The Hcf and Lcm of two numbers are 4 and 288. What are the two numbers?

The function is a square root function that has been transformed from the parent function. Three parameters are a = -1, h = -2, and k = 3.

A set that has a geometric structure by itself or another set constitutes the geometric transformation. A shape may change shape, but not appearance. Following then, the form could match or resemble its preimage. A change in something's appearance is what transformations actually signify. Planar transformations and spaces can be distinguished from one another using the dimensions of the operand sets. Their characteristics can also be used to classify them.

The given function is f(x) = -√(x+2) + 3.

The function is a square root function that has been transformed from the parent function.

The three parameters of the function are:

a = -1, which reflects the graph of the function over the x-axis.

h = -2, which shifts the graph of the function 2 units to the left.

k = 3, which shifts the graph of the function 3 units up.

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If in the summation formula n starts at n=n0 the starting index  n0 is 2

The given series can be written as:

∑n=n0[infinity]an = (x-7)³/(x-7)⁶³* 2! + (x-7)⁹⁹/(x-7)¹²²⁷ * 3! + ...

We can simplify this expression by canceling out the common factor of (x-7) in each term of the series. This gives:

∑n=n0[infinity]an = (x-7)⁻⁶⁰ * 2! + (x-7)⁻¹¹²⁸ * 3! + ...

Now, we can see that each term in the series has the form:

an = [tex]k!/[(x-7)^{(3k-60)}][/tex]

where k is the index of the term and k ≥ 2, since the first two terms in the original series were combined into the first term of the simplified series.

Therefore, the starting index of the series is n0 = 2.

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Answer:

x= y + 5

Step-by-step explanation:

This is simple linear equations!

y = x - 5  

Add 5 to both sides, and you get

y + 5 = x!

To determine whether there is a statistical difference in the proportion of students in the two groups who have used a performance-enhancing drug at the 5% level, we need to perform a hypothesis test.

are the null and alternative hypotheses: Null Hypothesis: There is no significant difference in the proportion of students in the two groups that have used a performance-enhancing drug. (p1=p2)Alternative Hypothesis: There is a significant difference in the proportion of students in the two groups that have used a performance-enhancing drug. (p1≠p2)Level of Significance: α=0.05In this scenario, we have two different samples of sizes 1679 and 1366, where 34 freshman athletes and 24 senior athletes used performance-enhancing drugs.

Using the two-proportion z-test formula, we get z = (p1 - p2) / sqrt[pq (1/n1 + 1/n2)]wherep1 = number of freshman athletes who have used a performance-enhancing drug / total number of freshman athletes = 34/1679p2 = number of senior athletes who have used a performance-enhancing drug / total number of senior athletes = 24/1366p = (p1 * n1 + p2 * n2) / (n1 + n2)q = 1 - p Now let's plug in the values to get z = (-0.00394) / 0.01267 = -0.311where we round the absolute value of the test statistic to two decimal places.

According to the standard normal distribution table, the critical values are ± 1.96 at a 5% level of significance. Since our test statistic, -0.311, does not fall outside this range, we fail to reject the null hypothesis. Hence, we conclude that there is no significant difference in the proportion of students in the two groups that have used a performance-enhancing drug at the 5% level.

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No, the expression -x^2 + 6x - 9 cannot be non-negative for all values of x.

The expression -x^2 + 6x - 9 cannot be non-negative for all values of x.

To see this, note that the leading coefficient of the quadratic term is negative, which means that the graph of the function is a downward-facing parabola.

The vertex of the parabola occurs at x = -b/(2a) = -6/(2*(-1)) = 3, and the value of the function at this point is -(-3)^2 + 6*(-3) - 9 = -18.

Since the value of the function at the vertex is negative, and the graph of the function is a downward-facing parabola, the function is negative for all x values to the left and right of the vertex, and thus can never be non-negative for all values of x.

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To find the sum of the first 100 terms of a geometric sequence, we can use the formula:

S = a(1 - f^n) / (1 - f)

where S is the sum of the first n terms, a is the first term, f is the common ratio, and n is the number of terms.

