suggest how ligand 7.30 coordinates to ru2 in the 6-coordinate complex ru(7.30)2]12 . how many chelate rings are formed in the complex? (7.30)

The solution of the equations is (2, -2)

Given is an equation we need to solve it,

[tex]3 ^ x * 3 ^ y = 1 \\\\2^{(2x - y)} - 64 = 0[/tex]

[tex]\begin{bmatrix}3^x\cdot \:3^y=1\\ 2^{2x-y}-64=0\end{bmatrix}[/tex]

[tex]\mathrm{Substitute\:}x=-y[/tex]

[tex]\begin{bmatrix}2^{2\left(-y\right)-y}-64=0\end{bmatrix}[/tex]

[tex]\begin{bmatrix}8^{-y}-64=0\end{bmatrix}[/tex]

[tex]x=-\left(-2\right)[/tex]

x = 2 and y = 2

Hence, the solution of the equations is (2, -2)

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The calculated value of the size of the angle m is 71 degrees

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

The quadrilateral

The sum of angles in a quadrilateral is 360

So, we have

86 + 180 - 67 + 90 + m = 360

Evaluate the like terms

So, we have

289 + m = 360

Subtract 289 from both sides

So, we have

m = 71

Hence, the size of the angle m is 71 degrees

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Elasticity in economics refers to the degree of responsiveness or sensitivity of a particular economic variable to a change in another variable.

More specifically, elasticity measures the percentage change in one variable resulting from a one percent change in another variable.

It is often used to describe the responsiveness of the quantity .

Demanded or quantity supplied of a good to a change in price, income, or other factors that affect demand or supply.

It shows the demands of good in the market as per the change in the price.

Elasticity is an important concept in economics because it helps to quantify the degree of responsiveness of economic variables .

And can be used to make predictions and inform decision-making.

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The wind has a magnitude of approximately 20.25 km/h and is blowing in a direction approximately 56.31° east of south. Magnitude is the hypotenuse of a right triangle with westerly and southerly components.

To find the magnitude and direction of the wind with a westerly component of 11 km/h and a southerly component of 17 km/h, we can use the Pythagorean theorem and trigonometry.

The magnitude of the wind is given by the hypotenuse of a right triangle with legs 11 km/h and 17 km/h. Using the Pythagorean theorem, we get:

magnitude = [tex]\sqrt{(11^2 + 17^2)} \approx 20.25 \;km/h[/tex]

To find the direction of the wind, we can use trigonometry. The angle θ between the wind direction and the east direction can be found using the inverse tangent function:

[tex]tan(\theta)[/tex] = opposite/adjacent = 17/11

[tex]\theta = atan(17/11) \approx 56.31^{\circ}[/tex]

Therefore, the wind has a magnitude of approximately 20.25 km/h and is blowing in a direction approximately 56.31° east of south.

In summary, to find the magnitude and direction of wind with given westerly and southerly components, we can use the Pythagorean theorem and trigonometry.

The magnitude is given by the hypotenuse of a right triangle with legs equal to the westerly and southerly components, while the direction is given by the angle between the wind direction and the east direction, which can be found using the inverse tangent function.

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The sample mean blood pressure of the 20 randomly selected people in China is more than 142.47 mmHg is approximately 0.042 or 4.2%

To find the probability that the sample mean blood pressure of the 20 randomly selected people in China is more than 142.47 mmHg, we need to use the central limit theorem.

Assuming that the population follows a normal distribution with a mean of 136 mmHg and a standard deviation of 17 mmHg (as calculated in previous parts of the question), the mean and standard deviation of the sample mean can be calculated as:

Mean of sample mean = population mean = 136 mmHg
Standard deviation of sample mean = population standard deviation / square root of sample size = 17 / square root of 20 = 3.804 mmHg

Now, we can standardize the sample mean using the formula z = (x - mean) / standard deviation, where x is the value of interest (142.47 mmHg in this case).

z = (142.47 - 136) / 3.804 = 1.722

Using a standard normal distribution table or calculator, we can find the probability that z is greater than 1.722, which is 0.042.

