Part A: Rhett made $250 washing cars with his mobile car wash company. He charges $70 per car and earned $50 in tips. Write an equation to represent this situation. (4 points)



Part B: Scarlett made a profit of $250. 00 with her mobile car wash company. She charged $75. 00 per car wash and received $35. 00 in tips, but also had to pay $5. 00 in cleaning supplies per car wash. Write an equation to represent this situation. (4 points)



Part C: Explain how the equations from Part A and Part B differ. (2 points)

Answer:

y=3x-10

Step-by-step explanation:

look at picture.

It looks like you want to evaluate the iterated integral of the function 2x(x + 2y) over a given region S. To evaluate the iterated integral, we first integrate with respect to y (dy) and then with respect to x (dx).

Let's first integrate with respect to y:

∫(2x(x + 2y)) dy = 2x(xy + y^2) + C₁

Now, we need to evaluate this expression for the limits of integration for y, which are not given in your question. I'll assume they are y = a and y = b, so we have:

[2x(xb + b^2) - 2x(xa + a^2)]

Next, we'll integrate this expression with respect to x:

∫(2x(xb + b^2) - 2x(xa + a^2)) dx = x^2(xb + b^2) - x^2(xa + a^2) + C₂

Finally, we need to evaluate this expression for the limits of integration for x, which are also not given in your question. Assuming they are x = c and x = d, we have:

[(d^2(dc + b^2) - d^2(da + a^2)) - (c^2(cc + b^2) - c^2(ca + a^2))]

This expression represents the value of the iterated integral for the function 2x(x + 2y) over the region S, given the limits of integration for x and y. Please provide the limits of integration for a more specific answer.

To learn more about integration visit;

brainly.com/question/30900582

#SPJ11

Answer:

D. 9

Step-by-step explanation:

Let

abc=100a+10b+c

bca=100b+10c+a

cab=100c+10a+b

abc + bca + cab=(100a+10b+c) + (100b+10c+a) + (100c+10a+b)

=100a + 10b + c + 100b + 10c + a + 100c + 10a + b

Collect like terms

=100a + a + 10a + 10b + 100b + b + c + 10c + 100c

=111a + 111b + 111 c

Factorise

=111(a+b+c)

abc + bca + cab = 111(a+b+c)

Factors of 111(a+b+c)= 1, 3, 37, 111, and (a+b+c)

abc + bca + cab is divisible by a+b+c because it is a factor of 111(a+b+c)

abc + bca + cab is divisible by 3 because 3 is a factor of 111(a+b+c)

abc + bca + cab is divisible by 37 because 37 is a factor of 111(a+b+c)

abc + bca + cab is not divisible by 9 because 9 is not a factor of 111(a+b+c)

When a sample has the same characteristics as the target population, the sample is said to be a representative sample.

When a sample has the same characteristics as the target population, the sample is said to be a representative sample.

Let's take a look at this definition in more detail below:

A representative sample is a sample that adequately represents the population, which is selected randomly, so that each member of the population has an equal chance of being included. A representative sample should have the same characteristics of the population it represents.

To get a representative sample, researchers must select participants randomly. A representative sample has the same distribution of variables as the population in which it was derived. It is generally believed that a representative sample is necessary for generalizing the results to the population

Therefore, the results obtained from a representative sample can be generalized to the larger population.In a nutshell, a representative sample is a sample that represents the population from which it was selected. It should have similar characteristics to the population from which it was drawn.

A representative sample is a necessary prerequisite for generalizing the results to the larger population. It ensures the accuracy of the study and the validity of the results obtained. The sample size should be large enough to provide adequate statistical power and enable the results to be generalized to the population.

Learn more about representative sample here https://brainly.com/question/21858881

#SPJ11

Answer:

Distance travelled = 91.61 feet

Step-by-step explanation:

Given the following data;

Diameter, d = 35 inches

Number of revolutions = 10 Rev

To find the distance travelled by the bike;

First of all, we would determine the circumference of the bike's tire, which is circular in shape.

Circumference of circle = πd

Circumference of circle = 35π

Next, we multiply by the number of revolutions to get the distance travelled.

Distance travelled = 35π * 10

Distance travelled = 350π inches

But pie, π = 3.142

Distance travelled = 350 * 3.142

Distance travelled = 1099.7 inches.

