what the slope and the y intercept for-12x+4y=-8

False. The derivative of the function f(x) = 5sin(x) + cos(x) is not equal to -5cos(x) - sin(x). The correct derivative of f(x) can be obtained by applying the rules of differentiation.

To find the derivative, we differentiate each term separately. The derivative of 5sin(x) is obtained using the chain rule, which states that the derivative of sin(u) is cos(u) multiplied by the derivative of u. In this case, u = x, so the derivative of 5sin(x) is 5cos(x).

Similarly, the derivative of cos(x) is obtained as -sin(x) using the chain rule.

Therefore, the derivative of f(x) = 5sin(x) + cos(x) is:

f'(x) = 5cos(x) - sin(x).

This result shows that the derivative of f(x) is not equal to -5cos(x) - sin(x).

In summary, the statement that f'(x) = -5cos(x) - sin(x) is false. The correct derivative of f(x) = 5sin(x) + cos(x) is f'(x) = 5cos(x) - sin(x).

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Answer:

we bought 12 $3 clearance mystery books and 5 $8 regular-priced science fiction books.

Step-by-step explanation:

The coefficient of variation for a)0.394 b)0.850 c)0.523 if standard deviation values are a $ 2,710 $ 1,070 b 2,140 1,820 c 2,160 1,130

The coefficient of variation as compared to  standard deviation is a factual proportion of the scattering of data of interest around the mean. The measurement is usually used to analyze the information scattering between particular series of information.

Dissimilar to the standard deviation that must continuously be viewed as with regards to the mean of the information, the coefficient of variation tells a somewhat straightforward and fast instrument to look at changed information series.

We know very well that coefficient of variation is defined as the ratio of standard deviation to the expected value, or in other words

Coefficient of variation=standard deviation/expected value

a)Standard deviation value=$1,070 and expected value is $2,710

Therefore, coefficient of variation=(1070/2710)=0.394

b)Standard deviation value=$1,820 and expected value is $2,140

Therefore, coefficient of variation=(1820/2140)=0.850

c)Standard deviation value=$1130 and expected value is $2,160

Therefore, coefficient of variation=(1130/2160)=0.523

Hence, coefficient of variation value is a)0.394 b)0.850 c)0.523

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Answer: x = 8

Step-by-step explanation:

No, f(x, y) = 3x - 3y^2 is not a harmonic function.

A harmonic function is a twice continuously differentiable function f(x, y) that satisfies the Laplace equation:
                                                                            ∂²f/∂x² + ∂²f/∂y² = 0.

Let's check if f(x, y) = 3x - 3y^2 satisfies the Laplace equation:
∂²f/∂x² = ∂/∂x(3) = 0
∂²f/∂y² = ∂/∂y(-6y) = -6
∂²f/∂x² + ∂²f/∂y² = 0 + (-6) = -6 ≠ 0

Since the Laplace equation is not satisfied, f(x, y) = 3x - 3y^2 is not a harmonic function.

As for finding a corresponding analytic function, an analytic function is a function that is locally given by a convergent power series. Since f(x, y) is not a harmonic function, there is no corresponding analytic function.

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Answer:

0.1456858

Step-by-step explanation:

Hope it help

Answer:

−1/2

Explanation:

cos(8π/3)

=cos(8π/3 − 2π)

=cos(2π/3) = − cos (π/3) = −1/2

The third word in the series is an a x r², and this is the answer to the given question based on the sequence.

A progression in mathematics is a particular form of sequence where the distance between succeeding terms is constant. A collection of numbers or other mathematical elements arranged in a specific order is called a sequence.

Arithmetic progressions, geometric progressions, and harmonic progressions are only a few of the several forms of progressions. The formula for the nth term of the sequence varies depending on the type of progression.

By dividing the first term by the common ratio r, one may get the second term in the sequence:

Second term = a x r

The second term can also be multiplied by the common ratio r to find the third term:

Third term = (a x r) x r = a x r²

As a result, an a x r² is the third term in the series.

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Given:

The rate of change in volume = 12 cubic feet per minute.

We need to find the rate of change of radius at radius = 6 feet.

Consider the formula to find the volume of the sphere.

Differentiate with respect to t.

Dividing both sides by 144pi, we get

Hence the radius of the balloon increases by 0.03 feet per minute.

The distance from the sun to the other focus is 5.01 × 10⁹ m.

