2/16 2 + x = 12
what is the answer to this equation?

The estimated probability of getting a green marble from this bag is 9/25.

The probability of getting a green marble from this bag.

We'll use the information provided on each student's results to calculate the probability.
Student 1: Picked a green marble 1 time in 4 experiments.
Student 2: Picked a green marble 5 times in 12 experiments.
Student 3: Picked a green marble 3 times in 9 experiments.
To estimate the probability, we'll add the number of successful green marble picks (1 + 5 + 3 = 9) and divide it by the total number of experiments (4 + 12 + 9 = 25).
Probability of getting a green marble = (Number of successful picks) / (Total number of experiments) = 9 / 25.

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Yes, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

A tessellation is a repeating pattern of shapes that covers a plane without any gaps or overlaps. In a semi-regular tessellation, multiple regular polygons are used to create the pattern.

For regular octagons and equilateral triangles to form a semi-regular tessellation, they must satisfy two conditions:

Vertex Condition: The same polygons meet at each vertex.

Edge Condition: The same polygons meet along each edge.

Let's examine these conditions for regular octagons and equilateral triangles:

Regular Octagon:

Each vertex of an octagon meets three other octagons.

Each edge of an octagon meets two other octagons.

Equilateral Triangle:

Each vertex of a triangle meets six other triangles.

Each edge of a triangle meets three other triangles.

The vertex condition is satisfied because each vertex of an octagon meets three equilateral triangles, and each vertex of an equilateral triangle meets three octagons.

The edge condition is satisfied because each edge of an octagon meets two equilateral triangles, and each edge of an equilateral triangle meets three octagons.

Therefore, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.Yes, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

A tessellation is a repeating pattern of shapes that covers a plane without any gaps or overlaps. In a semi-regular tessellation, multiple regular polygons are used to create the pattern.

For regular octagons and equilateral triangles to form a semi-regular tessellation, they must satisfy two conditions:

Vertex Condition: The same polygons meet at each vertex.

Edge Condition: The same polygons meet along each edge.

Let's examine these conditions for regular octagons and equilateral triangles:

Regular Octagon:

Each vertex of an octagon meets three other octagons.

Each edge of an octagon meets two other octagons.

Equilateral Triangle:

Each vertex of a triangle meets six other triangles.

Each edge of a triangle meets three other triangles.

The vertex condition is satisfied because each vertex of an octagon meets three equilateral triangles, and each vertex of an equilateral triangle meets three octagons.

The edge condition is satisfied because each edge of an octagon meets two equilateral triangles, and each edge of an equilateral triangle meets three octagons.

Therefore, regular octagons and equilateral triangles can form a semi-regular tessellation of the plane.

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The answer is 7/10 given that a book contains animal characters given that it is a graphic Nove. We have 24 books, of which 10 are graphic novels and 11 have animal characters.

Seven of them are graphic novels with animal characters. What we are looking for is the probability of an animal character being present, given that the book is a graphic novel. We can use the Bayes theorem to calculate this. Bayes' Theorem: [tex]P(A|B) = P(B|A)P(A) / P(B)P[/tex](Animal Characters| Graphic Novel) = P(Graphic Novel| Animal Characters)P(Animal Characters) / P(Graphic Novel)By looking at the question, P(Animal Characters) = 11/24,

P(Graphic Novel| Animal Characters) = 7/11, and P(Graphic Novel) = 10/24.P(Animal Characters| Graphic Novel) [tex]= (7/11) (11/24) / (10/24)P[/tex](Animal Characters| Graphic Novel) = 7/10The probability that a book contains animal characters given that it is a graphic novel is 7/10.

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The growth rate in elevation with respect to horizontal distance as you drive up the road is approximately 0.315 feet of elevation gained for every 1 foot of horizontal distance traveled.

Based on the information given, we know that the elevation at the base of the hill is 0 feet and the elevation at the top is 5000 feet (as the sign reads "Elevation 2500 feet" at the base and "Elevation 7500 feet" at the top). The horizontal distance from the base to the top is 3 miles.

To find the growth rate in elevation with respect to horizontal distance as you drive up the road, we can use the formula:

growth rate = change in elevation / horizontal distance

In this case, the change in elevation is 5000 feet (from 0 feet at the base to 5000 feet at the top), and the horizontal distance is 3 miles.