In this case, a = 10, f = 1.1, and n = 100. So we have:

S = 10(1 - 1.1^100) / (1 - 1.1)

S = 10(1 - 2.98551 x 10^7) / (-0.1)

S = 10(-2.98551 x 10^7 + 1) / 0.1

S = -2.98551 x 10^8 + 10

S = 3.7949 x 10^6

Therefore, the sum of the first 100 terms is approximately 3.7949 million. The closest answer choice to this value is C) 3.877 million, but none of the answer choices match the calculated value exactly.

The matching of the system of equations with the diagrams is:

1 → Diagram A

2 → Diagram B

3 → Diagram D

4 → Diagram C

In the given answered pairs, the systems of equations 1 – 4 have been numbered from left to right, and the diagrams A – D from top to bottom.

The attached the row-reduction of the first three systems (1 – 3). The last system (4) is obviously three repetitions of the same equation, so is the same plane 3 times, as in diagram C.

1. The last row of the given reduced matrix has a non-zero element in the rightmost column, which tells us that there is no solution. The two non-zero rows indicates that the system specifies planes that intersect in parallel lines. In a local area, the solution matches diagram A in that specific one plane intersects the two others in parallel lines. Despite the fact that the lines are parallel, the planes are not parallel.

The last attachment indicates a rendering of the particular first system of equations. Though the colors leave some doubts, but it is clear that they intersect in a way that provides a triangular tunnel. No (x, y, z) value is found on all three planes.

2. The reduced matrix shows there is a single solution, corresponding to the planes all intersecting at one point.

3. The last row of the reduced matrix being all zeros means the solution is a line, as shown in diagram D.

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An equation for the fish population, after years is P(t) ≈ 8000e-0.664t ± 0.00005

The equation for the fish population in a stocked lake can be determined using exponential growth.

The formula for exponential growth is

P(t) = [tex]P_0[/tex]ert

where P(t) is the population after time t,

[tex]P_0[/tex] is the initial population,

r is the rate of growth, and

e is the mathematical constant approximately equal to 2.71828.

The carrying capacity, K, is the maximum population that can be supported by the environment, and it is a limiting factor for population growth. When the population reaches K, the growth rate slows down until it reaches a point of equilibrium.

The carrying capacity can be estimated using a variety of methods, including the logistic model, which incorporates the carrying capacity into the equation for population growth.

However, for this problem, we are given an estimate of the carrying capacity, which we will use in the exponential growth equation.

We are also given the initial population,

[tex]P_0[/tex] = 1100, and we are asked to find an equation for the fish population, P(t), after t years.

The rate of growth, r, can be determined using the following formula:

r = ln(P/K) / t

where ln is the natural logarithm, and t is the time it takes to reach the carrying capacity, K.

Since we don't have an exact value for K, we will use the estimated value of 8000.

Thus,

r = ln(1100/8000) / 1

r ≈ -0.664

The negative sign indicates that the population is decreasing, since the growth rate is negative.

Therefore, the equation for the fish population after t years is:

P(t) = 8000e-0.664t

To round to 4 decimal places, we can use the formula:

P(t) ≈ 8000e-0.664t ± 0.00005

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It is important to keep both sides balanced when solving an equation. If you add 5 to one side you must add 5 to the other side. An equation is basically saying that both sides are equal. If you do something to one side but not the other the equation will no longer be equal. Another property we use when solving equations is the order of operations.

(a) The differential equation to model the growth rate for the fish population with harvesting is: dP/dt = kP - h, where P is the population of fish, k is the growth rate, and h is the harvesting rate.

(b) To calculate the maximum rate at which the farmer can harvest and maintain steady-state fish population, solve the differential equation for the steady-state solution, which is P = h/k.

(c) The minimum fish the farmer must buy so that the population will not die out is h/k, which is the same as the steady-state solution.

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The probability that less than 4 of them find their jobs stressful in the given binomial experiment would be 0.2919.

The probability of finding a job stressful in the given binomial experiment is p = 0.75 (given).The probability of not finding a job stressful is q = 1 - p = 1 - 0.75 = 0.25. The number of trials is n = 9 (given).We need to find the probability that less than 4 of them find their jobs stressful. Therefore, the required probability is:

P(X < 4)= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) where X is the number of workers who find their jobs stressful.

Using binomial probability distribution formula, we can calculate the probability for each value of X as follows:

P(X = x) = (nCx) p×q (n-x)

nCx = n! / (x!(n - x)!)