(a) The distance from the center of an ellipse to a focus is an where a is the semi major axis and e is the eccentricity. Thus, the separation of the foci ( in the case of Earth's orbit ) is;

2ae = 2(1.50 × 10¹¹)(0.0167) = 5.01 × 10⁹ m.

(b) To express this in terms of solar radii, we set up a ratio;

(5.01 × 10⁹)/(6.96 × 10⁸) = 7.2

Thus, we can conclude that the distance from the sun to the other focus is 5.01 × 10⁹ m.

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Complete question is;

The Sun's center is at one focus of Earth's orbit. How far from this focus is the other focus, (a) in meters and (b) in terms of the solar radius, 6.96 × 10⁸ m? The eccentricity is 0.0167, and the semimajor axis is 1.50 × 10¹¹ m.

The value of x is approximately 0.8 cm when rounded to the nearest tenth.

A triangle is a two-dimensional geometric shape that has three sides and three angles. It is the simplest polygon and is a fundamental building block in many areas of mathematics and science.

The three sides of a triangle can have different lengths and can be of different types, such as equilateral, isosceles, or scalene. An equilateral triangle has three sides of equal length, while an isosceles triangle has two sides of equal length and a third side of a different length. A scalene triangle has all three sides of different lengths.

In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Thus, we have:

sin(IHJ) = IJ/IH

Taking the sine of both sides and substituting the given values, we get:

sin(23°) = x/2

Multiplying both sides by 2, we have:

x = 2*sin(23°)

Using a calculator, we get:

x ≈ 0.83

Therefore, the value of x is approximately 0.8 cm when rounded to the nearest tenth.

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Answer:

Question 7: Company A has a lower rate

Question 8: Caiden's vehicle gets more miles per gallon

Step-by-step explanation:

Question 7:

According to the table for Company B's rates, if you divide the total cost by the square footage in each row, you would get $2.75/ft² each time. Therefore, Company A would charge the Smiths a lower cost per square foot.

$2.25/ft² < $2.75/ft²

Company A < Company B

Question 8:

Using the given equation, we already know that Caiden's vehicle has a gas mileage of 16.2 miles/gallon The graph for Ethan's gas mileage can be figured out using slope. Taking the points (0, 0) and (2, 30) on the line, we can figure out the line's slope.

slope = rise/run,

where "rise" is how much you add or subtract to get from the first y-value to the second, and "run" is how much you add or subtract to get from the first x-value to the second.

slope = rise/run

slope = 30/2

slope = 15/1

The slope of the line is 15, so that means the line's equation is y = 15x. Therefore, Ethan's gas mileage is 15 miles/gallon.

16.2 mpg > 15 mpg

Caiden > Ethan

Answer:

irst, 4865 rounded to the nearest ten is:

4870

Step-by-step explanation:

Here we will tell you what 4865 is rounded to the nearest ten and also show you what rules we used to get to the answer.

Remember, we did not necessarily round up or down, but to the ten that is nearest to 4865.

When rounding to the nearest ten, like we did with 4865 above, we use the following rules:

A) We round the number up to the nearest ten if the last digit in the number is 5, 6, 7, 8, or 9.

B) We round the number down to the nearest ten if the last digit in the number is 1, 2, 3, or 4.

C) If the last digit is 0, then we do not have to do any rounding, because it is already to the ten.

Answer:

The answer is 13.5

Step-by-step explanation:

Answer:

(x - 3)^2 + (y - 2)^2 = 4^2

Step-by-step explanation:

Looking at the graph its obviously a circle as it said.

We have to find (h,k) and the radius of that circle.

(h, k)  = (3,2)

Origin to the edge of the A.

radius = 4

(x - 3)^2 + (y - 2)^2 = 4^2

It looks like the system of ODEs is

\(\begin{cases} 2x' + y' - 4x - 4y = e^{-t} \\ x' + y' + 2x + y = e^{4t} \end{cases}\)

Differentiate both sides of both equations with respect to \(t\).

