Random variable X has a normal distribution with mean u and standard deviation 2. The pdf f(x) of X satisfies the following conditions: (A) f6 > f(16), (B) f(1)

The mass, in grams, of a piece of metal that has a volume of 3.83.8 cubic centimeters is approximately 0.3 g (rounded to the nearest tenth of a gram).

To determine the mass, in grams, of a piece of metal that has a volume of 3.83.8 cubic centimeters, we can use the proportional relationship between the mass and the volume of the pieces of metal. The table below lists the masses and volumes of several pieces of the same type of metal:

Volume (cubic centimeters)  Mass (grams)

72.7 31.29314.1 47.519112.1 140.239

We can find the mass of a piece of metal that has a volume of 3.83.8 cubic centimeters by using the proportional relationship between the masses and the volumes of the pieces of metal.

Here's how:

1.

We need to find the constant of proportionality that relates the masses and the volumes.

To do this, we can use any two pairs of values from the table.

Let's use the first and second pairs:

(mass) / (volume) = (31.293 g) / (72.7 cm³)

(mass) / (volume) = (47.519 g) / (14.1 cm³)

We can cross-multiply to get:

(31.293 g) × (14.1 cm³) = (72.7 cm³) × (mass)

(47.519 g) × (72.7 cm³) = (14.1 cm³) × (mass)

2.

We can solve for the mass in either equation.

Let's use the first one:

(31.293 g) × (14.1 cm³) = (72.7 cm³) × (mass)

mass = (31.293 g) × (14.1 cm³) / (72.7 cm³)

mass = 6.086 g

We have found that the mass of a piece of metal that has a volume of 72.7 cm³ is 6.086 g.

This means that the constant of proportionality is 6.086 g / 72.7 cm³ ≈ 0.08383 g/cm³.

3.

Finally, we can use the constant of proportionality to find the mass of a piece of metal that has a volume of 3.83.8 cubic centimeters.

We can use this formula:

(mass) / (volume) = 0.08383 g/cm³

mass = (volume) × 0.08383 g/cm³

mass = 3.83.8 cm³ × 0.08383 g/cm³

mass ≈ 0.321 g

Therefore, the mass, in grams, of a piece of metal that has a volume of 3.83.8 cubic centimeters is approximately 0.3 g (rounded to the nearest tenth of a gram).

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a) The function that would best model this situation is a quadratic function since the height of the lightsaber changes with time at a constant rate.

b) To evaluate h(4), we substitute s = 4 into the function:

h(4) = -15(4)^2 + 405

h(4) = -15(16) + 405

h(4) = -240 + 405

h(4) = 165

Therefore, the height of the lightsaber after 4 seconds is 165 feet.

what is function?

In mathematics, a function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. It can be represented using a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output.

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The product in standard form that is the area as sum of the generic rectangle is given by 6x² - 5x - 4.

Given the expression is:

(3x - 4)(2x + 1)

Multiplying the algebraic terms we get,

(3x - 4)(2x + 1)

= (3x)*(2x) - 4*(2x) + 1*(3x) - 4*1

= 6x² - 8x + 3x - 4

= 6x² + (3 - 8)x - 4

= 6x² + (-5)x - 4

= 6x² - 5x - 4

Hence the product of the algebraic expressions that is the area as sum of the generic rectangle is given by 6x² - 5x - 4.

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The equation f(x) = (-9 - 3x)(x + 4), as represented is in its factored form

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

f(x) = (-9-3x)(x+4)

Express properly

f(x) = (-9 - 3x)(x + 4)

The above equation is a quadratic function

As a general rule, a quadratic function in factored form is represented as

f(x) = (ax + b)(cx + d)

When the equation are compared, we have

a = -3, b = -9

c = 1 and d = 4

This means that the equation f(x) = (-9 - 3x)(x + 4) is in factored form

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The probability of committing a Type I error when the null hypothesis is true as an equality is d. The level of significance.

The level of significance, also known as alpha, is the threshold value that is used to determine if a result is statistically significant or not. It is the maximum probability of committing a Type I error that researchers are willing to accept.

