Two terms of a geometric sequence are a5=2500 and a8=312,500 Write an explicit rule for the nth term

The function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

To determine if the function f(x) = –2x^3 + 39x^2 -216x + 6 has a local minimum or maximum, we need to find the critical points of the function and then determine the nature of those critical points.

First, we take the derivative of the function to find the critical points:

f(x) = –2x^3 + 39x^2 -216x + 6

f'(x) = –6x^2 + 78x - 216

f'(x) = –6(x^2 - 13x + 36)

f'(x) = –6(x - 4)(x - 9)

Setting f'(x) = 0, we get:

–6(x - 4)(x - 9) = 0

This gives us two critical points at x = 4 and x = 9.

To determine the nature of these critical points, we need to look at the sign of the derivative on either side of each critical point.

When x < 4, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

When 4 < x < 9, we have:

f'(x) = –6(x^2 - 13x + 36) > 0

When x > 9, we have:

f'(x) = –6(x^2 - 13x + 36) < 0

This means that f(x) is decreasing on the interval (–∞, 4), increasing on the interval (4, 9), and decreasing on the interval (9, ∞). Therefore, we have a local minimum at x = 4 and a local maximum at x = 9.

To confirm this, we can evaluate the function at these critical points:

f(4) = –2(4)^3 + 39(4)^2 -216(4) + 6 = –26

f(9) = –2(9)^3 + 39(9)^2 -216(9) + 6 = 603

Therefore, the function f(x) = –2x^3 + 39x^2 -216x + 6 has one local minimum at x = 4 and one local maximum at x = 9.

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b. Nonresponse bias creates an impact on the confidence interval, but is not accounted for by the margin of error.

Given that, the dean of students at a large college is interested in learning about the opinions of students regarding the percentage of first-year students who should be given parking privileges in the main lot. He sends out an email survey to all students about this issue, but receives very few responses from sophomores, juniors, and seniors. Based on the responses he receives, he constructs a 90% confidence interval for the true proportion of students who believe first-year students should be given parking privileges in the main lot to be (0.71, 0.79).

Response bias refers to a systematic pattern of incorrect responses in a survey, which can be caused by factors such as question wording, social desirability bias, or interviewer bias.

Nonresponse bias, on the other hand, occurs when individuals who do not respond to a survey are systematically different from those who do respond, leading to a biased estimate of the population parameter.

Sampling variation refers to the fact that different samples from the same population can yield different estimates of the population parameter due to random variation.

Under coverage bias occurs when some members of the population are systematically excluded from the sample, leading to a biased estimate of the population parameter.

In this scenario, the fact that very few sophomores, juniors, and seniors responded to the survey could potentially introduce nonresponse bias, since those who did respond may not be representative of the entire population of students.

However, the confidence interval itself does not account for nonresponse bias or any other sources of bias. Instead, it reflects the range of values that is likely to contain the true proportion of students who believe first-year students should be given parking privileges in the main lot, based on the data that was collected.

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The rotation used in this problem is given as follows:

90º clockwise rotation.

The five more known rotation rules are given as follows:

A vertex and it's equivalent is given as follows:

A(1,3) and A'(3, -1).

Hence the rule is:

(x,y) -> (y, -x).

Which is the rule for a 90° clockwise rotation = 270º counterclockwise rotation.

The image is presented at the end of the answer.

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Answer:

90° clockwise rotation

Step-by-step explanation:

I did the exam and got it correct

The circumference of the circle is 6.28 kilometers if the diameter of the circle is 2 kilometers and assuming the value of π is 3.14 kilometers.

The diameter of the circle = 2 kilometers

The circumference of a circle is calculated by using the formula,

C = π *d

where,

C = circumference of a circle

d = diameter of the circle

π = Constant value = 3. 14 Km

Substituting the above-given values into the equation, we get:

C = π*d

C = 3.14 x 2 km

C = 6.28 km

Therefore, we can conclude that the circumference of the circle is 6.28 kilometers.

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The sum of all the angles in a quadrilateral is 360° ,So The angle measure indicated is 137°.

What is radius?

In classical geometry, the radius of a circle or sphere is any line segment that links the object's centre to its edge; in more modern usage, the term also refers to the length of such line segments. The Latin term "radius," which may also be used to describe a chariot wheel spoke, is where the word "radius" first appeared.

