If R is the unbounded region between the graph of [tex]y=\frac{1}{x(ln(x))^2}[/tex] and the x-axis for [tex]x\geq 3[/tex] then what is the area of R?
will give brainliest to answer with good explanation please i'm desperate

The work done to lift a 90 kg weight by 75 cm is approximately 661.5 joules.

Para calcular el trabajo realizado al elevar una pesa de 90 kg a una altura de 75 cm sobre el piso, necesitamos conocer la fuerza necesaria para levantar la pesa y la distancia que recorre la pesa para llegar a su altura máxima.

El trabajo se define como la energía necesaria para mover un objeto a través de una distancia determinada, y se calcula como el producto de la fuerza aplicada y la distancia recorrida.

En este caso, la fuerza necesaria para levantar la pesa es igual a su peso, que se calcula como su masa multiplicada por la aceleración debido a la gravedad (9.8 m/s²):

peso = masa x gravedad

    = 90 kg x 9.8 m/s²

    = 882 N

La distancia que recorre la pesa para elevarse 75 cm (o 0.75 m) es:

distancia = 0.75 m

Por lo tanto, el trabajo realizado al elevar la pesa es:

trabajo = fuerza x distancia

       = 882 N x 0.75 m

       = 661.5 J

Por lo tanto, se requiere un trabajo de 661.5 J para elevar una pesa de 90 kg a una altura de 75 cm sobre el piso.

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The red car is actually traveling along the road at a speed of approximately 26.67 feet per second.

We can start by drawing a diagram of the situation:

     P (police car)

      |

      |

      |

40    |    S (stop sign)

-------|--------------------

      | 160

      |    R (red car)

Let's use the Pythagorean theorem to find the distance between the police car and the red car at any time t:

d(t)² = 40² + (160 - v*t)²

Where v is the speed of the red car in feet per second, and d(t) is the distance between the police car and the red car at time t.

We want to find how fast the red car is actually traveling along the road, so we need to find v when the distance between the police car and the red car is decreasing at a rate of 100 feet per second:

d'(t) = -100

We can take the derivative of the equation for d(t) with respect to time:

2d(t)d'(t) = 0 + 2(160 - v*t)(-v)

Simplifying and plugging in d'(t) = -100, we get:

-4000 + 2v²t = -100(160 - vt)

Solving for v, we get:

v = 80/3 ≈ 26.67 feet per second

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Therefore, the solutions of tan x = -1/2 in the interval (0, 2) are:

x ≈ 2.034, 5.176

We can simplify the given equation as follows:

4 tan 2x - 4 cot x = 0

4(tan 2x - cot x) = 0

4[(2tan x)/(1 - tan^2 x) - (1)/(tan x)] = 0

Multiplying both sides by (1 - tan^2 x) * (tan x), we get:

8tan^3 x - 4tan^2 x - 8tan x + 4 = 0

Dividing both sides by 4 and rearranging, we get:

2tan^3 x - tan^2 x - 2tan x + 1 = 0

Factorizing, we get:

(tan x - 1)(2tan^2 x - tan x - 1) = 0

Using the quadratic formula to solve for the roots of 2tan^2 x - tan x - 1 = 0, we get:

tan x = [1 ± sqrt(1 + 8)] / 4 = [1 ± sqrt(9)] / 4 = 1, -1/2

Therefore, the solutions of the given equation in the interval (0, 2) are the values of x such that tan x = 1 or tan x = -1/2.

We know that tan (π/4) = 1 and tan (-π/4) = -1, so the solutions of tan x = 1 in the interval (0, 2) are:

x = π/4, 5π/4

We can find the solutions of tan x = -1/2 in the interval (0, 2) by finding the reference angle and using the signs of sine and cosine in the corresponding quadrants. We have:

tan x = -1/2

Let θ be the reference angle such that tan θ = 1/2. We know that θ is in the second or fourth quadrant.

In the second quadrant, sine is positive and cosine is negative, so we have:

sin θ = sqrt(1/(1 + tan^2 θ)) = sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = π - θ = π + arctan(1/2) ≈ 2.034

In the fourth quadrant, both sine and cosine are negative, so we have:

sin θ = -sqrt(1/(1 + tan^2 θ)) = -sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = 2π - θ = 2π + arctan(1/2) ≈ 5.176

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In the event of describing the likelihood that the team rebounds the next missed shot is likely, and the number of rebounds that the team should expect to have missed in 15  shots is 10.5 rebounds.