                             A lower level of significance will decrease the probability of committing a Type I error, but it will increase the probability of committing a Type II error (failing to reject a false null hypothesis). It is important to carefully select an appropriate level of significance in order to balance these two types of errors.

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To express 10cos30 - 3tan60 in the form of a square root of k, where k is an integer, we can use the fact that cosine and tangent are both periodic functions with a period of 2π.

Specifically, we can write:

10cos30 - 3tan60 = 10cos(30 + 2π) - 3tan(60 + 2π)

= 10cos(30) - 3tan(60)

= 10(cos(30) - sin(30)sin(60))

= 10(cos(30) - sin(60))

= 10cos(60)

Therefore, 10cos30 - 3tan60 is equal to 10cos(60), which is in the form of a square root of k, where k is an integer.

So the answer is:

10cos30 - 3tan60 = 10cos(60)

or in the form of a square root of k:

sqrt(10)(cos(60))

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The original length of the reed is 45.

Given: A reed was broken off a cubit. This reed fitted 60 times along the length of the field. After restoring what was broken off, it fitted 30 times along the width. The area of the field is 525 square nindas

To find: Original length of the reedIn order to solve the problem,

let’s first define the reed length as x. It means the length broken from the reed is x-1. We know that after the broken reed is restored it fits 30 times in the width of the field.

It means;The width of the field = (x-1)/30Next, we know that before breaking the reed it fit 60 times in the length of the field. After breaking and restoring, its length is unchanged and now it fits x times in the length of the field.

Therefore;The length of the field = x/(60/ (x-1))= x (x-1) /60

Now, we can use the formula of the area of the field to calculate the original length of the reed.

Area of the field= length x widthx

(x-1) /60 × (x-1)/30

= 525 2(x-1)2

= 525 × 60x²- 2x -1785

= 0(x-45)(x+39)=0

x= 45 (as x cannot be negative)

Therefore, the original length of the reed is 45. Hence, the answer in 100 words is: The original length of the reed was 45. The width of the field is given as (x-1)/30 and the length of the field is x (x-1) /60, which is obtained by breaking and restoring the reed.

Using the area formula of the field (length × width), we get x= 45.

Thus, the original length of the reed is 45. This is how the original length of the reed can be calculated by solving the given problem.

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The LP problem min z = -2x1 - x² s.t. x - x² = 2, x + x² = 6, x , x2 =(non-negativity), the basic feasible solution in standard form is (x, x², s, s²) = (0, 0, 2, 6).

For the linear programming (LP) problem. The given problem is:
Minimize z = -2x - x²
Subject to:
x - x² <= 2
x + x² <= 6
x, x² >= 0 (non-negativity)
First, let's convert the problem into standard form by introducing slack variables to eliminate inequalities:
x- x² + s = 2
x + x² + s² = 6
x, x², s, s² >= 0
Now, let's construct a basic feasible solution at which (x1, x2) = (0, 0):
0 - 0 + s = 2 => s = 2
0 + 0 + s² = 6 => s² = 6
So, the basic feasible solution in standard form is (x, x², s, s²) = (0, 0, 2, 6).

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One way that adding and subtracting polynomials is similar to adding and subtracting whole numbers and integers is that both operations follow the same basic rules for combining like terms.

In both cases, you add or subtract the coefficients (numbers) of the same type of term or same variable with the same exponent.

Just like adding and subtracting integers, you also need to consider the signs (+ or -) when combining the terms.

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The complete equation of (5x + ....)² = ....*x² + 70xy +  ....  is 25² + 70xy + 49y²

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

(5x + ....)² = ....*x² + 70xy +  ....

Rewrite the expression as

(5x + ay)² = ....*x² + 70xy +  ....

