measurement in simplest form.

Example from https://www.mathsisfun.com/numbers/ratio.html

A ratio says how much of one thing there is compared to another thing.

ratio 3:1

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

Use the ":" to separate the values:   3 : 1

Or we can use the word "to":   3 to 1

Or write it like a fraction:    31  

A ratio can be scaled up:

ratio 3:1 is also 6:2

Here the ratio is also 3 blue squares to 1 yellow square,

even though there are more squares.

y=14 I hope this helps!!

Answer:

i got 54ft^2

Step-by-step explanation:

Answer:

6 large and 4 small

Step-by-step explanation:

The volume of the solid obtained by revolving the region bounded by y = x - √x, y = 0, and x = 36 around the line x = 36 can be found using the method of cylindrical shells. The resulting volume is approximately 3,012 cubic units.

To calculate the volume, we integrate the formula for the volume of a cylindrical shell, which is given by V = 2π∫[a,b] x * h(x) dx, where [a,b] represents the range of x values.
In this case, the lower bound of integration is 0 and the upper bound is 36, since the region is bounded by y = 0 and x = 36. The height of the cylindrical shell, h(x), is given by the difference between the x-coordinate of the curve y = x - √x and the line x = 36.
To obtain the x-coordinate of the curve, we set x - √x = 0 and solve for x. This gives us x = 0 or x = 1.
Next, we calculate the difference between x and 36, which gives us  the height of the cylindrical shell. Then, we substitute the expressions for x and h(x) into the volume formula and integrate with respect to x.
After performing the integration, we find that the volume of the solid is approximately 3,012 cubic units.

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Answer:

5/6

Step-by-step explanation:

Hope this helped!!!

Answer:

i dont kno bestie.. :/

Step-by-step explanation:

Answer:

0.8 is the standard form

The solution to the Cauchy-Euler initial value problem is -3/2

To solve the Cauchy-Euler initial value problem, we need to find the general solution of the differential equation and then use the initial conditions to determine the specific solution.

The given Cauchy-Euler differential equation is:

y" + xy' + y = 0

To solve this equation, we assume a solution of the form [tex]y(x) = x^r[/tex]

Differentiating twice with respect to x, we have:

[tex]y' = rx^{r-1}[/tex] and y" = [tex]r(r-1)x^{r-2}[/tex]

Substituting these expressions into the differential equation, we get:

[tex]r(r-1)x^{r-2} + x(rx^{r-1}) + x^r = 0[/tex]

[tex]r(r-1)x^{r-2} + r*x^r + x^r = 0[/tex]

[tex]x^{r-2}(r(r-1) + r + 1) = 0[/tex]

For a non-trivial solution, the expression in parentheses must equal zero:

r(r-1) + r + 1 = 0

Expanding and rearranging, we have:

[tex]r^2 - r + r + 1 = 0\\r^2 + 1 = 0[/tex]

The roots of this equation are complex numbers:

r = ±i

Therefore, the general solution of the Cauchy-Euler differential equation is:

[tex]y(x) = c_1x^i + c_2x^{-i}[/tex]

To simplify the solution, we can rewrite it using Euler's formula:

[tex]y(x) = c_1x^i + c_2x^{-i}\\ = c_1(cos(ln(x)) + i*sin(ln(x))) + c_2(cos(ln(x)) - i*sin(ln(x)))\\ = (c_1 + c_2)cos(ln(x)) + (c_1 - c_2)i*sin(ln(x))[/tex]

Now, let's apply the initial conditions to find the specific solution. We are given:

y(t) = 3 and y'(1) = 4

Substituting x = t into the solution, we have:

[tex](c_1 + c_2)cos(ln(t)) + (c_1 - c_2)i*sin(ln(t)) = 3[/tex]

To satisfy this equation, the real parts and imaginary parts on both sides must be equal.

From the real parts:

[tex](c_1 + c_2)cos(ln(t)) = 3[/tex]

From the imaginary parts:

[tex](c_1 - c_2)i*sin(ln(t)) = 0[/tex]

Since sin(ln(t)) ≠ 0 for any t, we must have ([tex]c_1 - c_2[/tex]) = 0.