The length of tangents drawn from an external point is known to be constant. The circle's radius across the point of contact and any other point on the circle are perpendicular to the tangent. A quadrilateral has 360° of angles total. In light of this, 137° is the indicated angle measurement.

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Correct question is

Find the angle measure indicated. Assume that lines which appear to be tangent are tangent.

Answer:

(3-x)^2=y

Step-by-step explanation:

The mode of transportation that I will choose is motorized scooter this is because it is relatively cheap when compared to the other forms of transportation.

A motorized scooter  could be a moderately cheap mode of transportation compared to other alternatives such as cars, cruisers, or indeed bikes.

The starting price of obtaining a motorized bike is for the most part lower than that of a car or cruiser, and the continuous costs such as fuel, support, and protections are too ordinarily lower.

From the question, you can see that the price of the scooter at 5 miles is $10 and it is relatively cheap when compared to the rest.

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148.35 cubic feet is the volume of a cylinder with height of 7 inches and a base diameter of 18ft

We have to find the volume of a cylinder

V=πr²h

h is the height of cylinder and r is radius of the base.

Given height is 7 inches which is 0.583333 feet

Diameter is 18 ft

Radius is 9 ft

Now plug in value of height and radius

Volume=π(9)²×0.5833

=3.14×81×0.5833

=148.35 cubic feet

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The area k of the triangle a = 3, c = 2, b = 135 degrees is approximately 1.06125 square units.

The area k of the triangle, we can use the formula:
k = (1/2) * b * c * sin(A)
where A is the angle opposite side a.
Find A, we can use the fact that the angles in a triangle add up to 180 degrees:
A + B + C = 180
Substituting in the given values, we get:
A + 135 + 180 = 360
A = 45 degrees
Now we can plug in all the values into the area formula:
k = (1/2) * 2 * 3 * sin(45)
k = 1.5 * 0.707
k = 1.06125
Therefore, the area k of the triangle is approximately 1.06125 square units.

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Ethan's monthly salary is $1450.

Ethan's monthly salary, we can use the debt to income ratio formula, which is calculated by dividing monthly debt expenses by monthly income.

Given:

Monthly expenses = $1160

Debt to income ratio = 0.8

Let's assume Ethan's monthly salary as S.

We can set up the equation using the debt to income ratio formula:

Debt to income ratio = Monthly expenses / Monthly income

0.8 = $1160 / S

To solve for S (monthly salary), we can rearrange the equation:

S = $1160 / 0.8

Dividing $1160 by 0.8 gives us:

S ≈ $1450

Therefore, Ethan's monthly salary is approximately $1450.

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There are infinitely many possible solutions for the lengths of the three sides of the triangle.

What is triangle?

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry.

If we call x the length of the smallest side, then the other two sides are x+1 and x+2 (since they are three consecutive integers). According to the triangle inequality, the sum of any pair of sides must be greater than the length of the third side.

Therefore, we have:

x + (x+1) > (x+2) (and also x + (x+2) > (x+1) and (x+1) + (x+2) > x)

Simplifying each inequality, we get:

2x + 1 > x + 2 (and also 2x + 2 > x + 1 and 2x + 3 > x)

Which gives:

x > 1

So the smallest side must be greater than 1 cm.

Now, to find the length of the three sides, we can choose any value greater than 1 for x. For example, if we take x=2, then the three sides are:

2 cm, 3 cm, and 4 cm

If we take x=3, then the three sides are:

3 cm, 4 cm, and 5 cm

And so on. Therefore, there are infinitely many possible solutions for the lengths of the three sides of the triangle.

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The solution to the equation is m = 0.

What is an algebraic expression?

An algebraic expression is a mathematical phrase that contains variables, constants, and mathematical operations. It may also include exponents and/or roots. Algebraic expressions are used to represent quantities and relationships between quantities in mathematical situations, often in the context of problem-solving.

To solve the equation:

(-3m^ {5}) (-2m^ {4}) = (-6m^ {9})

We can simplify the left side of the equation by multiplying the terms:

(-3m^ {5}) (-2m^ {4}) = (6m^ {9})

Now we have:

6m^ {9} = (-6m^ {9})

To solve for m, we can divide both sides by 6m^ {9}:

m^ {9} = -m^ {9}

Since the powers of m on both sides are equal, we can simplify to:

2m^ {9} =0

Dividing both sides by 2, we get:

m^ {9} =0

Taking the ninth root of both sides, we get:

m = 0

Therefore, the solution to the equation is m = 0.