\(\begin{cases} 2x'' + y'' - 4x' - 4y' = -e^{-t} \\ x'' + y'' + 2x' + y' = 4e^{4t} \end{cases}\)

Eliminating the exponential terms, we have

\((2x' + y' - 4x - 4y) + (2x'' + y'' - 4x' - 4y') = e^{-t} + (-e^{-t}) \\\\ \implies (2x'' - 2x' - 4x) + (y'' - 3y' - 4y) = 0\)

\((x'' + y'' + 2x' + y') - 4 (x' + y' + 2x + y) = 4e^{4t} - 4\cdot e^{4t} \\\\ \implies (x'' - 2x' - 8x) + (y'' - 3y' - 4y) = 0\)

Now we can eliminate \(y\) and it derivatives.

\(\bigg((2x'' - 2x' - 4x) + (y'' - 3y' - 4y)\bigg) - \bigg((x'' - 2x' - 8x) + (y'' - 3y' - 4y)\bigg) = 0 - 0 \\\\ \implies x'' + 4x = 0\)

Solve for \(x\). The characteristic equation is \(r^2 + 4 = 0\) with roots at \(r=\pm2i\), hence the characteristic solution is

\(\boxed{x(t) = C_1 \cos(2t) + C_2 \sin(2t)}\)

Solve for \(y\). Substituting \(x\) into the second ODE gives

\(x' + y' + 2x + y = e^{4t} \\\\ \implies y' + y = e^{4t} + C_1 \cos(2t) + C_2 \sin(2t)\)

The characteristic equation this time is \(r + 1 = 0\) with a root at \(r=-1\), hence the characteristic solution is

\(y(t) = C_3 e^{-t}\)

Assume a particular solution with unknown coefficients \(a,b,c\) of the form

\(y_p = ae^{4t} + b \cos(2t) + c \sin(2t) \\\\ \implies {y_p}' = 4ae^{4t} - 2b\sin(2t) + 2c\cos(2t)\)

Substituting into the ODE gives

\(5ae^{4t} + (b+2c) \cos(2t) + (-2b+c) \sin(2t) = e^{4t} + C_1 \cos(2t) + C_2 \sin(2t) \\\\ \implies \begin{cases}5a = 1 \\ b+2c = C_1 \\ -2b+c = C_2\end{cases} \\\\ \implies a=\dfrac15, b=\dfrac{C_1-2C_2}5, c=\dfrac{2C_1+C_2}5\)

so that the general solution is

\(\boxed{y(t) = \dfrac15 e^{4t} + \dfrac{C_1-2C_2}5 \cos(2t) + \dfrac{2C_1+C_2}5 \sin(2t) + C_3 e^{-t}}\)

As per the information provided, a = 4√3/3 + 4, b = 4 - 4√3/3 the answer can be calculated with optimization method. it will be as follows:

Sum of a and b is 8, we get

a+b=8

b=8−a

Now, let x=a−b

Then we get,

\(x=a−(8−a) \\ x=2a−8 \\ x+8=2 \\ 1 \div 2x+4=a

\)

we use this to answer to solve for b

\(b=8−a \\ =8−(1 \div 2x+4) \\ =4−1 \div 2x

\)

Now we use the product of two numbers and its difference. This can be expressed as:

\(a⋅b⋅x=(1 \div 2x+4)(4−1 \div 2x) \\ x=2 {x}^{2} − \frac{1}{4} {x}^{3} +16x−2x^{2} \\ =−14x^{3} +16x

\)

Thus, this expression that we need to maximize. Take the derivative, set it equal to zero, and solve for x

\(−3 \div 4x ^{2} +16=0 \\ 16 =3 \div 4 x ^{2} \\ 643=x \\ 28√3 \div 3=x\)

Now for us to check that this is a maximum, we have to note that the second derivative is

\(−3 \div 2x

\\ At \\

x=8√3 \div 3

\)

the second derivative is −4√3. Since this number is negative, the point is a maximum.

Now we must find the values of a and b for this x. We have to use the relationship

\(a=1 \div 2x+4\)

\(a=1 \div 2 \times 8√3 \div 3+4 \\ =4√3 \div 3+4\)

now we use the relationship b=8−a

\(b=8−(4√3 \div 3+4) \\ =4−4√3 \div 3\)

The first step in determining a function's maximum or minimum value is differentiating it. Then, set this derivative to zero and conduct the computation.

x. This will reveal the location of a function's maximum or minimum, but it won't reveal which.

Take the second derivative to get more details. A local maximum occurs when both the first and second derivatives are negative. You have a local minimum when both the first derivative and the second derivative are zero.