Therefore, the probability that the sample mean blood pressure of the 20 randomly selected people in China is more than 142.47 mmHg is approximately 0.042 or 4.2% (rounded to 3 decimal places).

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Answer:

In fraction form:

4.06/100

In decimal form:

0.0406

To integrate the function G(x, y, z) = x over the parabolic cylinder defined by y = x^2, 0 ≤ x ≤ √2, and 0 ≤ z ≤ 2, we need to set up a triple integral over the specified region.

The integral is given by:

∫∫∫ G(x, y, z) dV

We can express the integral in terms of x, y, and z as follows:

∫∫∫ x dV

To evaluate this integral, we need to express the differential volume element dV in terms of x, y, and z. In this case, since we are integrating over a cylindrical region, we can express dV as dA dz, where dA represents the differential area element in the xy-plane.

The equation of the parabolic cylinder is y = x^2. To express the differential area element dA, we can use the Jacobian of the transformation from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z).

The Jacobian determinant is |J| = r, where r is the radial distance in the xy-plane.

Thus, dA = r dr dθ. However, since we are only interested in the region where 0 ≤ x ≤ √2, the limits of integration for r and θ will be determined by the given range of x.

For the given region, we have the following limits of integration:

0 ≤ x ≤ √2

0 ≤ z ≤ 2

To convert the function G(x, y, z) = x to cylindrical coordinates, we need to express x in terms of r and θ. In this case, x = r cos(θ).

Now we can rewrite the integral using cylindrical coordinates:

∫∫∫ x dV = ∫∫∫ (r cos(θ))(r dr dθ dz)

The limits of integration become:

0 ≤ r ≤ √2

0 ≤ θ ≤ 2π

0 ≤ z ≤ 2

We can now evaluate the integral:

∫∫∫ (r^2 cos(θ)) dr dθ dz

Integrating with respect to r first, we have:

∫∫ (r^3/3 cos(θ)) |r=0 to r=√2 dθ dz

Simplifying:

∫∫ (√2^3/3 cos(θ)) dθ dz

∫∫ (2√2/3 cos(θ)) dθ dz

Now integrating with respect to θ:

∫ (2√2/3 sin(θ)) |θ=0 to θ=2π dz

∫ (2√2/3)(0 - 0) dz

∫ 0 dz = 0

Therefore, the value of the integral ∫∫∫ G(x, y, z) dV over the given parabolic cylinder is 0.

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If the drawing of a circular water-cover, has the diameter of 1.5 feet, then it's circumference is 4.71 feet.

The "Circumference" of circle is the distance around the edge of a circle. It is the perimeter or the length of the boundary of the circle. The formula for the circumference(C) of a circle is given by : C = πd,

where "C" = circumference, "d" = diameter of circle, and π (pi) is a mathematical constant approximately equal to 3.14,

To find the circumference of a circular water-cover with a diameter of 1.5 feet, we substitute the value of diameter in formula of circumference:

We get,

⇒ C = π × d,

⇒ C = π × (1.5 feet),

Using  π as 3.14, We get,

⇒ C = 3.14 × (1.5 feet),

⇒ C = 4.71 feet,

Therefore, the circumference of the circular water-cover is 4.71 feet.

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The given question is incomplete, the complete question is

Find the circumference of the drawing of a circular water-cover, which has the diameter of 1.5 feet.

The final cost of the chair is $76.

Given that a chair has a sticker price of $100, with 20% discount, $20 off and $2 mail in rebate.

So, the price will be =

$100 - $20 - $100 × 0.20 - $2

= $80 - $2 - $2

= $80 - $4

= $76

Hence, the final cost of the chair is $76.

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a]Cannot occur due to violation of baryon number conservation, b] Can occur,  c] Can occur, d] Can occur but violates lepton number conservation, e] Can occur and f] Cannot o cur due to violation of charge and baryon number conservations.