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Complete Question:

Simplify the expression: $(-3m^ {5}) (-2m^ {4}) $.

Answer:

The radius of the circle is 5/√2 = (5√2)/2 inches.

Area of circle = π((5√2)/2)^2

= 25π/2 square inches

Area of triangle = (1/2)(5√2)((5√2)/2)

= 25/2 square inches

Area of shaded region

= (25/2)(π - 1) = 26.77 square inches

Answer 1 inch per hour

Step-by-step explanation:

Answer:

1 inch per hour

Step-by-step explanation:

Let x represent inches burned per hour

The equation to finding the rate of the candle burning:

3x=3

For those who dont know, whenever you have a variable with a term, its basically multiplying, and we want to do the opposite to get x by itself, so we divide

x=3/3

Which can be simplified as:

x=1

Where x is the inches burned per hour, which equal 1

The correct answer is 904.78 cm2

The problem presents a scenario where a square patio is placed in the centre of a 10m x 10m square backyard. The side length of the patio is given by x, and the remaining area of grass is expressed as A=-x^2+100.

To find the area of grass remaining, we can substitute different values of x into the equation.

For instance, if the patio is 5m x 5m, then x = 5 and the area of grass remaining is[tex]A = -5^2 + 100 = 75[/tex] square metres. Similarly, if the patio is 8m x 8m, then x = 8 and the area of grass remaining is [tex]A = -8^2 + 100 = 36[/tex] square metres.

As we can see, the area of grass remaining decreases as the size of the patio increases.

This problem illustrates the concept of content loaded and content remaining, where the initial content is the entire area of the square backyard, and the loaded content is the area of the square patio.

The remaining content is what is left after the loaded content is subtracted from the initial content. In this case, the loaded content is the patio area, and the remaining content is the grass area.

In summary, the area of grass remaining in the backyard after a square patio is placed in the centre can be calculated using the equation [tex]A=-x^2+100,[/tex] where x is the side length of the patio in metres.

The concept of content loaded and content remaining is also illustrated in this problem, where the loaded content is the patio area and the remaining content is the grass area.

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The upper and lower bound of the cross-sectional area of the insulation, would be A = [ 3129, 3137 ] cm².

The upper and lower bound of A would be found by the formula :

A = π x ( R ² - r ² )

The upper bound is therefore:

= π x  (( 48. 75 / 2) ² - ( 19.2 45 / 2) ²)

= π x ( 1183. 0625 - 184. 857025 )

= π x 998. 205475

= 3, 137 cm²

The lower bound will then be:

= π x ( ( 48. 65 / 2 ) ²- (19. 255 / 2) ²)

= π x ( 1180. 9225 - 184. 963025)

= π x 995. 959475

= 3, 129 cm²

The interval form is therefore A = [ 3129, 3137 ] cm²

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Answer:

Vertical Compression by factor of 1/4

Step-by-step explanation:

Two methods:

Method 1. Transformations

Method 2. Algebraic input-output tables

Method 1. Transformations

The Main concept of this question is about Transformations of functions -- specifically, multiplying on the outside by a positive number less than 1.

The transformation that occurs when multiplying a function by a positive number on the outside of the function is a vertical stretch or compression.

Positive numbers larger than 1 will stretch it vertically, whereas positive numbers smaller than 1 will compress it vertically.

Therefore, multiplying by 1/4 on the outside, a positive number less than 1, will vertically compress the function down to one-fourth the size.

This means that for g(x), all points on the original function f will have their heights reduced to 1/4 their original height (or depth) -- making all points on g(x) 1/4 their previous distance from the x-axis on the "f" function.

Method 2. Algebraic input-output tables

Observe on the graph three points on the function f:

Using the equation relating f and g, [tex]g(x)=\frac{1}{4}f(x)[/tex], we can find how those points would look like on the new function g(x).

For [tex]f(0)=0[/tex]

[tex]g(0)=\frac{1}{4}[f(0)]\\g(0)=\frac{1}{4}[0]\\g(0)=0[/tex]

For [tex]f(1)=4[/tex]

[tex]g(1)=\frac{1}{4}[f(1)]\\g(1)=\frac{1}{4}[4]\\g(1)=1[/tex]

For [tex]f(3)=0[/tex]

[tex]g(3)=\frac{1}{4}[f(3)]\\g(3)=\frac{1}{4}[0]\\g(3)=0[/tex]

These known points should correctly identify the graph from the possible choices.