Where n! = n × (n - 1) × (n - 2) × ... × 2 × 1

Using this formula and adding up the probabilities for all values of X, we get:

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)= (9C0)0.75^0 × 0.25^9 + (9C1)0.75^1 × 0.25^8 + (9C2)0.75^2 × 0.25^7 + (9C3)0.75^3 × 0.25^6= 0.00035 + 0.00664 + 0.05223 + 0.23255= 0.2919

Therefore, the probability that less than 4 of them find their jobs stressful in the given binomial experiment would be 0.2919.

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Thus, equation of line with same slope and passing point  (3,6) is :

y - 6 = 2/5(x - 3).

The difference between the change in y-values and the change in x-values is known as the slope of a line. This figure represents the slope of a line.

A quantity called the slope of a line is used to quantify how steep a line is. This number may be zero, positive, or negative. Moreover, it may be unreasonable or rational.

Given equation of line:

y + 4 = 2/5 ( x - 9 )

General equation in two point form:

y - y1 = m(x - x1)

On comparing both equation:

slope of line m = 2/5

Now, equation of line with same slope and passing point  (3,6).

y - 6 = 2/5(x - 3).

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the aggregate demand curve is downward-sloping because, other things being equal, With average price reductions, more people purchase goods and services.

This is known as the law of demand, which states that there is an inverse relationship between the price of a good or service and the quantity of that good or service demanded by consumers. When the price of a good or service goes up, consumers tend to demand less of it, and when the price goes down, consumers tend to demand more of it. Therefore, if all other factors affecting demand remain constant, an increase in price will lead to a decrease in the quantity demanded, and a decrease in price will lead to an increase in the quantity demanded. This is why the aggregate demand curve is downward-sloping.

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Compete Question

the aggregate demand curve is downward-sloping because, other things being equal, ____. FILL IN THE BLANKS

Answer:

178.50

Step-by-step explanation:

8.50 · 7

59.5 · 3

178.50

Aisha earned C178.50 in a week.

First, multiply the number of hours worked per day with the amount earned per hour.

7 × C8.50 = C59.50

Then, use that answer and multiply it by the number of days worked.

3 x C59.50 = C178.50

The store clerk is stocking the shelves. He places on the shelf a box of cereal that weighs 850 grams, a box of granola bars that weighs 385 grams, and a box of crackers that weighs 435 grams. The total weight in kilograms is 1.67 kilograms.

Given:

The weight of the box of cereal = 850 grams.

The weight of the box of granola bars = 385 grams.

The weight of the box of crackers = 435 grams.

The total weight in kilograms of the above boxes.

Total weight of the boxes = 850 + 385 + 435 grams= 1670 grams.

To convert grams to kilograms, we divide it by 1000.

So, the weight in kilograms= 1670/1000= 1.67 kilograms.

Hence, the total weight in kilograms of the above boxes is 1.67 kilograms.

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Answer:

14

Step-by-step explanation:

Answer:

b = 14

Step-by-step explanation:

In order to find the y intercept of the problem, we will have to put the equation of the line in point slope form and solve for y.

Point slope form: [tex]y-y_1=m(x-x_1)[/tex]

Lets put the equation of the line in point slope form.

[tex]y-2=4(x+3)[/tex]

Distribute 4

[tex]y-2=4x+12[/tex]

Add 2 to both sides

[tex]y=4x+14[/tex]

Now that the equation of the line is in slope intercept form, the y intercept is at the end of the equation, which is 14.

b = 14

The equation that can be used to find the nth term in the sequence is:

aₙ= 6n - 24

What is sequence ?

In mathematics, a sequence is a list of numbers, arranged in a specific order. Each number in the sequence is called a term, and the position of the term in the sequence is called its index. A sequence can be either finite or infinite.

For example, a sequence of even numbers can be written as:

2, 4, 6, 8, 10, ...

where each term in the sequence is obtained by adding 2 to the previous term. The first term of the sequence is 2, the second term is 4, and so on.

Sequences are used in various mathematical applications such as in number theory, calculus, statistics, and computer science. They are also used in real-world applications such as modeling the behavior of stocks in the stock market, modeling the spread of disease in a population, and so on.

We can notice that the given sequence is an arithmetic sequence with a common difference of 6.

So, to find the nth term of the sequence, we can use the formula:

aₙ= a₁+ (n-1)d

where a₁is the first term of the sequence, d is the common difference, and n is the term number we want to find.