We need to convert the units to be consistent, so let's convert 3 miles to feet:

3 miles = 3 x 5280 feet = 15,840 feet

Now we can plug in the values and solve for the growth rate:

growth rate = 5000 feet / 15,840 feet = 0.315

So the growth rate in elevation with respect to horizontal distance as you drive up the road is approximately 0.315 feet of elevation gained for every 1 foot of horizontal distance traveled.

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Answer: No Solution.

Step-by-step explanation:

To solve the system of equations 2x - y = -1 and 4x - 2y = 6 graphically, we can plot the two lines represented by each equation on the same coordinate plane and find the point of intersection, if it exists.

To graph the line 2x - y = -1, we can rearrange it into slope-intercept form:

y = 2x + 1

This equation represents a line with slope 2 and y-intercept 1. We can plot this line by starting at the y-intercept (0, 1) and moving up 2 units and right 1 unit to find another point on the line. Connecting these two points gives us the graph of the line (Look at the first screenshot).

To graph the line 4x - 2y = 6, we can rearrange it into slope-intercept form:

y = 2x - 3

This equation represents a line with slope 2 and y-intercept -3. We can plot this line by starting at the y-intercept (0, -3) and moving up 2 units and right 1 unit to find another point on the line. Connecting these two points gives us the graph of the line (Look at the second screenshot).

We can see from the graphs that the two lines are parallel and do not intersect. Therefore, there is no point of intersection and no solution to the system of equations.

On a Saturday afternoon, a family of five went to see a matinée movie. Everyone paid the same price for their movie tickets. The equation will be 57.50 = 5x + 25 and the value of x is $6.50.

Let the price of one movie ticket be x,

So if there are 5 members and the ticket price is same for all, the total price for 5 movie tickets = Price of one ticket × 5

= x × 5

= 5x

Money spent on drinks = $25

ow, the total money spent is $57.50

Total money spent = money spent on movie tickets + money spent on drinks

So, the equation will be:

57.50 = 5x + 25

Now, on solving the equation:

5x = 57.50 - 25

5x = 32.50

x = 32.50 / 5

x = 6.50

Hence, the price of one ticket is $6.50 and the equation for this question is 57.50 = 5x + 25.

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The first 3 non-zero terms only [tex]x^4 e^{-x/2} = x^4.[/tex]

We can use the Maclaurin series for the trigonometric functions and exponential function to obtain the Maclaurin series for the given functions.

Here are the solutions:

[tex]f(x) = cos(7x^4)[/tex]

The Maclaurin series for cosine is:

[tex]cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...[/tex]

Substituting 7x^4 for x, we get:

[tex]cos(7x^4) = 1 - (7x^4)^2/2! + (7x^4)^4/4! - (7x^4)^6/6! + .....[/tex]

Simplifying, we get:

[tex]cos(7x^4) = 1 - 49x^8/2! + 2401x^16/4! - 117649x^24/6! + ...[/tex]

The first three non-zero terms are:

[tex]cos(7x^4) ≈ 1 - 24.5x^8 + 168.1x^16 - 2042.5x^24 + ...[/tex]

f(x) = sin(-πx)

The Maclaurin series for sine is:

[tex]sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...[/tex]

Substituting -πx for x, we get:

[tex]sin(-\pi x) = -\pi x + (-\pi x)^3/3! - (-\pi x)^5/5! + (-\pi x)^7/7! -......[/tex]

Simplifying, we get:

[tex]sin(-\pix) = -\pi x + \pi ^3 x^3/3! - \pi^5 x^5/5! + \pi^7 x^7/7! - ...[/tex]

The first three non-zero terms are:

[tex]sin(-\pi x) \approx -\pi x + 5.17\pi ^3 x^3 - 10.8\pi ^5 x^5 + 14.7\pi ^7 x^7 - ...[/tex]

[tex]f(x) = tan^-1(4x)[/tex]

The Maclaurin series for the arctangent function is:

[tex]tan^-1(x) = x - x^3/3 + x^5/5 - x^7/7 + ...[/tex]

Substituting 4x for x, we get:

[tex]tan^-1(4x) = 4x - (4x)^3/3 + (4x)^5/5 - (4x)^7/7 + .....[/tex]

Simplifying, we get:

[tex]tan^-1(4x) = 4x - 64x^3/3 + 1024x^5/5 - 16384x^7/7 + ...[/tex]

The first three non-zero terms are:

[tex]tan^-1(4x) \approx 4x - 21.33x^3 + 163.84x^5 - 1866.28x^7 + ...[/tex]

[tex]f(x) = x^4 e^{-x/2}[/tex]

The Maclaurin series for the exponential function is:

[tex]e^x = 1 + x + x^2/2! + x^3/3! + ...[/tex]

Substituting -x/2 for x and multiplying by[tex]x^4[/tex], we get:

[tex]x^4 e^{-x/2} = x^4 (1 - x/2 + x^2/2! - x^3/3! + ...)[/tex]

Expanding the product, we get:

[tex]x^4 e^{-x/2} = x^4.[/tex]

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Answer:

$2.15

Step-by-step explanation:

There are 12 in a dozen.