This implies [tex]c_1 = c_2[/tex].

Substituting [tex]c_1 = c_2[/tex] into the real part equation, we get:

[tex]2c_1cos(ln(t)) = 3[/tex]

Solving for [tex]c_1[/tex], we find:

[tex]c_1 = 3/(2cos(ln(t)))[/tex]

Therefore, the specific solution of the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

Now, we can find y'(1) by differentiating the specific solution with respect to x and evaluating it at x = 1:

y'(x) = -(3/2)(ln(t)sin(ln(x)) + cos(ln(x)))

y'(1) = -(3/2)(ln(t)sin(ln(1)) + cos(ln(1)))

      = -(3/2)(ln(t)(0) + 1)

      = -3/2

Therefore, the solution to the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

y(t) = 3

y'(1) = -3/2

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Answer:

about 5.99 or D. 6 cm

Step-by-step explanation:

you can use this formula

[tex]V=4/3 * \pi *r^{3}[/tex]

Answer:

196.1

Step-by-step explanation:

Area of a circle is [tex]\pi r^{2}[/tex] so in order to find the radius you divide the diameter by 2 to get 7.9

Then you do [tex]7.9^{2}[/tex] x [tex]\pi[/tex] to get around 196.1

Answer:

5 * 5^(x - 1) ; (15/3)^x ; 15^x / 3^x

Step-by-step explanation:

From the options, equivalent expressions include :

(15/3)^x

This is the same as ;

(15/3)^x

15 ÷ 3 = 5 ; then to the power of x = 5^x

15^x / 3^x ; since they are both raised to the same power, we can divide directly to obtain :

5^x

5 * 5^(x - 1)

5 = 5^1

5^1 * 5^(x-1)

5^(1 + x - 1) = 5^x

Answer:

0.416667 ft

Step-by-step explanation:

To compute the probability P(X ≤ 2) for a binomial variable X with n = 6 and p = 0.4, we need to sum the probabilities of X taking on the values 0, 1, and 2.

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

To calculate these probabilities, we can use the binomial probability formula:

P(X = k) = (n choose k) [tex]* p^k * (1 - p)^(n - k)[/tex]

where (n choose k) represents the binomial coefficient, given by (n choose k) = n! / (k! * (n - k)!)

Let's calculate the probabilities step by step:

P(X = 0) = (6 choose 0) * [tex]0.4^0 * (1 - 0.4)^(6 - 0)[/tex]

P(X = 1) = (6 choose 1) * [tex]0.4^1 * (1 - 0.4)^(6 - 1)[/tex]

P(X = 2) = (6 choose 2) * [tex]0.4^2 * (1 - 0.4)^(6 - 2)[/tex]

Using the binomial coefficient formula, we can calculate the probabilities:

P(X = 0) = 1 * 1 * [tex]0.6^6[/tex] ≈ 0.04666

P(X = 1) = 6 * 0.4 * [tex]0.6^5[/tex] ≈ 0.18662

P(X = 2) = 15 * [tex]0.4^2 * 0.6^4[/tex] ≈ 0.31104

Now, let's sum these probabilities to find P(X ≤ 2):

P(X ≤ 2) ≈ 0.04666 + 0.18662 + 0.31104 ≈ 0.54432

Therefore, the probability P(X ≤ 2) is approximately 0.54432.

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The P-value of 0.343 is not significant and does not strongly suggest that the population standard deviation, sigma, is less than 7.2.

In hypothesis testing, the P-value is used to determine the strength of evidence against the null hypothesis. In this case, the null hypothesis is that the population standard deviation, sigma, is equal to 7.2. The alternative hypothesis is that sigma is less than 7.2.

To calculate the P-value, we need to compare the sample standard deviation, which is 5.985, to the hypothesized population standard deviation of 7.2. We can use the chi-square distribution to find the probability of observing a sample standard deviation as extreme as or more extreme than the one obtained, assuming the null hypothesis is true.