Given

Number of shots missed by the given team is 7

Total number of shots fired is 10

a) Then, moving on to the first part of the question

Here we have to apply probability to evaluate the likelihood of the given team rebounds the next missed shot.

Then,

Probability = no of shots attended / total number of shots fired

Probability = 7 /10

Then the event is likely

b) Now the second part

Then the number of rebounds the given team expect to have in the next 15 missed shots

= 7/10 ×15

= 105/10

= 10.5 rebounds

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Answer:

pls mrk me brainliest (⁠・⁠(⁠ｪ⁠)⁠・⁠）

A sample size of 40 attorneys is needed to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

To find the sample size, we use the formula:

n = (z*σ/E)²

where n is the sample size, z is the z-score for the desired level of confidence given as 98% , σ is the population standard deviation given as 15, and E is the margin of error given as 3 IQ points.

using the above values, we get:

n = (2.33*15/3)²

n = 39.05

Therefore, we need a sample size of at least 40 attorneys to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

Whether this sample size is practical or not depends on various factors, such as the availability of attorneys with the desired characteristics, the cost and time required to collect the data, and the resources available for analysis. In general, a sample size of 40 is considered moderate to large for many applications, and it may be feasible depending on the specific context.

A sample size of 40 attorneys is needed to estimate the mean IQ with 98% confidence and a margin of error of 3 IQ points.

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Answer:

100

Step-by-step explanation:

8000/80=100

8 thousands= 8000

8 tens= 80

Expert Answer, Mark AS BRAINLIEST!!!

Recall that the exponential growth function is defined as f(x) = [tex]a^x[/tex], where a is a positive constant greater than 1. Since any real number can be plugged in for x, the domain of the function is all real numbers.

Since the exponential growth function increases without bound as x goes to infinity, the range of the function is all positive real numbers (y > 0). Similarly, as x approaches negative infinity, the function approaches zero but never equals zero. Therefore, the range of the function does not include zero or any negative numbers (y > 0).

So, the complete answer is:

Domain: all real numbers; Range: y > 0.

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There are 1140 different predictions possible for this game of chance.

In this scenario, customers have an opportunity to predict three numbers out of 20, which will be drawn from a bin. The order in which the numbers are selected does not matter, which means the same set of numbers in different orders will be considered as the same prediction.

To solve this problem, we can use the formula for combinations, which is

=> [tex]^nC_x = \frac{n!}{ x! \times (n-x)!}[/tex]

where n is the total number of items, and x is the number of items to be selected.

In this case, we have 20 balls, and we want to select three balls without replacement. So, the formula becomes

=> [tex]^{20}C_3 = \frac{20!} { 3! \times (20-3)!}[/tex]

Using a calculator or simplifying the equation, we get:

[tex]= > ^{20}C_3 = \frac{201918} { 321} = 1140[/tex]

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Answer: 10

Step-by-step explanation:

just trust me

It's actually asking you to find the angles within the circles and match it with the angles it's supposed to be. For example if angle COE is 90° (im taking a fake angle value), you should put arc CE <——> 90

If you are still confused and need me to do it so that you can understand what I mean, reply and I'll help you!

Answer:

See below

Step-by-step explanation:

The objective of the question has been well explained by user vaishub1101.

I am just adding an additional hint and one answer to get you going

Since you just needed one example, I am providing just that
One thing to note in the figure is that segments DEF, ACD and ABF are all tangents to the circle. This fact is important since at the point of tangency (where the tangent touches the circle), the tangent to a circle is always perpendicular to the radius.

Using this knowledge and the given angles we can compute all the other angles but not the arc length [tex]\frown \atop {CE}[/tex] since to find arc length we need the value of the radius

As an example to help you get going,
[tex]\angle{DFA} \longleftrightarrow 58^\circ[/tex]

You would drag the tile with ∠DFA to the top left box and the tile with 58° to the top right box

I am sure you can figure out the rest or else user vaishub1101 can help you out with the rest

The lowest price she can charge per gift box to earn $2,000 in profit in her first month is approximately $3.2.

Jill is starting her own business called Fuzzy Socks Box, where she'll knit and sell gift boxes of fuzzy socks online.

Jill's profit per gift box is x - 2 dollars. To earn $2,000 in profit in her first month, she needs to sell

2000 / (x - 2) gift boxes.