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Answer:

x = 12

Step-by-step explanation:

67+42+6x-1=180

108+6x=180

6x=72

x=12

Answer:

11) 6

12) 3

Step-by-step explanation:

2 log 2 8

so 2(log2 8)

Base 2 so 2^x=8 so 3.

2*3 is 6.

log a a cubed

base a so a^ x = a^3

x=3

so 3.

Hope this helps ;D

Answer:

6 and 3

Step-by-step explanation:

Using the rules of logarithms

log\(x^{n}\) ⇔ n log x

\(log_{b}\) x = n ⇒ x = \(b^{n}\)

\(log_{a}\) a = 1

(11)

let 2 \(log_{2}\)8 = n , then

\(log_{2}\) 8² = n

\(log_{2}\) 64 = n

\(log_{2}\) \(2^{6}\) = n

\(2^{6}\) = \(2^{n}\)

Since bases on both sides are equal, equate the exponents, that is

n = 6

(12)

\(log_{a}\) a³

= 3 \(log_{a}\) a

= 3 × 1

= 3

a.we get, `A = √(2α³/π)`.

b.`⟨H⟩ = (3/4)hω - (h²/4ma²)` where `a = α/√(mω/h)`.

c.we can say that the variational principle is valid in this case.

(a) Let's find the normalization constant A.

We know that the integral over all space of the absolute square of the wave function is equal to 1, which is the requirement for normalization.  `∫⟨ψ|ψ⟩dx= 1`

Hence, using the given trial wavefunction, we get, `∫⟨ψ|ψ⟩dx = ∫ |A/(x^2+α²)|²dx= A² ∫ dx / (x²+α²)²`

Using a substitution `x = α tan θ`, we get, `dx = α sec² θ dθ`

Substituting these in the above integral, we get, `A² ∫ dθ/α² sec^4 θ = A²/(α³) ∫ cos^4 θ dθ`

Using the identity, `cos² θ = (1 + cos2θ)/2`twice, we can write,

`A²/(α³) ∫ (1 + cos2θ)²/16 d(2θ) = A²/(α³) [θ/8 + sin 2θ/32 + (1/4)sin4θ/16]`

We need to evaluate this between `0` and `π/2`. Hence, `θ = 0` and `θ = π/2` limits.

Using these limits, we get,`⟨ψ|ψ⟩ = A²/(α³) [π/16 + (1/8)] = 1`

Therefore, we get, `A = √(2α³/π)`.

Hence, we can now write the wavefunction as `ψ(x) = √(2α³/π)/(x²+α²)`.  

(b) Using the wave function found in part (a), we can now determine the expectation value of energy using the time-independent Schrödinger equation, `Hψ = Eψ`. We can write, `H = (p²/2m) + (1/2)mω²x²`.

The first term represents the kinetic energy of the particle and the second term represents the potential energy.

We can write the first term in terms of the momentum operator `p`.We know that `p = -ih(∂/∂x)`Hence, we get, `p² = -h²(∂²/∂x²)`Using this, we can now write, `H = -(h²/2m) (∂²/∂x²) + (1/2)mω²x²`

The expectation value of energy can be obtained by taking the integral, `⟨H⟩ = ⟨ψ|H|ψ⟩ = ∫ψ* H ψ dx`Plugging in the expressions for `H` and `ψ`, we get, `⟨H⟩ = - (h²/2m) ∫ψ*(∂²/∂x²)ψ dx + (1/2)mω² ∫ ψ* x² ψ dx`Evaluating these two integrals, we get, `⟨H⟩ = (3/4)hω - (h²/4ma²)` where `a = α/√(mω/h)`.

(c) Since we have an approximate ground state wavefunction, we can expect that the expectation value of energy ⟨H⟩ should be greater than the true ground state energy.

Hence, the value obtained in part (b) should be greater than the true ground state energy obtained by solving the Schrödinger equation exactly.

Therefore, we can say that the variational principle is valid in this case.

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Answer:

The length of the longest side is 9 inches

Step-by-step explanation:

The given parameters for the quadrilateral is that the length of the sides are;

The length of the sides are consecutive integers =  x, x + 2, x + 4, x + 6

The perimeter of the rectangle = 36 inches

The side length to be found = The length of the longest side = x + 3

2 6 10 14 18

∴ x + x + 2 + x + 4 + x + 6 = 36

4·x + 12 = 36

4·x = 36 - 12 = 24

4·x = 24

x = 24/4 = 6

x = 6 inches

The length of the longest side = x + 3 = 6 + 3 = 9

The length of the longest side = 9 inches.

