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If D be a region bounded by a simple closed path C in the xy-plane, then the centroid of the quarter circle is  (4a/3π , 4a/3π)   .

In the question ,

it is given that D is the region bounded by a simple closed path C in the xy-plane .

the radius of the circle is = "a" ,

let the equation of the circle be x² + y² = a² ,

So , the area of the quarter circle is (A)  = (1/4)*πa²

The coordinate of the centroid x will be :

x = [tex]\frac{1}{2A} \int\limits x^{2} dy[/tex]

x = [tex]\frac{1}{2(\frac{\pi a^{2} }{4})} \int\limits^a_0 {a^{2} -y^{2} } \, dy[/tex]

Simplifying further ,

we get ,

x = 2/πa² [{a²(a) - a³/3} - 0 ]

x = 2/πa² [a³ - a³/3 ]

x = 4a/3π

The coordinate of the centroid y will be :

y = [tex]\frac{1}{2A} \int\limits y^{2} dx[/tex]

y = [tex]\frac{1}{2(\frac{\pi a^{2} }{4})} \int\limits^a_0 {a^{2} -x^{2} } \, dx[/tex]

Simplifying further ,

we get ,

y = 2/πa² [{a²(a) - a³/3} - 0 ]

y = 2/πa² [a³ - a³/3 ]

y = 4a/3π

Therefore , the coordinates of the centroid is (4a/3π , 4a/3π) .

The given question is incomplete , the complete question is

Let D be a region bounded by a simple closed path C in the xy-plane. The coordinates of the centroid x, y of D are x = [tex]\frac{1}{2A} \int\limits x^{2} dy[/tex]  ,  y = [tex]\frac{1}{2A} \int\limits y^{2} dx[/tex] ?  

Find the centroid of a quarter-circular region of radius a.

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The cosine of the angle 199° will be less than zero that is cos 199° < 0. Then the correct option is C.

Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.

The angle of 199° lies in the third quadrant.

In the third quadrant, the sine and cosine of the angle are negative and the tangent of an angle is positive.

The cosine of the point 199° will be under zero that is cos 199° < 0. Then, at that point, the right choice is C.

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The indicated side of the right angle triangle is 183.

A right triangle is a triangle that has one of it's angles as 90 degrees. A right triangle sides are named according to the position of the angles.

The sides name are as follows:

let's find the value of side h using the trigonometric ratios as follows

tan ∅ = opposite / adjacent

Therefore,

tan 18.8  = h / x + 423

cross multiply

(x + 423) tan 18.8 = h

h = x tan 18.8 + 423 tan 18.8

h = 0.34042776856x + 144.000946101

tan 57.8 = h / x

cross multiply

x tan 57.8 = h

h = 1.58797305139x

Therefore,

1.58797305139x = 0.34042776856x + 144.000946101

1.58797305139x - 0.34042776856x = 144.000946101

1.24755305139x = 144.000946101

x = 144.000946101 / 1.24755305139

x = 115.426674549

Therefore,

h =  1.58797305139(115.426674549)

h = 183.294448434

h = 183

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Step-by-step explanation: To fill in the table, we will need to first count the number of times each wait time appears in the given data. We can then use these counts to fill in the Frequency column of the table. We will also need to calculate the relative frequency of each wait time, which is the number of times it appears in the data divided by the total number of data points. We can then use these relative frequencies to fill in the Relative Frequency column of the table, expressed as a percentage.

Here is the completed table:

Wait Time (minutes) Frequency Relative Frequency (%)

5                                     5                 11.36

10                                 4                  9.09

15                                 7                  15.91

20                                6                   13.64

25                                7                   15.91

30                                 4                   9.09

35                                5                    11.36

40                                3                    6.82

45                                 2                    4.55

50                                 1                    2.27

60                                  2                    4.55

75                                  1                       2.27

Note that the total number of data points is 44, so each relative frequency is calculated as the corresponding frequency divided by 44.

I hope this helps! Let me know if you have any other questions.

It takes 2 hrs to fill the fountain with both sprinklers operating

Two or more expressions with an Equal sign is called as Equation.