According to her survey, the number of gift boxes she'll sell is

-10x + 350

Setting these two expressions equal to each other, we can find the lowest price Jill can charge per gift box to earn $2,000 in profit

-10x + 350 = 2000 / (x - 2)

Multiplying both sides by (x - 2), we get

-10[tex]x^{2}[/tex] + 12x - 470 = 0

Solving this quadratic equation, we find that

x = (6 +[tex]\sqrt{196[/tex] )/5 ≈ 3.2

or

x = (6 -[tex]\sqrt{196[/tex] )/5 ≈ 0.7

Since Jill cannot charge a negative price for her gift boxes, the lowest price she can charge per gift box to earn $2,000 in profit in her first month is approximately $3.2.

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a. The expected proportion of significant errors per year at this hospital is 0.2%. b. We can expect the proportion of significant errors per year at this hospital to fall within the range of 0.15% to 0.25%.

a. The expected proportion of errors per year at this hospital can be calculated as follows.

Number of medications prescribed and dispensed per year = 60,000

Proportion of medications in error = 1/100 = 0.01

Expected number of medications in error per year = 60,000 x 0.01 = 600

Therefore, the expected proportion of errors per year at this hospital is 600/60,000 = 0.01 or 1%.

To calculate the expected proportion of significant errors per year, we need to know the proportion of errors that result in significant problems for the patient. From the given information, we know that 1 in 500 errors resulted in significant problems. Therefore, the proportion of significant errors is 1/500 = 0.002.

Expected number of significant errors per year = 60,000 x 0.002 = 120

Therefore, the expected proportion of significant errors per year at this hospital is 120/60,000 = 0.002 or 0.2%.

b. To calculate the 2-σ interval for the proportion of significant errors per year, we need to use the formula:

2-σ interval = expected proportion ± 2 x standard error

The standard error can be calculated as follows:

Standard error = sqrt(p(1-p)/n)

where p is the expected proportion of significant errors (0.002) and n is the number of medications prescribed and dispensed per year (60,000)

Standard error = sqrt(0.002 x 0.998/60,000) = 0.000246

Substituting the values in the formula, we get:

2-σ interval = 0.002 ± 2 x 0.000246

2-σ interval = 0.001509 to 0.002491 (rounded to 5 decimal places)

Therefore, we can expect the proportion of significant errors per year fall within the range of 0.001509 to 0.002491 or 0.15% to 0.25%.

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If regular hexagon, regular octagon are inscribed in circle, perimeter of hexagon is greater than perimeter of octagon, each perimeter is greater than circumference of circle. Therefore, statement B is true.

The perimeter of a polygon is the sum of the lengths of all its sides. In a regular polygon, all sides have equal length, so the perimeter is simply the number of sides multiplied by the length of one side. The circumference of a circle is the distance around its outer edge.

Since both polygons are inscribed in the same circle, they have the same circumcircle, which means that their perimeters are both less than the circumference of the circle.

To compare the perimeters of the two polygons, we need to know the number of sides of each polygon and the length of one side. A regular hexagon has six sides, and a regular octagon has eight sides. Since the circle is inscribed in both polygons, the sides of each polygon are tangents to the circle, forming right angles with the radii of the circle.

Thus, we can draw a right triangle with the radius of the circle as the hypotenuse, and the side of the hexagon (or octagon) as one leg. Using trigonometry, we can find the length of one side of the hexagon (or octagon) in terms of the radius of the circle.

After calculating the lengths of one side of each polygon, we can compare their perimeters. It turns out that the perimeter of the octagon is greater than the perimeter of the hexagon, since the octagon has more sides.

Therefore, statement B is true.

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Complete question is:

A regular hexagon and a regular octagon are both inscribed in the same circle. which of these statements is true?

A) the perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle.

B) the perimeter of the hexagon is greater than the perimeter of the octagon, and each perimeter is greater than the circumference of the circle.

Answer:

n = 10

Step-by-step explanation:

36 can be written as 6^2 because 6*6 = 36.

Since exponents multiply, 36^5 = (6^2)^5 = 6^(2*5) = 6^10 = 6^n.

n = 10.

Alternatively, 36^5 = 36 * 36 * 36 * 36 * 36.

If you replace each 36 with 6*6, the new equation is (6*6) * (6*6) * (6*6) * (6*6) * (6*6).

n = the number of 6's = 10.

After calculating the distance, Ranjan has driven 358.55 miles so far.