The answer are as follows-  a) This reaction can occur as it conserves charge, baryon number, and lepton number.

b) This reaction cannot occur as it violates conservation of charge. The right side has one more positive charge than the left side.

c) This reaction cannot occur as it violates conservation of charge. The left side has zero charge while the right side has one positive charge.

d) This reaction can occur as it conserves charge and lepton number. However, it violates conservation of baryon number as the left side has a baryon number of one while the right side has a baryon number of zero.

e) This reaction can occur as it conserves charge, lepton number, and baryon number.

f) This reaction cannot occur as it violates conservation of charge. The left side has a positive charge while the right side has a neutral charge.

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The length of the line AZ is 24.

We have,

If A is between Y and Z, then the distance between Y and A plus the distance between A and Z should be equal to the distance between Y and Z. In other words, YA + AZ = YZ.

Substituting the given values, we have:

6 + AZ = 30

Subtracting 6 from both sides, we get:

AZ = 24

Therefore,

The length of AZ is 24.

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The simplified value represented in the complex number form |(8+9i)(8+9i)| is equal to 145.

The product of the complex numbers are,

|(8+9i)(8+9i)|

We can expand the expression (8 + 9i)(8 + 9i) using the FOIL method,

(8 + 9i)(8 + 9i)

= 8(8) + 8(9i) + 9i(8) + 9i(9i)

= 64 + 72i + 72i + 81(i²)

Value of i² = -1.

= 64 + 144i - 81

= -17 + 144i

Then, to find the absolute value of this complex number, we take the square root of the sum of the squares of its real and imaginary parts .

Modulus of complex number is,

|(-17 + 144i)|

= √((-17)² + (144)²)

= √(289 + 20736)

= √(21025)

= 145

Therefore, the value of the complex number |(8 + 9i)(8 + 9i)| is equal to 145.

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Answer:

88

Step-by-step explanation:

180 - 92 (circle theorem) quadrilateral add up to 180

The value of y at x = π is  [tex]y(\pi) = -e^{(-\pi/2)}sin(\pi/2 + 1)[/tex].

The given differential equation is a second-order linear homogeneous equation with constant coefficients. The characteristic equation is r² + 2r + 2 = 0, which has complex conjugate roots -1 + i and -1 - i. Therefore, the general solution is:

[tex]y(x) = e^{(-x/2)}(c_1cos(x/2) + c_2sin(x/2))[/tex]

Using the initial conditions y(0) = 0 and y'(0) = 1, we can solve for c₁ and c₂ as follows:

y(0) = 0 => c₁ = 0

[tex]y'(x) = -1/2 * e^{(-x/2)*sin(x/2)} + 1/2 * e^{(-x/2)*cos(x/2)[/tex]

y'(0) = 1 => 1/2 * c₂ = 1 => c₂ = 2

Therefore, the particular solution is:

[tex]y(x) = e^{(-x/2)} * 2 * sin(x/2) = 2e^{(-x/2)} * sin(x/2)[/tex]

Plugging in x = π, we get:

[tex]y(\pi) = 2e^{(-\pi/2)} * sin(\pi/2) = -e^{(-\pi/2) }|* sin(\pi/2 + 1)[/tex]

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The height of the jar is 20 centimeters when the radius is 4 centimeters.

The area that any three-dimensional solid occupies is known as its volume. These solids can take the form of a cube, cuboid, cone, cylinder, or sphere.

Let V be the total volume of the jar. Since the jar is one-fourth full, we know that the remaining three-fourths are empty.

Thus, we can write:

V = (4/3)πr²h

We can also write the volume of the baby food as:

20π = (1/4)πr²h

Simplifying this equation, we get:

80 = r²h

Now, we can substitute this value of r²h in the equation for the total volume of the jar:

V = (4/3)πr²h

V = (4/3)πr²(80/r²)

V = (4/3)π(80)

V = 320π

Therefore, the total volume of the jar is 320π cubic centimeters.