1.5 dozen = 18.

$38.70/18 = $2.15

It's $2.15 for a single rose.

The measure of AB in the triangle is 15.29.

The measure of BC is 8.05.

We have,

Using sin identity.

Sin 58 = AC / AB

AB = AC / Sin 58

AB = 13 / 0.85

AB = 15.29

Now,

Using the Pythagorean theorem.

AB² = AC² + BC²

15.29² = 13² + BC²

233.78 = 169 + BC²

BC² = 233.78 - 169

BC² = 64.78

BC = 8.05

Thus,

The measure of AB is 15.29.

The measure of BC is 8.05.

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Answer:

angle

Step-by-step explanation:

since there is a < sign, that makes it an angle. I'm not sure if that is the whole problem, or if It is missing a picture. Hope this helps!

To find the number of integers between 400 and 851 inclusive that are divisible by four, we need to determine the number of multiples of four in that range. The first multiple of four in the range is 400, and the last multiple of four is 848. To find how many multiples of four there are, we can subtract the two numbers and divide by four, then add one (because we need to include the first multiple).


- First multiple of four in the range: 400
- Last multiple of four in the range: 848
- Difference between the two: 848 - 400 = 448
- Divide by four: 448 ÷ 4 = 112
- Add one: 112 + 1 = 113

Therefore, there are 113 integers between 400 and 851 inclusive that are divisible by four.

There are 113 integers between 400 and 851 inclusive that are divisible by four.

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Therefore, the probability that x bar is between 91 and 92.5 is approximately 1.

To find the probability that the sample mean x bar is between 91 and 92.5, we need to use the central limit theorem and assume that the sample mean follows a normal distribution with mean μ = 91.7 and standard deviation σ/√n = 0.5/√25 = 0.1.

Then, we can use a standard normal distribution with mean 0 and standard deviation 1 to find the probability:

P(91 ≤ x bar ≤ 92.5) = P[(91 - 91.7)/(0.1) ≤ (x bar - 91.7)/(0.1) ≤ (92.5 - 91.7)/(0.1)]

= P[-7 ≤ Z ≤ 8] where Z is a standard normal random variable.

Using a standard normal table or a calculator, we can find that the probability is approximately 1, since the range -7 ≤ Z ≤ 8 covers almost the entire area under the standard normal distribution.

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The density difference required for the flow at the exit of the contraction to have uniform velocity is simply -ρ₁.

Assuming steady, incompressible, and inviscid flow, the continuity equation states that the mass flow rate must be conserved, i.e.,

ρ₁A₁U₁ = ρ₂A₂U₂

where ρ₁ and ρ₂ are the densities of the two streams, A₁ and A₂ are the cross-sectional areas of the two streams, U₁ and U₂ are the velocities of the two streams, respectively.

Since the flow at the exit of the contraction has uniform velocity, we can set U₂ = U₁. Also, since the two streams are parallel, we can assume that A₁ = A₂ = A. Therefore, the continuity equation becomes:

ρ₁U₁ = ρ₂U₂ = ρ₂U₁

Now, we can express the density of the second stream in terms of the density of the first stream and the density difference:

ρ₂ = ρ₁ + Ap

Substituting this into the continuity equation, we get:

ρ₁U₁ = (ρ₁ + Ap)U₁

Simplifying this equation, we obtain:

Ap = -ρ₁(U₁/U₁ - 1) = -ρ₁

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The right response is option c, which states that an interval calculated in this method has a 95 percent chance of bracketing the population mean.

Because it correctly explains what a confidence interval is, choice c is the best one. A confidence interval is a set of values derived from a sample of data that, with a certain level of certainty, contains the true population parameter.

According to the 95% confidence interval, 95% of the sample means would fall between the ranges of 112.5 and 118.4 if we were to compute the means for several samples taken from the same population.

Instead of revealing the likelihood that this range contains the genuine population mean, it only indicates the likelihood that this range does.

Therefore, option c is the correct answer.