To solve this problem, we need to use the formula:

distance = fuel efficiency x fuel consumed

Let's start by calculating the total fuel consumed. At the first stop, Ranjan adds 6.7 gallons of gas, which means he consumed 6.7 gallons of gas since his tank was full at the beginning of the trip. At the second stop, he adds 3.4 gallons of gas, which means he consumed 3.4 gallons of gas between the first and second stops. Therefore, the total fuel consumed is:

6.7 + 3.4 = 10.1 gallons

Now we can calculate the distance driven using the fuel efficiency of 35.5 miles per gallon:

distance = 35.5 miles/gallon x 10.1 gallons = 358.55 miles

Therefore, Ranjan has driven 358.55 miles so far.

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The area of the 8-sided shape is 27.5 square centimeters. having a width of 0.875cm and a length of 7.875cm.

Let us assume that Wdth of rectangle= w

Length of rectangle = Width + 7cm

The perimeter of a rectangle is calculated by using the formula,

The perimeter of rectangle = 2(w + w + 7) = 4w + 14

It is given that there are 4 rectangles, which means there are 8 sides. Therefore, the Total perimeter of the area is 4 times that single rectangle.

Total perimeter = 4(4w + 14) = 16w + 56

16w + 56 = 70

16w = 14

w = 0.875

The width of the rectangle = 0.875cm

Length of rectangle = 7cm + 0.875cm =  7.875cm

The area of a triangle is calculated as:

The area of triangle = (1/2) x base x height

The area of triangle = (1/2) x 0.875 x 7.875 = 3.4375

The area of the triangle for an 8-sided shape = 8 x 3.4375 = 27.5 [tex]cm^2[/tex]

Therefore we can conclude that the area of the 8-sided shape is approximately 27.5    [tex]cm^{2}[/tex]

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Based on the survey results from Maryland households, approximately 62 percent (641/1,035) of households in Maryland purchase at least one prepared meal per week from the grocery store.

To determine if the national grocery chain should expand their selection of prepared meals, we need to compare the nationwide percentage of households that purchase at least one prepared meal per week (67%) with the percentage of Maryland households that do the same.

Here's a step-by-step explanation:

1. Calculate the percentage of Maryland households that purchase at least one prepared meal per week by dividing the number of households that do (641) by the total number of households surveyed (1,035).

Percentage of Maryland households = (641 / 1,035) * 100= 62%

2. Compare the percentage of Maryland households with the nationwide percentage (67%).

Based on the survey results from Maryland households, approximately 62 percent (641/1,035) of households in Maryland purchase at least one prepared meal per week from the grocery store.

This is slightly lower than the national estimate of 67 percent. However, it is still a significant portion of households and suggests that expanding the selection of prepared meals could be a viable option for the national grocery chain in Maryland.

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The dimensions of the package of largest volume are 18 inches by 18 inches by 36 inches. The largest possible volume is 11664 cubic inches.

To find the dimensions of the package of largest volume. Let the dimensions of the square end be x, and the length of the rectangular end be y. The girth of the package is 4x, and the length is y. According to the problem statement, the girth plus length may not exceed 108 inches, so we have:

We want to maximize the volume V(x,y) of the package, which is given by:

We can use the equation 4x + y = 108 to express y in terms of x:

Substituting this into the formula for V(x,y), we get:

The domain of V(x) is determined by the constraints that x and y must be positive and the girth plus length may not exceed 10 inches. Since the girth is 4x, we have:

Solving for x, we get:

Since x must be positive, the domain of V(x) is:

The maximum volume and the optimal dimensions

To find the absolute maximum of V(x) on the domain 0 < x ≤ 32/3, we take the derivative of V(x) with respect to x and set it equal to zero:

[tex]V'(x) = 216x - 12x^2 = 0[/tex]

Solving for x, we get:

To confirm that this is a maximum, we take the second derivative of V(x) with respect to x:

At x = 18, we have V''(18) = 0, which means that the second derivative test is inconclusive. However, we can see that V(x) is increasing on the interval 0 < x < 18 and decreasing on the interval 18 < x ≤ 32/3, which means that x = 18 is indeed the absolute maximum of V(x) on the domain.

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To cover a rectangular lawn of 160 feet by 160 feet with dichondra lawn food, you would need 210 pounds of fertilizer.