In feet;

1 inches = 0.0833 feet

1099.7 inches = x feet

Cross-multiplying, we have;

x feet = 1099.7 * 0.0833

x feet = 91.61 feet

Distance travelled = 91.61 feet

The inequality that represents all the solutions of -2(3x + 6) ≥ 4(x + 7) is x ≤ -4

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

-2(3x + 6) ≥ 4(x + 7)

Divide both sides of the inequality by -2

So, we have the following representation

3x + 6 ≤ -2(x + 7)

Open the brackets

This gives

3x + 6 ≤ -2x - 14

Add 2x to both sides of the inequality

So, we have the following representations

5x + 6 ≤ -14

Subtract 6 from both sides of the inequality

So, we have the following representations

5x ≤ -20

Divide both side of the inequality

x ≤ -4

Hence, the solution is x ≤ -4

Read more about inequality at

https://brainly.com/question/25275758

#SPJ1

The volume of the right rectangular prism is the amount of cubes in it

The formula to determine the volume of the right rectangular prism is V(n) = n

The question is incomplete; as the figure is not given.

So, I will provide a general solution

The volume of a one inch cube is:

\(V = 1^3\)

\(V = 1\)

Assume the number of one inch cubes used to fill the right rectangular prism is n.

Then the volume of the right rectangular prism would be:

\(V(n) = n * 1\)

This gives

\(V(n) = n\)

Hence, the general formula to determine the volume of the right rectangular prism is V(n) = n

Read more about volumes at:

https://brainly.com/question/1972490

Answer: 45 Seconds

Step-by-step explanation: Because if you do 3/4=0.75 or 75%. Knowing that a minute is 60 seconds long. 75% of 60 is 45 Seconds.

Hope this helped!

A confidence interval is made up of the mean of your estimate plus and minus the estimate's range. This is the range of values you expect your estimate to fall within if you repeat the test, within a given level of confidence.

A 95% confidence interval gives you a 5% chance of being wrong. A 90% confidence interval gives you a 10% chance of being wrong. In comparison to a 99% confidence interval, a 95% confidence interval is smaller (for example, plus or minus 4.5 percent instead of 3.5 percent).

When constructing a confidence interval for a propotion, if the condition of proportion requires 10 individuals in each category is not met, then confidence interval should be adjusted.

The number of individuals in each category is increased by 2 and sample size increases by 4.

To read more about confidence intervals, visit https://brainly.com/question/24131141

#SPJ4

Answer: scale to measure the weight

Step-by-step explanation:

A 3d cardboard box has 6 sides, each of which are rectangles. If you unfold the 3D box, and flatten it out, then you'll be left with 6 rectangles such as what you see in the attachment below. This is one way to unfold the box. This flattened drawing is the net of the 3D rectangular prism. You can think of it as wrapping paper that covers the exterior of the box. There are no gaps or overlapping portions. If you can find the area of each piece of the net, and add up those pieces, that gets you the total area of the net. This is the exactly the surface area of the box.

In the drawing below, I've marked the sides as: top, bottom, left, right, front, back. This way you can see how the 3D box unfolds and how the sides correspond to one another. Other net configurations are possible.

The probability that a random student will be taking both Algebra 2 and Chemistry is 0.0136 or 1.36%.

To find the probability that a random student will be taking both Algebra 2 and Chemistry, we need to use the concept of conditional probability.

Let's denote the event of taking Algebra 2 as A and the event of taking Chemistry as C. We are given that P(A) = 0.08 (8% probability of taking Algebra 2) and P(C|A) = 0.17 (17% probability of taking Chemistry given that the student is taking Algebra 2).

The probability of taking both Algebra 2 and Chemistry can be calculated using the formula for conditional probability:

P(A and C) = P(C|A) * P(A)

Substituting the given values:

P(A and C) = 0.17 * 0.08

P(A and C) = 0.0136

Therefore, the probability that a random student will be taking both Algebra 2 and Chemistry is 0.0136 or 1.36%.

It is important to note that the probability of taking both Algebra 2 and Chemistry is determined by the intersection of the two events, which means students who are taking both courses. In this case, the probability is relatively low, as it depends on the individual probabilities of each course and the conditional probability given that a student is taking Algebra 2.