When expanded, we have

(5x + ay)² = 25x² + 2 * 5x * ay + (ay)²

Evaluate the products

So, we have

(5x + ay)² = 25x² + 10axy + (ay)²

This means that

10axy = 70xy

So, we have

a = 7

The equation becomes

(5x + ay)² = 25x² + 10 * 7xy + (7y)²

Evaluate

(5x + ay)² = 25x² + 70xy + 49y²

Hence, the complete equation is 25² + 70xy + 49y²

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The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be 5625.

A permutation is an act of arranging items or elements in the correct order.

There are 3 yellow marbles, 5 blue marbles, and 5 green marbles.

The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be

[tex]\Rightarrow (3 \times 5 \times 5)^2[/tex]

[tex]\Rightarrow 75^2[/tex]

[tex]\Rightarrow 5625[/tex]

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Answer:

16 mm

Step-by-step explanation:

P = 2(L + W)

P = 2(2 mm + 6 mm)

P = 2(8 mm)

P = 16 mm

The original series ∑n=0[infinity]15 6n is always larger than the convergent series ∑n=0[infinity] 6n, we can also conclude that the original series converges by the direct comparison test.

To determine whether the series ∑n=0[infinity]15 6n converges or diverges, we can use the direct comparison test.
First, we need to find a series that is easier to analyze but still has a similar behavior as the original series.

In this case, we can compare the original series to the series ∑n=0[infinity] 6n.

We can see that the terms of the original series are always larger than the terms of the comparison series since the original series starts at n=0 and goes up to n=15 while the comparison series starts at n=0 and goes up to infinity.

Therefore, we can say that for all n ≥ 15,

6n ≤ 15 × 6n

Now, we can compare the two series using the direct comparison test. Since

∑n=0[infinity] 15 × 6n

converges (it is a geometric series with a ratio 6/15 < 1), we can conclude that
∑n=0[infinity] 6n

converges as well.

Since the original series ∑n=0[infinity]15 6n is always larger than the convergent series ∑n=0[infinity] 6n, we can also conclude that the original series converges by the direct comparison test.

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Thus, 1) The second derivative, with respect to x, of the wave function: d2dx2ψn(x) = -Cn^2(pi/L)^2sin(n*pi*x/L).

2) U(x)ψn(x) = 0

3) Eψn(x) = -Cn^2(pi/L)^2Esin(n*pi*x/L)

1) The second derivative, with respect to x, of the wave function ψn(x) in the interval 0≤x≤L can be found by applying the second derivative operator to the wave function:

d2dx2ψn(x) = -Cn^2(pi/L)^2sin(n*pi*x/L)

where n is the quantum number and C is the normalization constant.

2) U(x)ψn(x) is the product of the potential energy function U(x) and the wave function ψn(x) in the interval 0≤x≤L. If the potential energy function is zero in this interval, then U(x)ψn(x) is also zero.

Therefore, U(x)ψn(x) = 0.

3) Eψn(x) is the product of the energy E and the wave function ψn(x) in the interval 0≤x≤L. Substituting the wave function expression from part 1 into this product, we get:

Eψn(x) = -Cn^2(pi/L)^2Esin(n*pi*x/L)
where E is the energy of the particle.

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The estimated population of Georgia at the end of 2023 is 11,003,674.

To calculate Georgia's population at the end of 2023, we use the given information that Georgia has averaged approximately 1% growth per year for the last decade. This growth rate is applied to the population at the end of 2013, which was 9,975,592.

We calculate the number of years from 2013 to 2023, which is 10 years. Using the formula for compound interest with a growth rate of 1% (or 0.01), we can find the population after 10 years:

Population = Initial Population * (1 + Growth Rate)^Number of Years

Plugging in the values, we get:

Population = 9,975,592 * (1 + 0.01)^10

Simplifying the equation, we find:

Population ≈ 9,975,592 * (1.01)^10

Population ≈ 9,975,592 * 1.1046

Population ≈ 11,003,674

Therefore, based on the given growth rate, Georgia's population is estimated to be approximately 11,003,674 at the end of 2023. This estimation assumes that the 1% growth rate per year continues to hold true in the future.

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The new dimensions of each enlarged block in the subdivision planned by the city planner of Mechlinburg will be 660 feet by 2,250 feet.