Using the information given in the problem, we can find the value of a_1 by working backward from a₄

a₄= a₁ + 3d = 0

a₁= -3d

Substituting the values of a₁ and d, we get:

aₙ = -3d + (n-1)d

aₙ= (n-4)d

Since d = 6, we have:

aₙ= 6n - 24

So, out of the given options, the equation that can be used to find the nth term in the sequence is:

aₙ= 6n - 24

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The probability that a second call will not arrive in the next 20 seconds is approximately 0.2636 or 26.36%.

Poisson probability is a mathematical concept that describes the probability of a certain number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. The Poisson probability distribution is named after French mathematician Siméon Denis Poisson, who introduced it in the early 19th century to model the occurrence of rare events, such as errors in counting or measurement, accidents, or phone calls.

The Poisson probability distribution is a discrete probability distribution that gives the probability of a certain number of events (x) occurring in a fixed interval (t), when the average rate of occurrence (λ) is known. The Poisson probability distribution assumes that the events occur independently and at a constant average rate over time or space. The formula for Poisson probability is:

P(x; λ) = ([tex]e^{-\lambda}[/tex]) * λˣ) / x!

where:

P(x; λ) = the probability of x occurrences in a given interval, when the average rate is λ

e = a mathematical constant e (approximately 2.71828)

λ = it is the average rate of occurrence in the given interval

x = it is number of occurrences in the given interval

Given that calls for dial-in connections arrive at an average rate of four per minute and follow a Poisson distribution, we can use the Poisson probability formula to solve this problem. The Poisson probability formula is:

P(x; λ) =  ([tex]e^{-\lambda}[/tex]) * λˣ) / x!

where:

P(x; λ) = the probability of x occurrences in a given interval, when the average rate is λ

e = a mathematical constant e (approximately 2.71828)

λ = the average rate of occurrence in the given interval

x = it is the number of occurrences in the given interval

In this problem, we are interested in finding the probability that a second call will not arrive in the next 20 seconds, given that a call has already arrived at the beginning of a one-minute interval. Since we are given the average rate of calls per minute, we need to adjust the interval to 20 seconds, which is 1/3 of a minute. Therefore, the average rate of calls per 20 seconds is:

λ = (4 calls/minute) * (1/3 minute) = 4/3 calls/20 seconds

Using the Poisson probability formula, we can calculate the probability of no calls arriving in the next 20 seconds:

P(0; 4/3) = ( [tex]e^{-4/3}[/tex]* (4/3)⁰) / 0! = [tex]e^{-4/3}[/tex] ≈ 0.2636

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1. The cost per zinnia plant is: A. $0.04

2. The linear equation is D. C = 0.04z + 15

3. The cost of purchasing 700 zinnia plants is: C. $43.

Part 1: To find the cost per zinnia plant, derive two points from the info given which are:

(200, 23) and (500, 35)

The slope = cost per zinnia plant = 35 - 23 / 500 - 200

= 12/300

= $0.04.

Therefore, the correct answer: A.$0.04

Part 2: To derive a linear equation to represent the cost, C, to purchase z number of zinnia plants, substitute (z, C) = (200, 23) and m = 0.04 into C = mz + b to find the y-intercept (b):

23 = 0.04(200) + b

23 = 8 + b

23 - 8 = b

b = 15

Substitute m = 0.04 and b = 15 into C = mz + b:

C = 0.04z + 15

Therefore, the correct answer is D. C = 0.04z + 15

Part 3:

Using the equation C = 0.04z + 15, we can find the cost of purchasing 700 zinnia plants:

C = 0.04(700) + 15

C = 43

Therefore, the correct answer is: C. $43.

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Answer: 20.48

Step-by-step explanation: 12.8 x 1.6 = 20.48

Answer:

20.48

Step-by-step explanation:

If you do 12.8 multiplied by 1.6 you get 20.48.

The value of function f(4) is approximately = 32.23

To solve for f(4), we need to integrate the given second derivative twice and use the initial conditions to find the constants of integration. Then we can evaluate the function at x=4.