For more such questions on probability visit:

https://brainly.com/question/25839839

#SPJ8

(a) To find the Fourier transform F(u,v) of f(x,y), we can apply the 2D Fourier transform formula:

F(u,v) = ∫∫ f(x,y) exp(-i2π(ux+vy)) dx dy

where sinc(x) = sin(x)/x.

Plugging in the expression for f(x,y) and evaluating the integral yields:

F(u,v) = ∫∫ sinc(2x,2y) exp(-i2π(ux+vy)) dx dy + ∫∫ sinc() exp(-i2π(ux+vy)) dx dy

The first integral can be simplified by using the identity:

sinc(ax,ay) = (1/a) sinc(x/a, y/a)

So we have:

F(u,v) = (1/2) ∫∫ sinc(x/2,y/2) exp(-iπ(u x + v y)) dx dy + π δ(u,v)

where δ(u,v) represents the Dirac delta function. The second term arises from the second integral, which evaluates to a constant value of π.

Evaluating the first integral involves using the 2D convolution theorem, which states that the Fourier transform of a convolution is the product of Fourier transforms. Specifically, we have:

∫∫ sinc(x/2,y/2) exp(-iπ(u x + v y)) dx dy = (1/4) ∫∫ sinc(x',y') exp(-iπu x') exp(-iπv y') dx' dy'

where we have made the change of variables x' = x/2 and y' = y/2. The integral on the right-hand side is just the Fourier transform of sinc(x',y'), which can be evaluated exactly:

∫∞ -∞ ∫∞ -∞ sinc(x',y') exp(-iπu x') exp(-iπv y') dx' dy'

= ∫∞ -∞ sinc(x') exp(-iπu x') dx' ∫∞ -∞ sinc(y') exp(-iπv y') dy'

= 2/π (sin(πu)/u) (sin(πv)/v)

Therefore, we have:

F(u,v) = (1/2) (2/π) (sin(πu)/u) (sin(πv)/v) + π δ(u,v)

= π [δ(u,v) + (1/π) sin(πu)/u sin(πv)/v]

(b) If f(x,y) is the input signal of an ideal filter with transfer function H(u,v) = rect(u,v), then the output signal g(x,y) is given by the inverse Fourier transform of the product of F(u,v) and H(u,v):

g(x,y) = ∬ F(u,v) H(u,v) exp(i2π(ux+vy)) du dv

where rect(u,v) = 1 inside the rectangle [-1/2,1/2]x[-1/2,1/2] and zero elsewhere.

Plugging in the expression for F(u,v) and H(u,v) yields:

g(x,y) = π ∬ [δ(u,v) + (1/π) sin(πu)/u sin(πv)/v] rect(u,v) exp(i2π(ux+vy)) du dv

The integral over the rectangle can be simplified by noting that the product of two rectangular functions is itself a rectangular function:

rect(u,v) exp(i2π(ux+vy)) = rect(u-2Nx,v-2Ny)

where N is a positive integer and (2Nx,2Ny) is the closest point in the lattice of points with spacing 1/2 to the origin. Therefore, we have:

g(x,y) = π [1 + (1/π) ∑n,m sin(π(n-Nx))/π(n-Nx) sin(π(m-Ny))/π(m-Ny)] rect((x/2N)+1/2,(y/2N)+1/2)

where the sum ranges over all integers n and m except for n=m=0, and rect(a,b) = 1 if |a|<=1/2 and |b|<=1/2, and zero otherwise.

In other words, the output signal g(x,y) is the sum of a constant term (corresponding to the DC component of the input signal) and an infinite series of sinusoidal terms, each weighted by the product of two sinc functions. The amplitude of each term decays as 1/nm, so only a finite number of terms contribute significantly to the output signal.

Learn more about Fourier transform  here:

https://brainly.com/question/1542972

#SPJ11

The given function is \(y=cot⁻¹√(t−7). We are required to find dy/dt. The derivative of cot⁻¹(x) is -1/(1+x²).\) Using the chain rule, the derivative.

\(y=cot⁻¹√(t−7) is given asdy/dt = -1/(1+(√(t-7))²) * d/dt (√(t-7)).Therefore, dy/dt = -1/(1+(t-7)) * 1/(2√(t-7))= -1/(2t-15) * 1/√(t-7)Hence, dy/dt = -1/[√(t-7)*(2t-15)].\)

\(2. The given function is y=cos⁻¹(x)+x√(1−x²). cos⁻¹(x) is -1/√(1-x²).\)

Using the product rule, the derivative of y=cos⁻¹(x)+x√(1−x²) is given asdy/dx = -1/√(1-x²) + √(1-x²)*d/dx (x) + x*d/dx (√(1-x²)).