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The correct answer is (a) i.e. the probability that the sum of the numbers on the two dice is greater than 4 is 30/36.

To find the probability that the sum of the numbers on two dice is greater than 4, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.

We can start by listing all the possible outcomes when rolling two dice:

1-1, 1-2, 1-3, 1-4, 1-5, 1-6

2-1, 2-2, 2-3, 2-4, 2-5, 2-6

3-1, 3-2, 3-3, 3-4, 3-5, 3-6

4-1, 4-2, 4-3, 4-4, 4-5, 4-6

5-1, 5-2, 5-3, 5-4, 5-5, 5-6

6-1, 6-2, 6-3, 6-4, 6-5, 6-6

Out of these 36 possible outcomes, the outcomes where the sum is greater than 4 are:

1-4, 1-5, 1-6

2-3, 2-4, 2-5, 2-6

3-2, 3-3, 3-4, 3-5, 3-6

4-1, 4-2, 4-3, 4-4, 4-5, 4-6

5-1, 5-2, 5-3, 5-4, 5-5, 5-6

6-1, 6-2, 6-3, 6-4, 6-5, 6-6

There are 30 favorable outcomes. Therefore, the probability that the sum of the numbers on the two dice is greater than 4 is 30/36.

So the correct answer is (a) 30/36.

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The lengths of the various foam pieces are represented here in inches according to the supplied specs. The following information is provided on a separate sheet of paper, which can be used to answer the questions that are there: 10 in, 20 in, 2 in, 10 in, 5 in, 20 in, 20 in, 8 in, 5 in, 2 in, and 26 in.

The provided measurements suggest that the foam play toy is made up of a number of different foam pieces, each of which has a different length.

One would need to conduct an analysis of the provided measures and give careful consideration to the particular questions that are being asked in order to answer the questions on the separate paper. Because the questions themselves are not included in the information that is provided, it is required to evaluate the prompts that are on the separate page and respond to them in the appropriate manner.

The lengths of the foam pieces can be determined by using the specified measures, which can also be used to answer any queries regarding the arrangement of the foam pieces, the overall length, or any other special inquiries that are mentioned in the https://brainly.com/question/28170201.

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Answer:

A, B, and D

Step-by-step explanation:

Answer: Your answer is D. 3.5

Step-by-step explanation: 28 divided by 8 is 8.5. I learned how to divide numbers on paper in last years math class.

Hope it helped :D

Answer:

D. 3.5

Step-by-step explanation:

We just take 28 divided by 8 and get 3.5

True, This Continuity Editing/Cutting device is used in Classical Hollywood Cinema.

Match on action is a common continuity editing/cutting technique used in Classical Hollywood Cinema, where the editor cuts from one shot to another while maintaining visual continuity between the two shots by showing the continuation of an action or movement from one shot to the next. This helps to create a smooth and seamless flow of action on screen and maintains the illusion of reality for the viewer.

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The value of x in the equation is -3.

The given equation is 4 = (3x+17)/2

We have to find the value of x

Apply cross multiplication

4×2 = 3x+17

8=3x+17

Subtract 17 from both sides

8-17=3x

-9=3x

Divide both sides by 3

x=-3

Hence, the value of x in the equation is -3.

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The x-intercepts are approximately 0.54 and 11.46. Since the coefficient of x^2 is positive, the graph opens upwards. Combining all of this information, we can sketch a graph of f(x) that looks like a "U" shape with vertex at (6, -12) and x-intercepts at approximately 0.54 and 11.46.

(a) To express f(x) in standard form, we need to complete the square. First, we can factor out the coefficient of x^2 to get:

f(x) = x^2 - 12x + 24

Next, we add and subtract (12/2)^2 = 36 to the expression inside the parentheses to get:

f(x) = (x^2 - 12x + 36) - 36 + 24

The expression inside the parentheses can be rewritten as (x - 6)^2, so we have:

f(x) = (x - 6)^2 - 12

Therefore, the standard form of the quadratic function f(x) is f(x) = (x - 6)^2 - 12.

(b) To sketch a graph of f, we can first identify the vertex as (6, -12) from the standard form. This is the lowest point on the graph since the coefficient of x^2 is positive. We can also find the x-intercepts by setting f(x) = 0:

(x - 6)^2 - 12 = 0

(x - 6)^2 = 12

x - 6 = ±√12

x = 6 ± 2√3.

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The period of the following functions. g ( x ) = cos ( x 4 ) is that it doesn't have any.