The standard size of a city block in Manhattan is 264 feet by 900 feet. To enlarge these dimensions by 2.5 times, we need to multiply each side of the block by 2.5.

So, the new length of each block will be 264 feet * 2.5 = 660 feet, and the new width will be 900 feet * 2.5 = 2,250 feet.

Therefore, the new dimensions of each enlarged block in the subdivision planned by the city planner of Mechlinburg will be 660 feet by 2,250 feet. These larger blocks will provide more space for buildings, streets, and public areas, allowing for a potentially larger population and accommodating the city's growth and development plans.

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A truth table with 4 rows (one for each combination) and at least 3 columns (one for R, one for B, and one for the statement itself).

The truth table for Statement 1B will have 4 lines.

To see why, we can look at the number of possible combinations of truth values for the variables involved in the statement. In this case, there are two variables: R and B. Each variable can take on one of two truth values (true or false).

So, there are 2 × 2 = 4 possible combinations of truth values for R and B. These are:

R = true, B = true

R = true, B = false

R = false, B = true

R = false, B = false

We need to evaluate the given statement for each of these combinations, which will require us to create a truth table with 4 rows (one for each combination) and at least 3 columns (one for R, one for B, and one for the statement itself).

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The equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is 17xy is [tex]y = e^{\frac{17}{2} } x^{2}[/tex]

Identify the given information: The point is (0, 1), and the slope at (x, y) is 17xy.
Understand that the slope is the derivative of the function: [tex]\frac{dy}{dx} =  17xy[/tex]

Separate variables to integrate: [tex]\frac{dy}{y} = 17 x dx[/tex]
Integrate both sides with respect to their variables: [tex]\int\limits {\frac{1}{y} } \, dy  = \int\limits {17x} \, dx[/tex]  .

Evaluate the integrals: [tex]ln|y| = (\frac{17}{2} )x^2 + C_{1}[/tex],  where C₁ is the constant of integration.
Solve for y by exponentiating both sides: [tex]y = e^{\frac{17}{2} } x^{2} +C_{1}[/tex].
Use the initial condition (0, 1) to find the value of [tex]C_{1}:1  = e^{0+C_{1}  }[/tex], so C₁ = 0.
Plug the value of C₁ back into the equation: [tex]y = e^{\frac{17}{2} } x^{2}[/tex].

So, the equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is 17xy is [tex]y = e^{\frac{17}{2} } x^{2}[/tex].

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The Laplace transform of a cos(ωt) b sin(ωt) is [(a + ib)s]/[(s^2) + ω^2].

We can use the identity cos(a)sin(b) = (1/2)(sin(a+b) - sin(a-b)) to write:

a cos(ωt) b sin(ωt) = (a/2)(e^(iωt) + e^(-iωt)) + (b/2i)(e^(iωt) - e^(-iωt))

Taking the Laplace transform of both sides, we get:

L{a cos(ωt) b sin(ωt)} = (a/2)L{e^(iωt)} + (a/2)L{e^(-iωt)} + (b/2i)L{e^(iωt)} - (b/2i)L{e^(-iωt)}

Using the fact that L{e^(at)} = 1/(s-a), we can evaluate each term:

L{a cos(ωt) b sin(ωt)} = (a/2)((1)/(s-iω)) + (a/2)((1)/(s+iω)) + (b/2i)((1)/(s-iω)) - (b/2i)((1)/(s+iω))

Combining like terms, we get:

L{a cos(ωt) b sin(ωt)} = [(a + ib)/(2i)][(1)/(s-iω)] + [(a - ib)/(2i)][(1)/(s+iω)]

Simplifying the expression, we obtain:

L{a cos(ωt) b sin(ωt)} = [(a + ib)s]/[(s^2) + ω^2]

Therefore, the Laplace transform of a cos(ωt) b sin(ωt) is [(a + ib)s]/[(s^2) + ω^2].

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The Laplace transform of acos(ωt) + bsin(ωt) is (as + bω) / (s^2 + ω^2).