In this case, the P-value is 0.343. This means that if the null hypothesis is true, there is a 34.3% chance of obtaining a sample standard deviation of 5.985 or more extreme. Since the P-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. Therefore, we do not have strong evidence to suggest that the population standard deviation is less than 7.2.

In conclusion, the correct choice is: The P-value 0.343 is not significant and does not strongly suggest that sigma is less than 7.2.

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The solution to the given system of equations y = -x + 2 are option A. (-2,4) and D. (4,-2).

A system of equations is two or more equations that can be solved together to get a unique solution. the power of the equation must be in one degree.

The equation is given as

y = -x + 2

here, we need to find the solutions to the equation, we can apply the given options one by one to satisfy the equation.

For the solution

A. (-2,4)

y = -x + 2

Substitute the value x = -2 and y = 4

y = 4

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

Thus, the given solution are the system of equation.

For the solution

B. (0,-4)

y = -x + 2

Substitute the value x = 0 and y = -4

y = -4

-x + 2 = 0 + 2 = 2

Thus, both the sides are not equal so, the given solution are not the system of equation.

For the solution

C. (2,-4)

y = -x + 2

Substitute the value x = 2 and y = -4

y = -4

-x + 2 = 2 + 2 = 4

Thus, both the sides are not equal so, the given solution are not the system of equation.

For the solution

D. (4,-2)

y = -x + 2

Substitute the value x = 4 and y = -2

y = -2

-x + 2 = -4 + 2 = -2

Thus, the given solution are the system of equation.

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Answer:

B

Step-by-step explanation:

25/100 is 25%.

Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.

For a normal distribution of SAT critical reading scores with a mean of 505 and a standard deviation of 118, approximately 79% of the SAT verbal scores are less than 600. If 1000 SAT verbal scores are randomly selected, it is expected that approximately 722 of them would be greater than 575.

To determine the percentage of SAT verbal scores that are less than 600, we need to find the area under the normal distribution curve to the left of 600. We can use the standard normal distribution table or a statistical software to find the corresponding z-score.

First, we calculate the z-score using the formula:

z = (x - μ) / σ

Substituting the values:

z = (600 - 505) / 118

z ≈ 0.8051

Using the standard normal distribution table, we can find the area to the left of z = 0.8051, which is approximately 0.7910.

To determine the percentage, we multiply the result by 100, giving us approximately 79% of SAT verbal scores that are less than 600.

For part (b), we can apply the same approach. We calculate the z-score for x = 575:

z = (575 - 505) / 118

z ≈ 0.5932

Using the standard normal distribution table, we find the area to the left of z = 0.5932, which is approximately 0.7242. This means that approximately 72.42% of SAT verbal scores are less than 575.

To estimate the number of SAT verbal scores greater than 575 in a sample of 1000, we multiply the percentage by the sample size:

Number of scores greater than 575 = 0.7242 * 1000 ≈ 722.

Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.

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Answer:

12, 25, 26, 26, 26, 34, 35, 39, 42, 42, 50, 72.

Step-by-step explanation:

A stem and leaf plot works like a digit separator. The left is the first number, which is usually repeated, and the right is the number you add to it.

In this example, 3 is used three times for the numbers 34, 35, and 39.

The upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm

To find the upper boundary for a 95% confidence interval for the mean length of corn ears, we can use the formula:

Upper Boundary = Mean + (Critical Value * Standard Error)

The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be obtained from the standard normal distribution, which is approximately 1.96.

The standard error is calculated by dividing the standard deviation by the square root of the sample size:

Standard Error = s / [tex]\sqrt{(n)}[/tex]

Given that the mean length was 31.5 cm (Mean) and the standard deviation was s = 5.8 cm, and the sample size was n = 26, we can calculate the upper boundary as follows:

Standard Error = 5.8 / [tex]\sqrt{26}[/tex] ≈ 1.138

Upper Boundary = 31.5 + (1.96 * 1.138) ≈ 33.8

Therefore, the upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm.

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