To find the period of the function g(x) = cos(x^4), we need to find the smallest positive value of p such that g(x + p) = g(x) for all values of x. That is, we need to find the most minor p such that cos((x + p)^4) = cos(x^4) for all values of x.

Using the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b), we can expand the left-hand side of the equation:

cos((x + p)^4) = cos(x^4 + 4px^3 + 6p^2x^2 + 4p^3x + p^4)

= cos(x^4)cos(4px^3) - sin(x^4)sin(4px^3)cos(6p^2x^2)

=cos(x^4)sin(4px^3)sin(6p^2x^2) - sin(x^4)cos(4px^3)cos(6p^2x^2) + cos(x^4)cos(4px^3)sin(6p^2x^2)

Let's consider the second condition first. Since x is an integer multiple of π^(1/4), we have: 4px^3 = (4pπ^(3/4))x^3

For this to be an integer multiple of π, we must have p = q/π^(3/4), where q is an integer. Substituting this value of p into the second condition, we get 4qx^3 = rπ

where r is an integer. This implies that x is a multiple of π, which contradicts our assumption that x is an integer multiple of π^(1/4). Therefore, there is no value of p for which g(x + p) = g(x) for all values of x.

In other words, the function g(x) = cos(x^4) does not have a period.

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The population will reach 10,000 after about 166.68 minutes.

We can use the formula for exponential growth: N = N0 * e^(rt), where N is the population at time t, N0 is the initial population, r is the growth rate, and e is Euler's number.

Let's use the first two data points to find the growth rate and initial population. We know that after 15 minutes, N = 400, so:

400 = N0 * e^(r*15)

Similarly, after 30 minutes, N = 1400, so:

1400 = N0 * e^(r*30)

Dividing the second equation by the first, we get:

3.5 = e^(r*15)

Taking the natural logarithm of both sides, we get:

ln(3.5) = r*15

So the growth rate is:

r = ln(3.5)/15

r ≈ 0.0918

Using the first equation above, we can solve for N0:

400 = N0 * e^(0.0918*15)

N0 ≈ 98.51

So the initial population was about 98.51.

The doubling period is the time it takes for the population to double in size. We can use the formula for doubling time: T = ln(2)/r, where T is the doubling time.

T = ln(2)/0.0918

T ≈ 7.56 minutes

So the doubling period is about 7.56 minutes.

To find the population after 120 minutes, we plug in t = 120:

N = 98.51 * e^(0.0918*120)

N ≈ 22601.27

So the population after 120 minutes is about 22,601.27.

To find when the population reaches 10,000, we set N = 10,000 and solve for t:

10,000 = 98.51 * e^(0.0918*t)

t = ln(10,000/98.51)/0.0918

t ≈ 166.68 minutes

So the population will reach 10,000 after about 166.68 minutes.

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Then, describe the steps involved in calculating taxable income including pension contributions, and provide examples to illustrate how these calculations work.

Finally, you can conclude by emphasizing the importance of proper tax planning and compliance to avoid penalties and other legal issues.

When calculating taxable income, you need to include pension contributions.

Here's how to calculate taxable income including pension contributions:

Step 1: Add up your income for the year, including all sources such as salary, bonuses, rental income, and investment income.

Step 2: Subtract your allowable deductions such as mortgage interest, charitable contributions, and state taxes.

Step 3: Subtract your personal exemptions, which are based on the number of dependents you have.

Step 4: Subtract your pension contributions from your income. These contributions reduce your taxable income, so the higher your contributions, the lower your taxable income.

Step 5: The result of these calculations is your taxable income. You can use this figure to determine how much tax you owe.

Then, describe the steps involved in calculating taxable income including pension contributions, and provide examples to illustrate how these calculations work.

Finally, you can conclude by emphasizing the importance of proper tax planning and compliance to avoid penalties and other legal issues.

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Answer:

700

Step-by-step explanation:

4(175)=700

The length of the radius of cone A. is [tex]\frac{\pi}{6}[/tex].

The lateral surface area of cone A is a sector with central angle 6 radians and radius π.

We can use the formula for sector area to find the lateral surface area of the cone.

Area of sector = θ/2π×π²

where θ is the central angle and π is the radius.

Area of cone’s lateral surface area (L) =θ/2π×2πr=rθ.

So, r = L/θ = π/6 (when L=π and θ=6 radians).

The length of the radius of cone A is π/6 which is approximately 0.524.