Given,

A park has two sprinklers that are used to fill a fountain

One sprinkler can fill the fountain in 3 h

second sprinkler can fill the fountain in 6 h

We need to find the time required to fill the fountain by two sprinklers.

Let t = time required with both sprinklers working

t/3+t/6=1

2t+t/6=1

3t/6=1

Apply cross multiplication

3t=6

Divide both sides by 3

t=2

Hence, it takes 2 hrs to fill the fountain with both sprinklers operating

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Answer:

Step-by-step explanation:

14545

Make a basic proportion

16,000/x=110/100

Cross multiply

1,600,000=110x

Divide

x=14545

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{2}{h}~~,~~\underset{-3}{k})}\qquad \stackrel{radius}{\underset{4}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - 2 ~~ )^2 ~~ + ~~ ( ~~ y-(-3) ~~ )^2~~ = ~~4^2\implies {\large \begin{array}{llll} (x-2)^2+(y+3)^2=16 \end{array}}[/tex]

Answer:

136

Step-by-step explanation:

The formula of the volume of a cone is v=b*h.First you need to find the diameter, simple, 6*2=12, you get the 12 because the diameter is always twice as large as the radius. Next you multiply 18*12 which gives you 136!

The elements of the required sets would be:

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

A set, denoted by the curly brackets , is a grouping of items, numbers, elements or objects.

A set of numbers, for instance, is 1, 2, 3, 4.

Solution:

P' = {2, 4, 6, 8, 10}

Q' = {1, 4, 6, 8, 9, 10}

R' = {6, 7, 8, 9, 10}

P∩Q = {3, 5, 7}

R'∪P' = {2, 4, 6, 7, 8, 9, 10}

P∪R = {1, 2, 3, 4, 5, 7, 9}

P∩R = {1, 3, 5}

P∪Q∪R = {1, 2, 3, 4, 5, 7, 9}

P'∩Q'∩R' = S - P∪Q∪R

               = {6, 8, 10}

Now we can find the elements of the required sets:

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

Hence, the elements of the required sets would be

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

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The slope of the line passing through the points (10) 2, and (14), is -9/-13.

A line's steepness and direction are measured by the line's slope.

Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all

Any two different points on a line can be used to calculate the slope of any line.

A line's slope is determine by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively.

Two locations located on a straight line can be used to determine a line's slope. We can use the slope of line formula if we know the two spots' coordinates. These two points' coordinates should be-

P1 = (x1, y1) and P2 = (x2, y2)

Hence, The slope of the line passing through the points (10) 2, and (14), is -9/-13.

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The no. ways roll a 1 on the die and draw a 7 from the bag of tickets is  1 And the probaility is [tex]\frac{1}{42}[/tex]

Probability is the ratio of favorable outcomes to all other potential outcomes of an event. The symbol x can be used to signify how many positive findings there are in an experiment with 'n' results. The following formula can be used to determine the likelihood of an event:

Probability (event) equals a favorable result/the total of outcomes divided by n.

The no. ways roll a 1 on the die and draw a 7 from the bag of tickets is  1

And the probaility is -

[tex]\frac{1}{6} X \frac{1}{7}[/tex]

[tex]= \frac{1}{42}[/tex]

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When renting a car for $40.49 with a fixed mileage charge of $0.05, a businessperson can go 1590.2 miles.

An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles. Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression in mathematics is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) You can think of expressions as being comparable to phrases.

Here,

The cost of car rent=$40.49

Cost for per mile=$0.05

Let him travel x miles,

40.49+0.05x=120

0.05x=120-40.49

0.05x=79.51

x=79.51/0.05

x=1590.2 miles

The business person can travel 1590.2 miles when the rent of car cost $40.49 fixed and $0.05 for every mile travelled.

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Answer:

y = -100/13

Step-by-step explanation:

To solve this equation, we can first get all the terms containing the variable y on one side of the equation and all the constant terms on the other side. We can do this by subtracting 25 from both sides of the equation:

5/13y + 25 - 25 = 0 - 25

This gives us:

5/13y = -25

Next, we can divide both sides of the equation by 5/13 to get rid of the fraction:

(5/13y) / (5/13) = (-25) / (5/13)

This simplifies to:

y = -100/13

Therefore, the solution to the equation is y = -100/13.