Now, we can use the formula for the volume of a cylinder to find the height of the jar:

320π = πr²h

320 = 16h

h = 20

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Answer:

5 centimeters

Step-by-step explanation:

The formula for the volume of a cylinder is:

[tex]\boxed{V=\pi r^2 h}[/tex]

where:

If a cylindrical jar is one-fourth full of baby food, and the volume of the baby food is 20π cm³, then the volume of the jar is 80π cm³.

[tex]\begin{aligned}\textsf{Volume of the jar}& = 4 \cdot 20\pi \\&=80 \pi \; \sf cm^3 \end{aligned}[/tex]

To calculate the height of the jar when its radius is 4 cm, substitute r = 4 and V = 80π into the formula for the volume of a cylinder, and solve for h:

[tex]\begin{aligned}V&=\pi r^2 h\\\implies 80\pi & = \pi (4)^2h\\80\pi & = 16\pi h\\80& = 16 h\\h&=80 \div 16\\h&=5\; \sf cm\end{aligned}[/tex]

Therefore, the height of the jar when the radius of the jar is 4 cm is:

The interval of continuity for the function f(x) is (1, 3) ∪ (5, ∞).

To find the interval of continuity for the function f(x), we need to check if it is continuous at all points within the domain of the function.

The given function is a composition of two continuous functions, sin(x) and ln(x-1)+√(x-5)/(x-3), which are continuous on their respective domains.

However, there are some restrictions on the domain of the function f(x) to ensure that the function is well-defined and continuous.

The expression inside the square root must be non-negative: x-5 ≥ 0, which gives x ≥ 5.

The denominator (x-3) cannot be equal to zero, so we have x ≠ 3.

Similarly, the argument of the natural logarithm must be positive: x-1 > 0, which gives x > 1.

Therefore, the domain of f(x) is the interval (1, 3) ∪ (5, ∞).

Now we need to check the continuity of f(x) at the endpoints of the intervals.

At x = 1, the function is undefined because the expression inside the logarithm becomes negative.

At x = 3, the function is also undefined because the denominator becomes zero.

Therefore, the interval of continuity for the function f(x) is (1, 3) ∪ (5, ∞).

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The dealers included in the sample would be the 5th dealer, the 9th dealer, the 13th dealer, the 17th dealer, and so on, depending on the total number of dealers on the list. This method of sampling is a systematic approach that helps ensure a representative and unbiased sample while still being efficient and random.

Using systematic random sampling, every fourth dealer is selected starting with the 5th dealer in the list. This means that the first dealer in the sample would be the 5th dealer on the list. Then, every fourth dealer after that would also be included in the sample. In this case, you will start with the 5th dealer and select every fourth dealer afterward. Here's the step-by-step explanation:

1. Start with the 5th dealer on the list (since that's your starting point).
2. Move 4 dealers down the list (because you're selecting every 4th dealer) and select the next dealer.
3. Repeat step 2 until you reach the end of the list.

By following these steps, you'll get the dealers included in the sample.

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The equation of the line is expressed in slope-intercept form as:

y = -5/6x - 7.

The equation of a line can be written in slope-intercept form as y = mx + b, where we have:

m = the slope

b = the y-intercept.

Find the slope (m):

Slope (m) = rise/run = -5/6

The y-intercept (b) is -7.

Substitute m = -5/6 and b = -7 into y = mx + b:

y = -5/6x - 7

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Answer:

55 total quizes

Step-by-step explanation:

45 divided by 9 = 5 which means Celine was taking five tests every week for nine weeks. After 11 weeks it had been two weeks since the 9 weeks which means 10 quizes. 45+10=55

To find the value of k for g(x) = f(x) + k, we compared the graphs of f(x) and g(x). We estimated the distance between the graphs at a common point, x=2, and found k to be approximately 3.25. So, the correct option is A).

We can determine the value of k by comparing the graphs of f(x) and g(x).

The graph of f(x) is a vertical asymptote at x=2, and it approaches zero as x moves away from 2 in either direction.