To find the area of the rectangular lawn

Area = Length x Width

Area = 160 ft x 160 ft

Area = 25,600 sq ft

Since one bag of lawn food can cover 4000 square feet, we need to divide the total area of the lawn by the coverage of one bag

Number of bags = Total area ÷ Coverage of one bag

Number of bags = 25,600 sq ft ÷ 4000 sq ft

Number of bags = 6.4

Since we cannot buy a fraction of a bag, we need to round up to the nearest whole number of bags, which is 7.

Therefore, we need 7 bags of lawn food to cover the rectangular lawn. To find the total weight of fertilizer needed, we multiply the number of bags by the weight of one bag

Total weight of fertilizer = Number of bags x Weight of one bag

Total weight of fertilizer = 7 bags x 30 pounds/bag

Total weight of fertilizer = 210 pounds

Thus, we need 210 pounds of fertilizer to cover the rectangular lawn.

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The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.

We are given that the radius is increasing at a rate of 5 mm per second. This means that the rate of change of the radius with respect to time is dr/dt = 5 mm/s.

To find the rate at which the area is changing, we need to find dA/dt, the derivative of the area with respect to time. We can use the chain rule to find this derivative:

dA/dt = dA/dr * dr/dt

We can find dA/dr by taking the derivative of the area formula with respect to r:

dA/dr = 2πr

Now we can substitute the values we know into the chain rule formula:

dA/dt = dA/dr * dr/dt = 2πr * 5

When the radius is 28 mm, the rate of change of the area is:

dA/dt = 2π(28) * 5 = 280π ≈ 879.64 mm^2/s

Therefore, the area is changing at a rate of approximately 879.64 mm^2/s when the radius is 28 mm.

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We know that the measure of AB is 240°.

Given that AC is a diameter of D and MBDC = 150°, we know that angle ABC is a right angle (90°) because it is inscribed in a semicircle.

Using the fact that the sum of angles in a triangle is 180°, we can find angle MBC:

MBC + 150° + 90° = 180°
MBC  = -60°

Since angle MBC is negative, we know that it must actually be 360° - 60° = 300°.

Finally, using the fact that angles on a straight line add up to 180°, we can find angle ABM:

ABM + MBC = 180°
ABM + 300° = 180°
ABM = -120°

Again, since angle ABM  is negative, we know that it must actually be 360° - 120° = 240°.

Therefore, the measure of AB is 240°.

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The use of instant replay in professional soccer has been a controversial issue, as it raises questions about the balance between maintaining the flow of the game and ensuring accurate officiating.

Supporters of instant replay, often referred to as Video Assistant Referee (VAR), argue that it helps referees make better decisions, leading to fairer outcomes. By reviewing footage of crucial incidents such as goals, penalties, and red card situations, VAR can help correct errors that could have a significant impact on the final result.

On the other hand, critics of instant replay believe that it disrupts the natural flow of the game and can cause confusion among players, fans, and officials. Soccer is a fast-paced sport, and many argue that interruptions for video reviews can negatively affect the momentum and excitement of the game. Additionally, the technology can still be subjective, as referees have the final say in interpreting the video footage. This can lead to inconsistencies in decision-making, further fueling controversy.

Another concern is that the use of instant replay may undermine the authority of on-field referees. If their decisions are consistently questioned and overturned, their credibility may be damaged. Furthermore, not all soccer leagues and tournaments have the resources to implement VAR, which could lead to disparities between competitions.

In conclusion, while instant replay in professional soccer can contribute to fairer and more accurate officiating, it also has the potential to disrupt the game's flow, create confusion, and challenge the authority of referees. As the debate continues, it is important for soccer's governing bodies to carefully consider the implications of VAR and strive for a solution that upholds the integrity of the sport while minimizing its negative impacts.

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The approximate area of the figure is 40 square meters. So, the correct answer is B).

Recall the formula for the area of a kite, which is

Area = (1/2) x Base x Height

where "Base" is the length of one of the diagonals and "Height" is the length of the other diagonal.

Identify the base and height of the given kite from the problem statement. Here, it is given that the height is 10 meters and the base is 8 meters.

Substitute the values of the base and height into the formula for the area of a kite

Area = (1/2) x 8 meters x 10 meters

Simplify the expression by multiplying the base and height together and dividing by 2

Area = 40 square meters

Round the answer to the nearest whole number or keep the answer as a decimal, depending on the instructions of the problem.

Therefore, the approximate area of the given kite is 40 square meters. So, the correct answer is B).