To find the Laplace transform of a function, we can use the standard formulas and properties of Laplace transforms.

Let's start with the Laplace transform of a cosine function:

L{cos(ωt)} = s / (s^2 + ω^2)

Next, we'll find the Laplace transform of a sine function:

L{sin(ωt)} = ω / (s^2 + ω^2)

Using these formulas, we can find the Laplace transform of the given function acos(ωt) + bsin(ωt) as follows:

L{acos(ωt) + bsin(ωt)} = a * L{cos(ωt)} + b * L{sin(ωt)}

= a * (s / (s^2 + ω^2)) + b * (ω / (s^2 + ω^2))

= (as + bω) / (s^2 + ω^2)

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8.66 years for 3,850 to double if it is invested at 8% compounded continuously.

Rounded to two decimal places, the answer is 8.66 years.

The continuous compounding formula is given by:

A =[tex]P\times e^{(rt)[/tex]

A is the amount of money at time t, P is the principal, r is the annual interest rate, and e is the base of the natural logarithm.

P = 3850, r = 0.08, and we want to find the time t it takes for the money to double, means A = 2P = 7700.

Plugging in these values, we get:

7700 = [tex]3850\times e^{(0.08t)[/tex]

Dividing both sides by 3850, we get:

2 = [tex]e^{(0.08t)[/tex]

Taking the natural logarithm of both sides, we get:

ln(2) = 0.08t

Solving for t, we get:

t = ln(2)/0.08 ≈ 8.66

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Solving an exponential equation we can see that it takes 8.66 months.

The formula for continuous compound is:

[tex]P = A*e^{r*t}[/tex]

Where A is the initial amount, r is the rate (in this case 8% as a decimal, so it is 0.08) and t is the time (in this case we don't know the units for time, let's say that it is in months).

The doubling time is the value of t such that the second factor is equal to 2, then we need to solve:

[tex]e^{0.08*t} = 2\\[/tex]

Now apply the natural logarithm in both sides and solve for t:

[tex]ln(e^{0.08*t}) = ln(2)\\0.08*t = ln(2)/ln(e)\\t = ln(2)/0.08[/tex]

Where we used that ln(e) = 1

t = 8.66

It takes 8.66 months.

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The final answer is ∫C y^2z ds = 178/3. the line integral, where C is the given curve. ∫C y^2z ds, C is the line segment from (3, 3, 3) to (1, 2, 5).

The line integral of a scalar function f(x, y, z) along a curve C can be expressed as:

∫C f(x, y, z) ds = ∫C f(x(t), y(t), z(t)) ||r'(t)|| dt

where r(t) = x(t)i + y(t)j + z(t)k is the parameterization of the curve C.

In this case, the curve C is the line segment from (3, 3, 3) to (1, 2, 5), which can be parameterized as:

x(t) = 3 - 2t

y(t) = 3 - t

z(t) = 3 + 2t

with 0 ≤ t ≤ 1.

The derivative of r(t) is:

r'(t) = -2i - j + 2k

The length of r'(t) is ||r'(t)|| = sqrt(9) = 3.

So the line integral becomes:

∫C y^2z ds = ∫0^1 (3 - t)^2 (3 + 2t)^2 3 dt

which can be evaluated by expanding the integrand and integrating each term. The final answer is:

∫C y^2z ds = 178/3.

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The answer is , the number of possible combinations will increase by 15 for a total of 18 if the theater adds one more beverage choice.

A movie theater sells 5 different beverages in small, medium, or large cups.

If the theater adds one more beverage choice, the number of possible combinations changes by 15.

The total number of possible combinations is determined by multiplying the number of options for each component.

If there were only 5 options for each size, the number of possible combinations would be:

3 (sizes) x 5 (drinks) = 15 combinations

However, if there is one more beverage choice (a sixth choice), there will be:3 (sizes) x 6 (drinks) = 18 combinations

Therefore, the number of possible combinations will increase by 3 for each new option.

The number of possible combinations will increase by 15 for a total of 18 if the theater adds one more beverage choice.