If she continues to increase the length of her runs the same way, it will not be enough to reach the half-marathon distance of 13.1 miles within the remaining time of the training program.

Sariah has just begun training for a half-marathon, which is 13.1 miles. There are 6 weeks of training runs remaining before the race. In her first week of training, Sariah ran 3 miles. She ran 4.5 miles the second week and 6 miles the third week.

To determine if there is enough time left in the training program for her to get up to half-marathon distance, let's analyze the pattern of her weekly increases in distance:

Week 2 - Week 1 = 4.5 miles - 3 miles = 1.5 miles increase
Week 3 - Week 2 = 6 miles - 4.5 miles = 1.5 miles increase

Sariah is consistently increasing her weekly mileage by 1.5 miles. With 3 weeks of training already completed, she has 3 more weeks to go. Let's see if she can reach the half-marathon distance of 13.1 miles:

Week 4: 6 miles + 1.5 miles = 7.5 miles
Week 5: 7.5 miles + 1.5 miles = 9 miles
Week 6: 9 miles + 1.5 miles = 10.5 miles

After 6 weeks of training, Sariah will have increased her longest run to 10.5 miles. Unfortunately, this is not enough to reach the half-marathon distance of 13.1 miles within the remaining time of the training program.

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The probability that the smallest bell rings the first three days in a row is 1/27. when Each day three church bells are rung in a random order

In the given data there are 3 bells in the church in which there is a small bell and the three bells are rung in a random order. we need to find the probability that the smallest bell rings the first three days in a row.

The probability that the smallest bell rings on any given first day can be given as  = 1/3

Because there are three bells and each bell has an equal chance of being rung first. The probability that this happens three days in a row is given as

= (1/3) × (1/3) × (1/3)

= 1/27

Therefore, the probability that the smallest bell rings the first three days in a row is 1/27

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The value of the tangent TU for the circle with secant through U which intersect the circle at points V and W is equal to 12

Circle theorems are a set of rules that apply to circles and their constituent parts, such as chords, tangents, secants, and arcs. These rules describe the relationships between the different parts of a circle and can be used to solve problems involving circles.

For the tangent TU and the secant through U which intersect the circle at points V and W;

TU² = UV × VW {secant tangent segments}

(5x)² = 9 × 16

(5x)² = 144

5x = √144 {take square root of both sides}

5x = 12

x = 12/5

so;

TU = 5(12/5)

TU = 12

Therefore, the value of the tangent TU for the circle with secant through U which intersect the circle at points V and W is equal to 12

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S = lim[n → ∞] sn
 = lim[n → ∞] (306n^2 + 35n + 306)
 = ∞
So unfortunately, the series (n + 16)(n + 18) diverges to infinity and does not have a finite sum.To find the sum S of the series (n + 16)(n + 18), we need to take the limit of the sequence of partial sums as n approaches infinity. So let's first find the formula for the nth partial sum sn:

sn = (1 + 16)(1 + 18) + (2 + 16)(2 + 18) + ... + (n + 16)(n + 18)
  = ∑[(k + 16)(k + 18)] (from k = 1 to n)

Using the formula for the sum of squares, we can expand each term in the sum:

(k + 16)(k + 18) = k^2 + 34k + 288

So now we have:

sn = ∑(k^2 + 34k + 288) (from k = 1 to n)
  = ∑k^2 + 34∑k + 288n (from k = 1 to n)
  = n(n + 1)(2n + 1)/6 + 34n(n + 1)/2 + 288n
  = 306n^2 + 35n + 306

Now we can take the limit of sn as n approaches infinity to find S:

S = lim[n → ∞] sn
 = lim[n → ∞] (306n^2 + 35n + 306)
 = ∞

So unfortunately, the series (n + 16)(n + 18) diverges to infinity and does not have a finite sum.

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The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample.

To create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out in the opening, we can follow these steps:

The probability of the mouse finding its way out before reaching a dead end or going out in the opening can be estimated by dividing the number of successful outcomes by the total number of trials in the sample. This simulation approach can help us understand the probability of success in a random maze environment, and also explore the impact of various factors such as maze complexity, size and starting position of the mouse on the outcome.