Answer:

The length of the longest side is 9 inches

Step-by-step explanation:

The given parameters for the quadrilateral is that the length of the sides are;

The length of the sides are consecutive integers =  x, x + 2, x + 4, x + 6

The perimeter of the rectangle = 36 inches

The side length to be found = The length of the longest side = x + 3

2 6 10 14 18

∴ x + x + 2 + x + 4 + x + 6 = 36

4·x + 12 = 36

4·x = 36 - 12 = 24

4·x = 24

x = 24/4 = 6

x = 6 inches

The length of the longest side = x + 3 = 6 + 3 = 9

The length of the longest side = 9 inches.

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Answer:

the desired equation is y = (-1/3)x + 6

Step-by-step explanation:

Let the given line be A:  y=3x+2

The slope of line A is m = 3.

The slope of any line B which is perpendicular to line A is the negative reciprocal of the slope of A:  m = -(1/3).

The particular perpendicular line that passes through (3, 5) is then

5 = (-1/3)(3) + b, which simplifies to 5 = -1 + b, or b = 6.

Thus, the desired equation is y = (-1/3)x + 6

First, we want to note two things:

We can describe this with an inequality:  x ≥ -10
Be sure you use ≥ and not >, since -10 is included.

We can describe this with interval notation: [ -10, infty )
Be sure you use [ and not ( on -10, since -10 is included.

You can also use set-builder notation: { x | x ≥ -10 }

Check the picture below.

\(\cfrac{17.5+14}{17.5}~~ = ~~\cfrac{x}{12.5}\implies \cfrac{(17.5+14)(12.5)}{17.5}~~ = ~~x\implies 22.5=x\)

Answer:√2/2

Step-by-step explanation:

Let's label the sides of the right triangle as follows:

The side adjacent to the angle θ (cosine is adjacent/hypotenuse): 6

The hypotenuse (the longest side): 6√2

The side opposite to the angle θ (sine is opposite/hypotenuse): 6√3

Using the Pythagorean theorem, we can find the length of the missing side:

a² + b² = c²

6² + (6√3)² = (6√2)²

36 + 108 = 72

144 = 72

√144 = √72

12 = 6√2

Now that we know the length of all three sides, we can use the cosine ratio to find the value of cos(θ):

cos(θ) = adjacent/hypotenuse = 6/6√2 = √2/2

Therefore, the value of cos(θ) in the right triangle with sides of 6, 6√2, and 6√3 is √2/2.

A systematic sample is a sampling method in which members from a population are selected in a regular and systematic way.

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Hi! I'd be happy to help you write the sum in sigma notation. Given the sum: 3 - 3x + 3x^2 - 3x^3 + , + (-1)^n * 3x^n, the sigma notation would be:

Σ[(-1)^k * 3x^k] from k=0 to n

Here's a step-by-step explanation:

1. Identify the pattern in the sum: It alternates between positive and negative terms, and each term has a power of x multiplied by 3.
2. Assign the variable k for the index of summation.
3. Determine the range of k: The sum starts with k=0 and goes up to k=n.
4. Represent the alternating sign using (-1)^k.
5. Combine all components to form the sigma notation: Σ[(-1)^k * 3x^k] from k=0 to n.

The sum can be written in sigma notation as:

\($\displaystyle\sum_{n=1}^\infty (-1)^n 3x^n$\)

The given series is:

\(3 - 3x + 3x^2 - 3x^3 + ...\)

To write it in sigma notation, we first notice that the terms alternate in sign, and each term is a power of x multiplied by a constant (-3). We can write the general term of the series as:

\((-1)^n * 3 * x^n\)

where n is the index of the term, starting from n = 0 for the first term.

Using sigma notation, we can express the sum of the series as:

\($\displaystyle\sum_{n=1}^\infty (-1)^n 3x^n$\)

where the summation is over all values of n starting from n = 0.

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Answer: Choice B)   sin (33) = x/14

This is because the sine of an angle is equal to the opposite side over the hypotenuse

sin(angle) = opposite/hypotenuse

sin(33) = x/14

We can't use cosine because we don't have a variable for the adjacent side.