Therefore,\(dy/dx = -1/√(1-x²) + √(1-x²)*1 + x * (-1/2)(1-x²)-½ * (-2x) = -1/√(1-x²) + √(1-x²) + x²/√(1-x²).Therefore, dy/dx = (x²-1)/√(1-x²)\).

Hence, the derivative of \(y=cos⁻¹x+x√(1−x²) with respect to x is dy/dx=(x²-1)/√(1-x²).\)

To know more about  function visit:

brainly.com/question/21426493

#SPJ11

The measure of angle x to the nearest tenth of a degree is 57.7°.

The figure in the image is a right triangle.

From the figure;

To determine measure of angle x, we use one of the six (6) trigonometric ratio.

Cosine = Adjacent / Hypotenuse

Plug in adjacent = 4.6 units and hyptenuse = 8.6 unit, then solve for angle x.

cos( x ) = 4.6 units / 8.6 units

Take the cosine inverse.

x = cos⁻¹( 4.6 / 8.6 )

x = 57.66397

x = 57.7 degree.

Therefore, the value of x is 51.8 degrees.

Learn more about trigonometric ratio here: brainly.com/question/28016662

#SPJ1

Answer: y=9

Step-by-step explanation:

Plug in (6,9) into the slope intercept form equation:

9=0(6) +b

Then get

9=b

When you make the equation you would end up with:

y=0x+9

which simplifies to y=9

The gas station serves 16 vehicles per hour, and 72 vehicles arrive in 4 hours. The vehicles will spend 4.50 hours at the gas station.



To find the total time the vehicles will spend at the gas station, we need to calculate the total number of vehicles that arrive and then divide it by the rate at which the gas station serves vehicles.

Given:

Arrival rate: 18 vehicles per hour

Service rate: 16 vehicles per hour

Time: 4 hours

First, let's calculate the total number of vehicles that arrive during the 4-hour period:

Total number of vehicles = Arrival rate * Time

                      = 18 vehicles/hour * 4 hours

                      = 72 vehicles

Since the gas station can serve 16 vehicles per hour, we can determine the time it takes to serve all the vehicles:

Time to serve all vehicles = Total number of vehicles / Service rate

                         = 72 vehicles / 16 vehicles/hour

                         = 4.5 hours

Therefore, the vehicles will spend 4.5 hours at the gas station. Rounded to 2 decimal places, the answer is 4.50 hours.

To learn more about hours click here brainly.com/question/29121756

#SPJ11

Answer:

MCK: 50

MCM: 32

Step-by-step explanation:

The function g(x) can be derived from f(x) by vertical dilation and vertical translation. f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2 and g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502.

In this question we must infer what kinds of rigid transformations were used to obtain g(x) in terms of f(x). Points A and B belong to the function f(x) and points A' and B' lie on the curve of function g(x). Rigid transformations are transformations applied on functions such that Euclidean distance is conserved.

We noticed the use of rigid transformations on f(x) such that g(x) is found:

Vertical dilation

f'(x) = k · f(x), where k > 0

Vertical translation

g(x) = k · f(x) + c, where c is the translation factor.

We notice that both f(x) and g(x) are both cubic functions. The former has a vertex at (x, y) = (0, 1.5). First, we determine the coefficients of f(x) by solving the following system of linear equations:

(x₁, y₁) = (1, 6), (x₂, y₂) = (- 2, - 30), (x₃, y₃) = (0, 1.5), (x₄, y₄) = (- 1, - 4)

a + b + c + d = 6

- 8 · a + 4 · b - 2 · c + d = - 30

d = 1.5

- a + b - c + d = - 4

The solution of the system is (a, b, c, d) = (13 / 4, - 1 / 2, 7 / 4, 3 / 2). Then, the expression for the function f(x) is f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2.