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The exact volume of the composite figure with a cylinder of height and diameter 3.5 feet and a hemisphere on top is 49.92 cubic feet.

To find the volume of the composite figure, we need to add the volumes of the cylinder and the hemisphere on top of it.

The formula for the volume of a cylinder is:

V_cylinder = π[tex]r^2[/tex]h

where r is the radius of the cylinder and h is its height.

The formula for the volume of a hemisphere is:

V_hemisphere = (2/3)π[tex]r^3[/tex]

where r is the radius of the hemisphere.

In this case, the diameter of the cylinder is given as 3.5 feet, so the radius is half of that, or 1.75 feet. The height of the cylinder is also given as 3.5 feet. Therefore, the volume of the cylinder is:

V_cylinder = π(1.75[tex])^2[/tex](3.5) ≈ 32.67 cubic feet

To find the volume of the hemisphere, we need to first find its radius. Since the diameter of the cylinder is also the diameter of the hemisphere, the radius of the hemisphere is also 1.75 feet. Therefore, the volume of the hemisphere is:

V_hemisphere = (2/3)π(1.75[tex])^3[/tex] ≈ 17.25 cubic feet

Finally, we add the volumes of the cylinder and hemisphere to get the total volume of the composite figure:

V_total = V_cylinder + V_hemisphere

≈ 32.67 + 17.25

= 49.92 cubic feet

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The probability model for choosing a marble from the box is P(green) = 36/50 and P(blue) = 14/50.

To create this probability model, first, count the total number of marbles chosen, which is 50. Then, count the number of green and blue marbles chosen, which are 36 and 14, respectively.

Divide the number of each color by the total number of marbles to find the probability of choosing a green or blue marble.

P(green) is calculated as 36/50 or 0.72, and P(blue) is calculated as 14/50 or 0.28. This model represents the likelihood of choosing a green or blue marble from the box based on Yosef's experiment.

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The sum of the arrivals and departures from these three cities is 3x + 18.1 million.

If we let x be the number of arrivals and departures at Frankfurt airport, then the number of arrivals and departures at London airport is x + 12.9 million

The number of arrivals and departures at Paris airport is x + 5.2 million.

The sum of the arrivals and departures from these three cities is:

x + (x + 12.9 million) + (x + 5.2 million)

Simplifying this expression, we can combine like terms:

3x + 18.1 million

Therefore, the sum of the arrivals and departures from these three cities is 3x + 18.1 million.

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The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

We have to given that;

A satellite orbiting Earth travels 4.95×10⁷ meters for each orbit.

And, It takes 6,600 seconds to make one orbit.

Hence, the speed of the satellite in meters per second is,

⇒ Speed = Distance / Time

⇒ Speed = 4.95×10⁷/ 6,600

⇒ Speed = 0.00075 x 10⁷

⇒ Speed = 7.5 x 10⁻⁴ x 10⁷

⇒ Speed = 7.5 x 10³ m /sec

Thus, The value of the speed of the satellite in meters per second is,

⇒ Speed = 7.5 x 10³ m /sec

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Answer: That made no sense

Step-by-step explanation:

The direction of the vector is (-5, -8).

To find the direction in which the function is increasing most rapidly at point P(2, -1),

we need to find the gradient vector of the function at that point.

The gradient vector of the function f(x, y) = xy^2 - yx^2 is given by:

∇f(x, y) = ( ∂f/∂x , ∂f/∂y ) = ( y^2 - 2xy , 2xy - x^2 )

So, at point P(2, -1), we have:

∇f(2, -1) = ( (-1)^2 - 2(2)(-1) , 2(2)(-1) - 2^2 ) = (-5, -8)

The direction of greatest increase is in the direction of the gradient vector.

So, the direction in which the function is increasing most rapidly at point P(2, -1) is in the direction of the vector (-5, -8).

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The value of the given expression is 44. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four basic operations, also referred to as "arithmetic operations," are thought to explain all real numbers. Operations like division, multiplication, addition, and subtraction come before operations like quotient, product, sum, and difference in mathematics.

We are given an expression as [tex]\frac{1}{2}[/tex] (7) (4) + 6 (5).

We know that when there is no sign in between two numbers, it denotes multiplication.

So, we get

⇒  [tex]\frac{1}{2}[/tex] * (7)  * (4) + 6 * (5)

⇒ 14 + 30

⇒ 44                                            (Using addition operation)

Hence, the value of the given expression is 44.

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