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The torque which is being created by the biceps is: O 27Nm flexion torque.

To calculate the torque created by the biceps, you need to consider the force and the perpendicular distance from the elbow joint.

The biceps are concentrically contracting with a force of 900N at a perpendicular distance of 3cm (0.03m) from the elbow joint.

To calculate the torque, you can use the formula: torque = force × perpendicular distance.

Torque = 900N × 0.03m = 27Nm

Therefore, the biceps are creating a 27Nm flexion torque. Answer is: O 27Nm flexion torque.

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This limit equals (7/6) < 1, therefore the series is convergent by the Ratio Test.

Using the Ratio Test, we have lim n → [infinity] |((-1)ⁿ⁺¹ * 7(n+1) * 6n³) / ((-1)ⁿ⁺¹ * 7n * 6(n+1)³)| = lim n → [infinity] (7/6) * (n/(n+1))³.

To evaluate lim n → [infinity] an + 1 / an, we substitute an with (-1)ⁿ⁺¹ * 7n / 6n³. This gives lim n → [infinity] |((-1)ⁿ⁺¹ * 7(n+1) * 6n³) / ((-1)ⁿ⁻¹ * 7n * 6(n+1)³) * (6n³ / 7n)|.

Simplifying this expression yields lim n → [infinity] |((-1)ⁿ⁺¹ * n/(n+1))³|. This limit equals 1, therefore the Ratio Test is inconclusive and we cannot determine convergence or divergence using this test.

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A business loan differs from a personal loan in terms of documentation, collateral, and repayment sources.

To distinguish between a business and a personal loan, several factors come into play.

The loan's purpose is key; if it finances business-related expenses, it is likely a business loan, while personal loans serve personal needs.

Documentation requirements, collateral, and repayment sources also offer clues. Business loans demand business-related documentation, may require business assets as collateral, and rely on business revenue for repayment.

Personal loans, however, focus on personal identification, income verification, personal assets, and personal income for repayment. Loan terms, including duration and loan amount, can also help differentiate between the two types.

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The correct answer is d. homogeneous.

A completely randomized design is useful when the experimental units are

homogeneous.

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2 is the constant in the expression 4y+2+x

The given expression is 4y+2+x

four times of y plus two plus x

x and y are the variables in the expression

We have to find the constant in the expression

The constant in the expression is the term which doesnot have any variable.

2 is the constant.

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The equation y = 2x represents the relationship between the number of cans of black paint, x, and the number of cans of white paint, y, that Franklin mixes together.

To find out how much black paint Franklin would mix with 8 cans of white paint, we need to substitute y = 8 into the equation and solve for x.

y = 2x

8 = 2x

To isolate x, we divide both sides of the equation by 2:

8/2 = 2x/2

4 = x

Therefore, Franklin would mix 4 cans of black paint with 8 cans of white paint to create storm clouds.

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The domain of the new function h(x) = f(x) + g(x) = 1/(x-2) + 5x is (-∞, 2) U (2, ∞), where x cannot be equal to 2.

The sum of two functions f(x) and g(x) is defined as h(x) = f(x) + g(x). In this case, we have f(x) = 1/(x-2) and g(x) = 5x.

Thus, h(x) = f(x) + g(x) = 1/(x-2) + 5x.

To determine the domain of h(x), we need to consider the domains of f(x) and g(x) separately. The domain of f(x) is all real numbers except x=2, because the denominator (x-2) cannot be zero.

The domain of g(x) is all real numbers, because there are no restrictions on x in the expression 5x.

Now, to find the domain of h(x), we need to consider where both f(x) and g(x) are defined. The only restriction is that x cannot be equal to 2, because f(x) is undefined at x=2.

Therefore, the domain of h(x) is all real numbers except x=2. In interval notation, we can write the domain of h(x) as (-∞, 2) U (2, ∞).

In conclusion, the domain of the new function h(x) = f(x) + g(x) = 1/(x-2) + 5x is (-∞, 2) U (2, ∞), where x cannot be equal to 2.

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