The graph of g(x) is also a vertical asymptote, but it occurs at x=-3. Moreover, the graph of g(x) is identical to the graph of f(x) shifted upwards by k units.

To find the value of k, we need to find the difference in y-values between the two graphs at any point. Let's take the point x=2, which is on the graph of f(x).

f(2) = 1 / (3(2) - 2) = 1/4

g(2) = f(2) + k = 1/4 + k

Since the graphs of f(x) and g(x) have the same shape and differ only by a vertical shift, we can see that the distance between the graphs at x=2 is equal to k.

Looking at the graph, we can estimate that the distance between the graphs at x=2 is approximately 3 units. Therefore, we have

k = g(2) - f(2) = (1/4 + 3) - 1/4 = 3 1/4

So the value of k is approximately 3.25. So, the correct answer is A).

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We can use the substitution u = 3x, which means du/dx = 3 and dx = du/3. Then we can rewrite the integral as:

∫f(3x) dx = ∫f(u) (du/3)

Using the substitution, we have changed the variable from x to u and also adjusted the differential to match. Now we can apply the substitution to the limits of integration:

When x = 0, u = 3(0) = 0

When x = 6, u = 3(6) = 18

So the new limits of integration are 0 and 18. Substituting these into the integral, we get:

∫f(3x) dx = ∫f(u) (du/3) from 0 to 18

Using the Fundamental Theorem of Calculus, we can evaluate this integral by finding an antiderivative of f(u). However, we don't actually need to find the antiderivative, because we know that:

∫f(x) dx = 40

This means that the definite integral of f(x) from any lower limit a to any upper limit b is just f(b) - f(a). Applying this to our integral, we get:

∫f(3x) dx = ∫f(u) (du/3) from 0 to 18
= (1/3) [f(18) - f(0)]

We don't know what f(18) or f(0) are, but we do know that the integral of f(x) from 0 to 6 is 40. That is:

∫f(x) dx = 40 from 0 to 6

Using the Fundamental Theorem of Calculus, we can write:

f(6) - f(0) = 40

Rearranging this equation, we get:

f(6) = 40 + f(0)

So we can substitute this expression into our earlier result to get:

∫f(3x) dx = (1/3) [f(18) - f(0)]
= (1/3) [f(6) + 40 - f(0)]

We don't know what f(0) is, but it doesn't matter because we're only interested in the difference f(6) - f(0). So we can simplify the expression as:

∫f(3x) dx = (1/3) [f(6) + 40 - f(0)]
= (1/3) [f(6) - f(0)] + 40/3

Recall that f(6) - f(0) = 40, so we can substitute this expression in:

∫f(3x) dx = (1/3) [40] + 40/3
= 80/3

Therefore, the value of the integral ∫f(3x) dx is 80/3.

To find the integral of f(3x) with respect to x from 1 to 6, you can use the substitution method. Let u = 3x. Then, du/dx = 3, so du = 3dx. When x = 1, u = 3, and when x = 6, u = 18.

Now, we substitute and adjust the integral:

∫(1 to 6) f(3x) dx = (1/3)∫(3 to 18) f(u) du

Since ∫f(x) dx = 40 (from x=a to x=b), we can use this information in our integral:

(1/3)∫(3 to 18) f(u) du = (1/3) * 40 = 40/3

So, the integral of f(3x) with respect to x from 1 to 6 is 40/3.

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The length of secant segment ED = 39 units

We know that the Intersecting Secants Theorem states that 'when two secants of a circleintersect at an exterior point, then the product of the one secant segment and its external secant segment is equal to the product of  the other secant segment and its external secant segment.'

Here, ABC and EDC are secants of a circle.

Using  Intersecting Secants Theorem,

AB × BC = ED × DC

Here, BC = 13, DC = 12 and AB = ED - 3

Substituting values in above equation we get,

(ED - 3) × 13 = ED × 12

13ED - 39 = 12ED

ED = 39 units

Therefore, ED = 39 units

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If Drew completes 11 laps around the circular track with a diameter of approximately 1/4 mile, he runs approximately 8.635 miles.