Now we evaluate the function f(x) at x = 7 and x = - 8:

f(7) = (13 / 4) · 7³ - (1 / 2) · 7² + (7 / 4) · 7 + 3 / 2

f(7) = 1104

f(- 8) = (13 / 4) · (- 8)³ - (1 / 2) · (- 8)² + (7 / 4) · (- 8) + 3 / 2

f(- 8) = - 3417 / 2

Now we determine the coefficient of dilation and the translation factor:

f(7) = 1104, f(- 8) = - 3417 / 2, g(7) = 2.5, g(- 8) = - 6.5

2.5 = 1104 · k + c

- 6.5 = - 3417 · k / 2 + c

The solution of the system of linear equations is (k, c) = (250 / 47489, 2.494). Then, we find that the expression for g(x) is g(x) = (250 / 47489) · [(13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2] + 2.494, whose simplest form is:

g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.494

g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502

The function g(x) can be derived from f(x) by vertical dilation and vertical translation. f(x) = (13 / 4) · x³ - (1 / 2) · x² + (7 / 4) · x + 3 / 2 and g(x) = (125 / 7306) · x³ - (125 / 47789) · x² + (875 / 94978) · x + 375 / 47489 + 2.502.

To learn more on rigid transformations: https://brainly.com/question/1761538

#SPJ1

The average annual return on this investment from 2011 to 2015 is approximately 0.8%.

To calculate the value of a $1 investment made at the beginning of 2011 and its average annual return by the end of 2015, we need to multiply the successive annual returns and calculate the cumulative value.

The successive annual returns on small U.S. stocks from 2011 to 2015 are:

-3.80%, 19.15%, 45.91%, 3.26%, and -3.80%.

To calculate the cumulative value, we multiply the successive returns by the initial investment value of $1:

(1 + (-3.80%/100)) * (1 + (19.15%/100)) * (1 + (45.91%/100)) * (1 + (3.26%/100)) * (1 + (-3.80%/100))

Calculating this expression, we find that the cumulative value is approximately $1.044, rounded to three decimal places.

Therefore, a $1 investment made at the beginning of 2011 would have been worth approximately $1.044 at the end of 2015.

To calculate the average annual return, we need to find the geometric mean of the annual returns. We can use the following formula:

Average annual return = (Cumulative value)^(1/number of years) - 1

In this case, the number of years is 5 (from 2011 to 2015).

Average annual return = (1.044)^(1/5) - 1

Calculating this expression, we find that the average annual return is approximately 0.008 or 0.8% per year, rounded to two decimal places.

Therefore, the average annual return on this investment from 2011 to 2015 is approximately 0.8%.

Learn more about compound interest here:

https://brainly.com/question/24274034

#SPJ11

The estimated population of octopuses in Lizbeth Rich's study is approximately 0.9968.

The population of Lizbeth Rich's study is the total number of octopuses that she is interested in studying, which is not explicitly stated in the given information. However, we can estimate the population based on the sample size and the proportion of octopuses maintaining gardens.

In the study, Lizbeth surveys 312 randomly selected octopuses to see if they maintain a garden. Out of these 312 octopuses, 23 maintained gardens.

To estimate the population, we can use the concept of sampling proportion. We know that 23 out of 312 octopuses maintained gardens. We can set up a proportion:

23/312 = x/total population

We can cross-multiply and solve for the total population:

23 * total population = 312 * x

23 * total population = 312x

total population = (312x) / 23

To find the value of x, we need to divide the number of octopuses maintaining gardens (23) by the proportion of octopuses maintaining gardens in the sample (312):

x = 23 / 312

x ≈ 0.0737

Now we can substitute this value back into the equation to find the total population:

total population = (312 * 0.0737) / 23

total population ≈ 0.9968

So, the estimated population of octopuses in Lizbeth Rich's study is approximately 0.9968.

Learn more about sampling proportion:

https://brainly.com/question/11461187

#SPJ11

Answer: Hugo should buy car A

Step-by-step explanation:

Car A

Take 702 divided by 12 = 58.5 miles per gallon of gas

Car B

Take 873 divided by 15 = 58.2 miles per gallon of gas.

So Hugo should buy car A

The total cost before the sales tax was 19.828 pounds.

For a party, Jordan purchased 3.8 pounds of turkey, 2.2 pounds of cheese, and 3.6 pounds of egg salad.

We have to determine the total cost before sales tax.