First, we integrate f′′(x) to get f′(x):

f′(x) = ∫ f′′(x) dx = ∫ 9x 6sin(x) dx = -9x 6cos(x) + C1

Next, we integrate f′(x) to get f(x):

f(x) = ∫ f′(x) dx = ∫ (-9x 6cos(x) + C1) dx = -9x 6sin(x) + C1x + C2

Using the initial conditions, we can solve for C1 and C2:

f(0) = 4 = C2

f′(0) = 3 = C1

Therefore, we have:

f(x) = -9x 6sin(x) + 3x + 4

Finally, we can evaluate f(4):

f(4) = -9(4) 6sin(4) + 3(4) + 4 = -36sin(4) + 16

So the value of f(4) is approximately -32.23.

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The vertical support cable length that is 60 feet from the towers is 56.5 feet.

The given information:

Towers are 400 ft apart

The supporting cable is 100 ft high

The supporting cable runs 30 ft above the road

The shape of the cable is a parabola.

Find the length of the vertical support cables that are 60 feet from the towers.

First, let’s set up a coordinate system with the origin at the lowest point of the cable and the x-axis along the road. The towers are 400ft apart, so their x-coordinates will be -200 and 200.

The equation of a parabola is y = ax^2 + bx + c. Since the cable is 30ft above the road at its lowest point, c = 30.

The towers are 100ft tall, so when x = -200 and x = 200, y = 100. Substituting these values into the equation of the parabola gives us two equations: 100 = a(-200)^2 + b(-200) + 30 and 100 = a(200)^2 + b(200) + 30.

Solving these equations simultaneously for a and b gives us a = 0.001875 and b = 0.

So the equation of the parabola is y = 0.001875x^2 + 30.

The towers are 400ft apart, so their x-coordinates are -200 and 200. The vertical support cable that we want to find the length of is 60 feet from one of the towers. So if we move 60 feet from one of the towers along the x-axis towards the center of the bridge, we will reach the point where the vertical support cable is attached to the road. This point will have an x-coordinate of -200 + 60 = -140 or 200 - 60 = 140.

Now we can find the length of the vertical support cable that is 60 feet from one of the towers. When x = -140 or x = 140,

y = 0.001875(140)^2 + 30 ≈ 56.5.

So the length of the vertical support cable that is 60 feet from one of the towers is approximately 56.5 feet.

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The height of the water in cylinder after inversion is h = R(2 - √3) cm.

The curved surface of a liquid in a container is known as a meniscus and is created by the intermolecular interactions between the liquid and the material of the container. The type of the liquid and the container, as well as outside variables like temperature and pressure, all affect the meniscus' form. A concave meniscus has a liquid surface that is lower in the middle than it is at the edges, whereas a convex meniscus has a liquid surface that is higher in the middle than it is at the edges. Many scientific and technical applications, such as figuring out a liquid's surface tension or how fluids behave in microfluidic devices, might benefit from understanding the geometry of the meniscus.

Let us suppose the height of the water level after inversion = h.

The volume of the figure 1 is:

V1 = πR²(R-h)

where, (R-h) represents the height of the cylinder filled with water.

When inversion takes place water forms a smaller hemisphere with radius h.

The volume is:

V2 = 2/3 πh³ + πh² (R-h)

Setting the equations:

V2 = 2/3 πh³ + πh² (R-h)

h = R(2 - √3)

Hence, the height of the water after inversion is h = R(2 - √3) cm.

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Answer: 37

Step-by-step explanation:

STEP 1: Write out the equation; 89 - 4^2 (4) + 12

STEP 2: Determine 4^2 [16]

STEP 3: Rewrite the equation to reflect the above step; 89 - 16 (4) + 12
STEP 4: Determine 16 (4) [64]

STEP 5: Rewrite the equation to reflect the above step; 89 - 64 + 12

STEP 6: Determine 89 - 64 + 12 [37]

Therefore, the answer is 37.

NOTE: Following PEMDAS for these types of problems is very helpful. Feel free to message me if you have any questions.

Answer:

Step-by-step explanation:

Therefore, the length of each side of the original square is 5 cm.

Area is a measure of the size of a two-dimensional surface or region. It is the amount of space enclosed by a boundary in two dimensions. In simple terms, area is the size of a flat surface, such as the floor, a wall, or a piece of paper. It is usually measured in square units such as square meters (m²), square centimeters (cm²), square feet (ft²), or acres.

Here,

Let x be the length of each side of the original square.

When the length of each side is increased by 10 cm, the new length is x + 10, and the area of the new square is (x + 10)².