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IF the median height is exactly 68 inches, then exactly 50% of the heights are at least 68 inches.

Without seeing the box-and-whisker plot, it is difficult to determine the exact fraction of heights that are at least 68 inches. However, we can make an estimate based on the general characteristics of a box-and-whisker plot.

In a box-and-whisker plot, the "box" represents the middle 50% of the data, with the median (50th percentile) marked by a line inside the box. The "whiskers" represent the minimum and maximum values within 1.5 times the interquartile range (IQR) of the data. Any points outside the whiskers are considered outliers.

Assuming that there are no outliers in the data, we can estimate that at least 50% of the heights are above the median, which is marked by the line inside the box. If the median height is at least 68 inches, then at least 50% of the heights are at least 68 inches.

If we assume that the median height is exactly 68 inches, then exactly 50% of the heights are at least 68 inches.

If the median height is less than 68 inches, then we can estimate that slightly less than 50% of the heights are at least 68 inches.

In summary, without more information about the box-and-whisker plot or the data it represents, we can estimate that at least 50% of the heights are at least 68 inches.

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If 75 people chose tulips as their favourite flower then Yelina surveyed 500 people.

What is surveying?

Surveying is a branch of mathematics that deals with the measurement, analysis, and representation of physical features on land or in space. It uses geometry, trigonometry, and infinitesimal calculus to accurately measure distances, angles, elevations, and positions. Surveying is used in various fields such as civil engineering, construction, mapping, and geology to obtain valuable information about the earth's surface and to create maps and plans for various purposes.

According to the given information

Since the central angle for tulips is 54°, we can set up a proportion:

54/360 = 75/x

Simplifying this proportion, we get:

0.15 = 75/x

Multiplying both sides by x, we get:

x * 0.15 = 75

Dividing both sides by 0.15, we get:

x = 500

Therefore, Yelina surveyed 500 people.

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Answer:

134° + (2x)° = 180°

(2x)° = 46°

x = 23

So angles 1, 2, 3, and 4 all measure 23°.

The demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

The demand equation is given as:

q = 55.1 - 0.023p

where q is the quantity demanded and p is the price of the airfare.

This equation shows an inverse relationship between price and quantity demanded. As the price of the airfare increases, the quantity demanded decreases, and vice versa.

For example, if the airfare price is $100, we can calculate the quantity demanded as:

q = 55.1 - 0.023(100) = 52.8

This means that at a price of $100, the quantity demanded is approximately 52.8 units.

Similarly, if the airfare price is $200, we can calculate the quantity demanded as:

q = 55.1 - 0.023(200) = 50.4

This means that at a price of $200, the quantity demanded is approximately 50.4 units.

So, demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

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To find the number of minutes it will take for both Aquarium A and Aquarium B to contain the same amount of water, we can set up an equation using the given information.

Aquarium A starts with 6 gallons and is filled at 2 gallons per minute. The equation for Aquarium A will be:

A = 6 + 2t

Aquarium B starts with 54 gallons and is filled at 1 gallon per minute. The equation for Aquarium B will be:

B = 54 + 1t

We want to find the time 't' when the amount of water in both aquariums is equal, so we can set the equations equal to each other:

6 + 2t = 54 + 1t

Now, solve for 't':
2t - 1t = 54 - 6
t = 48

After 48 minutes, both Aquarium A and Aquarium B will contain the same amount of water.

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This circle with center C will be circumscribed around triangle △DEF, touching each vertex. The translated and dilated ⊙O will coincide with ⊙P, proving that they are similar.

1. To construct a circle with center C circumscribed about triangle △DEF, first find the perpendicular bisectors of all three sides of the triangle. Next, locate the point where the three perpendicular bisectors intersect, which will be the center C of the circumscribed circle. Finally, draw the circle using the distance from point C to any vertex of the triangle as the radius.

2. To prove that ⊙O and ⊙P are similar using similarity transformations on ⊙O, follow these steps:
  a. Translate ⊙O by the vector <-2+12, 7-(-1)> or <10, 8> so that its center coincides with the center of ⊙P. The translated ⊙O will have the center (10, 15).
  b. Since the ratio of the radii of ⊙O and ⊙P is 5:12, perform a dilation of ⊙O with a scale factor of 12/5 and center (10, 15). The dilated ⊙O will have a radius of 12, which is equal to the radius of ⊙P.
 

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