As per the question, we have prices as:

cost of turkey = 3.95 per pound

cost of egg cheese = 1.3 per pound

cost of egg salad = 0.89 per pound

The total cost of turkey = 3.8 × 3.95  = 15.01 pounds

The total cost of cheese = 2.2 × 1.3 = 2.86 pounds

The total cost of egg salad = 2.2 × 0.89 = 1.958 pounds

The total cost before sales tax = 15.01 + 1.958 +2.86

Apply the addition operation, and we get

The total cost before sales tax = 19.828 pounds

To learn more about the total cost click here:

https://brainly.com/question/2433744

#SPJ1

The question seems to be incomplete the correct question would be:

Jordan bought 3.8 pounds of turkey, 2.2 pounds of cheese, and 3.6 pounds of egg salad for a party. If prices are 3.95 per pound turkey, 1.3 per pound cheese and 0.89 per pound egg salad What was the total cost before sales tax?

Client will have improved oral hygiene practices, including regular brushing, flossing, and using mouthwash, resulting in healthier oral mucous membranes.

L. Directions: Choose the outcomes that are written correctly below. Identify what's wrong with the statements that are written incorrectly.

1. John will know the four basic food groups by I/14.

  - Incorrect: The date format is incorrect; it should be written as 1/14 instead of I/14.

2. Mry. Eipert will demoastrate how to use her walker unassisted by Saturday.

  - Incorrect: There is a spelling error in "Mry. Eipert"; it should be "Mrs. Eipert." Additionally, "demoastrate" should be "demonstrate."

3. Mr. McKillop will improve his appetite by 11/5.

  - Incorrect: The date format is incorrect; it should be written as 11/5 instead of 11/5.

4. Erica will list the equipment needed to change sterile dressings by 9/5.

  - Incorrect: The date format is incorrect; it should be written as 9/5 instead of 9/5.

5. Mrs. Baylis will understand the importance of maintaining a salt-free diet.

  - Correct: No issues found. The statement is written correctly.

6. Nurse will bathe client every day during hospitalization.

  - Correct: No issues found. The statement is written correctly.

IL. Directions: For each diagnosis or problem below, write an appropriate client goal and outcome.

1. Constipation related to insufficient roughage intake in the diet

  - Client Goal: Increase regular bowel movements and relieve constipation.

  - Outcome: Client will have a bowel movement at least once daily by incorporating high-fiber foods into the diet.

2. Altered oral mucous membranes related to poor oral hygiene

  - Client Goal: Improve oral health and maintain healthy mucous membranes.

  - Outcome: Client will have improved oral hygiene practices, including regular brushing, flossing, and using mouthwash, resulting in healthier oral mucous membranes.

To know more about mucous refer here:

https://brainly.com/question/30315177#

#SPJ11

Answer:

I can't really give an answer but I can describe how to get the answer

Step-by-step explanation:

First, we need the corrdinates of Point P which should be an answer of (x,y). After we got the coordinates, move 10 lines towards the left and replace the x with the number you have gotten by doing that and then you should get your answer (hopefully)

Answer:

One factor is x+1.

Step-by-step explanation:

f(x) = x^3 - x^2 - 3x - 1.

The factor theorem states that if x-a is a factor of f(x) then f(a) = 0.

Try x = 1 (to see if (x-1) is a factor:

f(1) = 1 - 1 - 3 - 1 = -4.  This is not zero so x-1 NOT a factor.

Try f(-1) = -1 - 1 + 3 - 1 = -3 + 3 = 0

So x+1 is a factor.

The one factor of the given polynomial is (x + 1).

A polynomial is an expression that consists of variables (or indeterminate), terms, exponents and constants.

According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number then, (x-a) is a factor of f(x), if f(a)=0.

According to the given question

We have a polynomial

\(f(x) = x^{3} -x^{2} -3x-1\)

For finding the factors of the given polynomial, substitute f(x) = 0.

⇒\(x^{3} -x^{2}-3x-1=0\)

⇒ \((x+1)(x^{2} -2x-1)=0\)

Therefore, the one factor of the given polynomial is (x + 1).

Learn more about factor theorem here:

https://brainly.com/question/17092136

#SPJ2

The following initial condition is y(9) ≈ 2.286

The given differential equation is:

\(dy/dx = (1+x^2y^8)/x\)

We can start by separating the variables:

\(dy/(1+y^8) = dx/x\)

Integrating both sides, we get:

\((1/8) arctan(y^4) = ln(x) + C1\)

where C1 is the constant of integration.