The distance that Drew runs can be calculated using the formula: distance = circumference x number of laps. The circumference of a circle can be found by multiplying its diameter by pi (π), which is approximately equal to 3.14.

Given that the diameter of the track is approximately 1/4 mile, its radius is 1/8 mile (since the radius is half of the diameter). Therefore, the circumference of the track is 2 x pi x 1/8 mile, which simplifies to pi/4 mile or approximately 0.785 miles.

To find the distance Drew runs in 11 laps, we simply multiply the circumference of the track by 11.

distance = 0.785 miles/lap x 11 laps

distance = 8.635 miles

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Complete question:

What is the approximate distance that Drew runs if he completes 11 laps around a circular track with a diameter of approximately 1/4 mile?

This shows that the difference between the eigenvalues of x for vector A and B is related to the commutator [A, B] and the eigenvector of x for matrix B.

To show that if x is an eigenvector of matrix A belonging to an eigenvalue λ, then x is also an eigenvector of matrix B belonging to an eigenvalue μ, we can start with the eigenvector equation for matrix A:

A x = λ x

Multiplying both sides by matrix B, we get:

B (A x) = B (λ x)

Using the associative property of matrix multiplication, we can rewrite the left side as:

(B A) x = (A B) x

Substituting the eigenvector equation for matrix A, we get:

(λ B) x = (A B) x

Since x is nonzero, we can divide both sides by x:

λ B = A B

This shows that if x is an eigenvector of matrix A belonging to eigenvalue λ, then it is also an eigenvector of matrix B belonging to eigenvalue μ = λ.

The matrices A and B are related through the commutator [A, B] = AB - BA. We can rewrite the equation λ B = A B as:

λ B - A B = [A, B] B

Since x is nonzero, we can multiply both sides by x:

λ B x - A B x = [A, B] B x

Using the eigenvector equation for matrix A and the fact that x is an eigenvector of matrix A, we get:

λ x - μ x = [A, B] B x

Simplifying, we get:

(λ - μ) x = [A, B] B x

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Answer: x = 3

Step-by-step explanation:

[tex]2x^2 - 12x + 18 = 0[/tex]

a = 2, b = -12, c = 18

plugging into the quadratic formula, which is:

[tex]\frac{-b +/- \sqrt{b^2 - 4ac} }{2a}[/tex]

we get two answers: x = 3. and x = 3.

as you can tell, theyre the same answer, so x = 3.

To find the relative and absolute extrema of the function f(x) = x^3/(x^2 - 48), we first find the derivative:

f'(x) = (3x^2(x^2 - 48) - 2x(x^3))/(x^2 - 48)^2

= (x^4 - 144x)/(x^2 - 48)^2

We can see that f'(x) is defined for all x except x = 0 and x = ± 6√2. To find the critical points, we set f'(x) = 0:

(x^4 - 144x)/(x^2 - 48)^2 = 0

x(x^3 - 144)/(x^2 - 48)^2 = 0

The numerator is zero when x = 0 or x = ±6, but x = 0 and x = ±6 are not in the domain of f(x). Therefore, there are no critical points in the domain of f(x).

Next, we check the endpoints of the domain of f(x), which are x = ±∞. We take the limit as x approaches infinity:

lim x→∞ f(x) = lim x→∞ (x^3/(x^2 - 48))

= lim x→∞ (x/(1 - 48/x^2)) (by dividing numerator and denominator by x^2)

= ∞

Similarly, we take the limit as x approaches negative infinity:

lim x→-∞ f(x) = lim x→-∞ (x^3/(x^2 - 48))

= lim x→-∞ (x/(1 - 48/x^2))

= -∞

Therefore, there is no absolute maximum but there is an absolute minimum at x = -∞.

Since there are no critical points in the domain of f(x), there are no relative extrema. Therefore, the function has an absolute minimum at x = -∞ and does not have any maximums or minimums in the domain.