According to the problem, the increase in area is 200 cm², so we can set up the equation:

(x + 10)² - x² = 200

Expanding the left side of the equation, we get:

x² + 20x + 100 - x² = 200

Simplifying, we get:

20x + 100 = 200

Subtracting 100 from both sides, we get:

20x = 100

Dividing both sides by 20, we get:

x = 5

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Answer:

Annie jumps 34.56 cm (approximately)

Step-by-step explanation:

Since we know that Annie jumps 8% higher than Rosie, we can add the 8% increase to the height that Rosie jumped.

32 cm + 2.56 cm = 34.56 cm

Therefore, Annie jumped approximately 34.56 cm in the air, which is 8% higher than the height that Rosie jumped.

Step-by-step explanation:

To find out how high Annie jumped, you first need to calculate what 8% of Rosie's jump height is, and then add this amount to Rosie's jump height.

To calculate 8% of Rosie's jump height, you can multiply 32cm by 8% expressed as a decimal, which is 0.08:

8% of 32cm = 0.08 x 32cm = 2.56cm

So Annie jumped 2.56cm higher than Rosie's jump of 32cm.

To find out how high Annie jumped, you can add 2.56cm to Rosie's jump height:

Annie's jump height = Rosie's jump height + 2.56cm

= 32cm + 2.56cm

= 34.56cm

Therefore, Annie jumped 34.56cm in the air.

The perimeter of the rectangle is 2(sqrt(82) + 4) units.

what is perimeter?

Perimeter is the total distance around the outside of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape. In other words, if you were to walk along the edge of a shape, the distance you would cover would be the perimeter of the shape. The units used to measure perimeter are the same as those used to measure length, such as inches, centimeters, or meters.

Let's call the two points on the x-axis (a,0) and (b,0), where a and b are both positive.

Since the rectangle has an area of 54 square units, we know that:

length x width = 54

We also know that one point is located at (5,9), so the length of the rectangle must be the distance between (5,9) and (a,0) or (b,0). Similarly, the width must be the distance between (5,9) and (a,0) or (b,0).

Let's first find the length. The distance formula gives us:

length = sqrt((5-a)^2 + 9^2) or sqrt((5-b)^2 + 9^2)

Next, let's find the width. Again, using the distance formula, we have:

width = sqrt((a-b)^2 + 0^2)

Now we can use the fact that the area of the rectangle is 54 to solve for a and b.

54 = length x width

54 = sqrt((5-a)^2 + 9^2) x sqrt((a-b)^2 + 0^2)

54 = sqrt((5-a)^2 x (a-b)^2 + 0^2)

2916 = (5-a)^2 x (a-b)^2

Since a and b are both positive, we know that 5 > a > b. Let's try some values of a and b that satisfy this condition and see which one gives us an equation that works out to 2916.

If a = 4 and b = 2, we get:

2916 = (5-4)^2 x (4-2)^2 = 4

This doesn't work. Let's try a = 3 and b = 2:

2916 = (5-3)^2 x (3-2)^2 = 4

Still doesn't work. Let's try a = 6 and b = 2:

2916 = (5-6)^2 x (6-2)^2 = 16

This works! So the two points on the x-axis are (2,0) and (6,0).

Now we can find the length and width:

length = sqrt((5-6)^2 + 9^2) = sqrt(82)

width = sqrt((6-2)^2 + 0^2) = 4

Finally, we can find the perimeter:

perimeter = 2 x length + 2 x width

perimeter = 2 x sqrt(82) + 2 x 4

perimeter = 2(sqrt(82) + 4)

Therefore, the perimeter of the rectangle is 2(sqrt(82) + 4) units.

To learn more about perimeter from the given link

https://brainly.com/question/6465134

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Answer:

  use (1 -cos²x) for sin²x and expand

Step-by-step explanation:

You want to verify sin⁴x·cos²x = cos²x -2cos⁴x +cos⁶x.

You can get the desired result by substituting (1 -cos²x) for sin²x.

  [tex]\sin^4{x}\,\cos^2{x}=(1-\cos^2{x})^2\,\cos^2{x}=(1-2\cos^2{x}+\cos^4{x})\cos^2{x}\\\\=\boxed{\cos^2{x}-2\cos^4{x}+\cos^6{x}}[/tex]