Multiplying both sides by 8 and taking the tangent of both sides, we get:

\(y^4 = tan(8(ln(x)+C1))\)

Applying the initial condition y(1) = 6, we get:

\(6^4 = tan(8(ln(1)+C1))\)

C1 = (1/8) arctan(1296)

Substituting this value of C1 in the above equation, we get:

\(y^4 = tan(8(ln(x) + (1/8) arctan(1296)))\)

Taking the fourth root of both sides, we get:

\(y = [tan(8(ln(x) + (1/8) arctan(1296)))]^{(1/4)\)

Using this equation, we can find y(9) as follows:

\(y(9) = [tan(8(ln(9) + (1/8) arctan(1296)))]^{(1/4)\)

y(9) ≈ 2.286

for such more question on differential equation

https://brainly.com/question/25731911

#SPJ11

To solve the separable differential equation dy/dx = (1+x^2)y^8, we first separate the variables by dividing both sides by y^8 and dx. Integrate both sides: ∫ dy / (1 + xy^8) = ∫ dx

1/y^8 dy = (1+x^2) dx

Next, we integrate both sides:

∫1/y^8 dy = ∫(1+x^2) dx

To integrate 1/y^8, we can use the power rule of integration:

∫1/y^8 dy = (-1/7)y^-7 + C1

where C1 is the constant of integration. To integrate (1+x^2), we can use the sum rule of integration:

∫(1+x^2) dx = x + (1/3)x^3 + C2

where C2 is the constant of integration.

Putting it all together, we get:

(-1/7)y^-7 + C1 = x + (1/3)x^3 + C2

To find C1 and C2, we use the initial condition y(1) = 6. Substituting x=1 and y=6 into the equation above, we get:

(-1/7)(6)^-7 + C1 = 1 + (1/3)(1)^3 + C2

Simplifying, we get:

C1 = (1/7)(6)^-7 + (1/3) - C2

To find C2, we use the additional initial condition y(9). Substituting x=9 into the equation above, we get:

(-1/7)y(9)^-7 + C1 = 9 + (1/3)(9)^3 + C2

Simplifying and substituting C1, we get:

(-1/7)y(9)^-7 + (1/7)(6)^-7 + (1/3) - C2 = 9 + (1/3)(9)^3

Solving for C2, we get:

C2 = -2.0151

Substituting C1 and C2 back into the original equation, we get:

(-1/7)y^-7 + (1/7)(6)^-7 + (1/3)x^3 - 2.0151 = 0

To find y(9), we substitute x=9 into the equation above and solve for y:

(-1/7)y(9)^-7 + (1/7)(6)^-7 + (1/3)(9)^3 - 2.0151 = 0

Solving for y(9), we get:

y(9) = 3.3803

To solve the given separable differential equation, let's first rewrite it in a clearer format:

dy/dx = 1 + xy^8, with x > 0, and initial condition y(1) = 6.

Now, let's separate the variables and integrate both sides:

1. Separate variables:

dy / (1 + xy^8) = dx

2. Integrate both sides:

∫ dy / (1 + xy^8) = ∫ dx

3. Apply the initial condition y(1) = 6 to find the constant of integration. Unfortunately, the integral ∫ dy / (1 + xy^8) cannot be solved using elementary functions. Therefore, we cannot find an explicit solution to this differential equation with the given initial condition.

Learn more about integration at: brainly.com/question/18125359

#SPJ11

Answer:

\(\boxed{\bold{\huge{\boxed{Area\ of\ Trapezium = 150\ cm\²}}}}\)

Step-by-step explanation:

Using Pythagorean Theorem first to find EC

\(\sf CB^2 = EB^2+EC^2\\Where \ CB = 13 \ and \ EB = 5 \\13^2 = 5^2 + EC^2 \\169 - 25 = EC^2 \\EC^2 = 144\\Taking \ sqrt \ on \ both \ sides\\EC = 12 \ cm\)

Given that:

AE = EC

AE = 12 cm

Now,

AB = AE + EB

AB = 12 + 5

AB = 17 cm

ABCD is a trapezoid / trapezium

So,

Area of Trapezium = \(\sf \frac{AB+DC}{2} (EC)\)

Where AB = 17 , DC = 8 and EC = 12

Area of Trapezium = \(\sf \frac{17+8}{2} ( 12)\)

Area of Trapezium = (25)(6)

Area of Trapezium = 150 cm²