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To write an expression that selects all words with at least five letters from a list, you can use a list comprehension in Python and that are word, list, filtered, for, kite.

List comprehensions provide a concise way to create new lists by filtering or modifying elements from an existing list.
Here's an example using the words you provided:
```python
words_list = ['being', 'for', 'the', 'benefit', 'of', 'mister', 'and', 'kite']
filtered_words = [word for word in words_list if len(word) >= 5]
```

In this example, the list comprehension iterates through each word in `words_list` and checks if its length (`len(word)`) is greater than or equal to 5. If the condition is met, the word is added to the new `filtered_words` list. The result will be `['being', 'benefit', 'mister']`, which are the words with at least five letters in the original list.

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Answer:

C

Step-by-step explanation:

note that

[tex]6^{0}[/tex] = 1

[tex]6^{1}[/tex] = 6

[tex]6^{2}[/tex] = 36

6³ = 216

that is 6 raised to the power of d gives the corresponding values of c

then explicit formula is

[tex]a_{d}[/tex] = [tex]6^{d}[/tex]

The probability of exactly two of five independently selected droplets exceeding 1500 μm is approximately 0.0000072.

We can use the normal distribution model to calculate the probability of a droplet having a size larger than a certain value. Let X be the size of a droplet, then X follows a normal distribution with mean μ = 1050 μm and standard deviation σ = 150 μm.

To calculate the probability that exactly two of five droplets exceed 1500 μm, we can use the binomial distribution. Let Y be the number of droplets that exceed 1500 μm, then Y follows a binomial distribution with parameters n = 5 and p = P(X > 1500), where P(X > 1500) is the probability of a droplet having a size larger than 1500 μm.

To find P(X > 1500), we need to standardize the distribution by subtracting the mean and dividing by the standard deviation. Let Z = (X - μ) / σ be the standardized random variable, then Z follows a standard normal distribution with mean 0 and standard deviation 1.

P(X > 1500) = P(Z > (1500 - 1050) / 150)

= P(Z > 3.67)

= 1 - P(Z < 3.67)

Using a standard normal table or calculator, we can find that P(Z < 3.67) = 0.9998

So P(X > 1500) = 1 - 0.9998

= 0.0002.

Now we can use the binomial distribution to calculate the probability of exactly two droplets exceeding 1500 μm:

P(Y = 2) = (5 choose 2) * (0.0002)² * ( 0.0002)³

= 0.0000072

Therefore, the probability of exactly two of five independently selected droplets exceeding 1500 μm is approximately 0.0000072.

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The initial value problem y''+ty'-2y=6-t, y(0) =0, y'(0) =1 whose Laplace transform exists by taking the Laplace transform of the given differential equation, simplifying it, and then using partial fractions to separate the terms.  The solution to the initial value problem is: y(t) = 2 - 4e^{-2t} + 2e^{t} where y(0) = 0 and y'(0) = 1.

To solve the initial value problem, we first need to take the Laplace transform of the given differential equation:

L{y''} + L{ty'} - L{2y} = L{6-t}

Using the properties of Laplace transforms, we can simplify this equation to: s^2 Y(s) - s y(0) - y'(0) + s Y(s) - y(0) - 2 Y(s) = 6/s - L{t}

Substituting in the initial values y(0) = 0 and y'(0) = 1, we get: s^2 Y(s) + s Y(s) - 2 Y(s) = 6/s - L{t} Simplifying further, we can write this equation as: Y(s) = (6/s - L{t}) / (s^2 + s - 2)

To find the inverse Laplace transform of this equation, we need to factor the denominator as (s+2)(s-1) and then use partial fractions to separate the terms: Y(s) = (2/s) - (4/(s+2)) + (2/(s-1))

Taking the inverse Laplace transform of each term, we get: y(t) = 2 - 4e^{-2t} + 2e^{t} Therefore, the solution to the initial value problem is: y(t) = 2 - 4e^{-2t} + 2e^{t} where y(0) = 0 and y'(0) = 1.

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