Therefore, the length of the radius of cone A is [tex]\frac{\pi}{6}[/tex], when unwrapped, given that the lateral surface area of cone A is a sector with central angle 6 radians and radius pi.

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the simplified expression is 1 / (200[tex]x^4[/tex]).

No problem! Let's break down the expression step by step and simplify it.

The expression you provided is:

[tex](20x^{(-3)} / 10x^{(-1)})^{(-2)}[/tex]

To simplify this, we can start by simplifying the numerator and denominator separately.

Numerator:

20[tex]x^{(-3)}[/tex]

Since we have a negative exponent, we can move the term to the denominator and change the sign of the exponent:

1 / (20[tex]x^3[/tex])

Denominator:

10[tex]x^{(-1)}[/tex]

Similarly, we move the term to the numerator and change the sign of the exponent:

10x

Now, we can rewrite the original expression as:

(1 / (20[tex]x^3[/tex])) / (10x)

To divide by a fraction, we multiply by its reciprocal:

(1 / (20[tex]x^3[/tex])) * (1 / (10x))

Multiplying the numerators and the denominators, we get:

1 / (200[tex]x^4[/tex])

Finally, we have:

1 / (200[tex]x^4[/tex])

So, the simplified expression is 1 / (200[tex]x^4[/tex]).

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Hello !

Answer:

[tex]\Large \boxed{\sf f'(x)=-\frac{7}{(x+6)^2} }[/tex]

Step-by-step explanation:

Let's remember !

Given : [tex]\sf f(x) = \frac{7}{x+6}[/tex]

We have :

Let's derivate u and v with the previous formulas:

Now we can apply the first formula !

[tex]\sf f'(x)=\frac{0\times(x+6)-7\times1}{(x+6)^2} \\\boxed{\sf f'(x)=-\frac{7}{(x+6)^2} }[/tex]

Have a nice day ;)

The salary of Jamal, for a month where he made no sales, will be $3,400.

The base monthly salary of Jamal is $3,400, and he gets a commission of $0.05 for every dollar in sales.

In this equation, x represents the amount of sales he makes, and y represents his total earnings.Jamal has not made any sales this month, so x will be equal to zero. To determine his salary, we will substitute x = 0 in the given equation to get:

y = 3,400 + 0.05(0)y = 3,400 + 0y = 3,400

As per the given equation, if Jamal does not make any sales, his salary will be $3,400. He earns a base monthly salary of $3,400, and his salary increases by $0.05 for every dollar of sales he makes.

This is a linear equation with a slope of $0.05, indicating that his salary will increase by $0.05 for each dollar of sales he makes.

The y-intercept is $3,400, indicating that his base monthly salary is $3,400. We can plot this line on a graph with the y-axis representing Jamal's salary and the x-axis representing the amount of sales he makes. The slope will be 0.05, and the y-intercept will be 3,400

The salary of Jamal, for a month where he made no sales, will be $3,400.

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The sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.

To prove that for all n ≥ 0, the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation, we can use mathematical induction.

Base case:

For n = 0, we have h0 = 5/3 which is equal to the given initial value.

For n = 1, we have h1 = 11/3 which is also equal to the given initial value.

Inductive step:

Assume that the recurrence relation holds for some k ≥ 1, i.e., hk = 3hk-1 - 4hk-2 + 6k.

We want to show that it also holds for k+1, i.e., h(k+1) = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1).

Using the recurrence relation for hk, we have:

hk+1 = 3hk - 4hk-1 + 6k+3 (by substituting k+1 for n in the given recurrence relation)

Similarly, we have:

hk = 3hk-1 - 4hk-2 + 6k (by assumption)

hk-1 = 3hk-2 - 4hk-3 + 6(k-1) (by assumption)

Substituting these values into the expression for hk+1, we get:

hk+1 = 3(3hk-1 - 4hk-2 + 6k) - 4(3hk-2 - 4hk-3 + 6(k-1)) + 6(k+1)

Simplifying the expression, we get:

hk+1 = 9hk-1 - 12hk-2 + 18k - 12hk-2 + 16hk-3 - 24(k-1) + 6(k+1)

hk+1 = 9hk-1 + 4hk-3 - 12hk-2 - 6(k-1) + 6(k+1)

hk+1 = 3(3hk-1 - 4hk-2 + 6k+1) - 4(3hk-2 - 4hk-3 + 6k) + 6(k+1)

hk+1 = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1)

This shows that the recurrence relation holds for all n ≥ 0 by mathematical induction, and hence the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.

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