On solving the provided question, we got the area of triangle  - A =  1/2 B X H= 15 sq. units.

A triangle is a shape formed when three straight lines meet. All triangles have three sides and three corners (angles). The point where two sides of a triangle meet is called a vertex. The base of a triangle can be any one of its three sides, but it is usually the bottom one. The longest side of a triangle is opposite its biggest angle, and the shortest side of a triangle is opposite its smallest angle. The three angles inside a triangle always add up to 180°.

To find the Area of triangle -

By help formula,

[tex]A = \frac{1}{2} B X H\\\\[/tex]

[tex]\\A = \frac{1}{2} (6 X 5)\\[/tex]

[tex]\\A = 15 sq.unit[/tex]

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Answer:

(2, 5 )

Step-by-step explanation:

5x - y = 5 → (1)

- x + 3y = 13 → (2)

multiplying (1) by 3 and adding to (2) will eliminate y

15x - 3y = 15 → (3)

add (2) and (3) term by term to eliminate y

14x + 0 = 28

14x = 28 ( divide both sides by 14 )

x = 2

substitute x = 2 into either of the 2 equations and solve for y

substituting into (2)

- 2 + 3y = 13 ( add 2 to both sides )

3y = 15 ( divide both sides by 3 )

y = 5

solution is (2, 5 )

Answer:

  9

Step-by-step explanation:

You want to know the least number of visits Jonathan must make to a movie theater to have at least 135 points if he has 25 points and earns 13.5 for each visit.

After x visits to the theater, Jonathan will have earned 13.5x points in addition to the signing bonus. His total will be ...

  13.5x +25

Jonathan wants that total to be at least 135:

  13.5x +25 ≥ 135

  13.5x ≥ 110 . . . . . . . subtract 25

  x ≥ 110/13.5 ≈ 8.148

The smallest integer greater than 8.148 is 9.

Jonathan needs to make at least 9 visits to earn a free movie ticket.

<95141404393>

Total profit - Total cost = Profit

The profit made is $1078.

Option C is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 7 = 8 is an equation.

We have,

The total profit, in dollars per day, after x days can be modeled by the function

R(x) = 3x³ – 21x² + 21x + 45.

The total cost, in dollars per day, after x days, can be modeled by the function

C(x) = x² – 2x – 3.

Now,

(R - C) (10):

= R(10) - C(10)

= 3x³ – 21x² + 21x + 45 - x² + 2x + 3

= 3x³ - 22X² + 23x + 48

= 3 x 1000 - 22 x 100 + 230 + 48

= 3000 - 2200 + 230 + 48

= $1078

Thus,

Total profit - Total cost = Profit

The profit made is $1078.

Option C is the correct answer.

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The shell from the fireworks will reach the water after 0.75 seconds or -0.625 seconds. Since the time cannot be negative, the shell will reach the water after 0.75 seconds.

To find the time it takes for the shell from the fireworks to reach the water, we need to solve the equation h = 0 for t.

The given equation for h is h = -16t^2 + 64t + 36, so we can substitute this into the equation h = 0 to get:

0 = -16t^2 + 64t + 36

To solve this equation, we can use the quadratic formula, which is:

t = (-b +/- sqrt(b^2 - 4ac)) / (2a)

In this case, a = -16, b = 64, and c = 36, so the formula becomes:

t = (-64 +/- sqrt(64^2 - 4(-16)(36))) / (-32)

Simplifying this gives us:

t = (-64 +/- sqrt(4096 - 2304)) / (-32)

which simplifies to:

t = (-64 +/- sqrt(1792)) / (-32)

Finally, simplifying gives us:

t = (-64 +/- 44) / (-32)

t = (-20 or 24) / (-32)

t = -0.625 or 0.75

Thus, the shell from the fireworks will reach the water after 0.75 seconds or -0.625 seconds. Since the time cannot be negative, the shell will reach the water after 0.75 seconds.

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Answer:

They are all prime numbers

Explanation:

17, 19, 46, and 55 are all prime numbers. A prime number is a whole number greater than 1 that is only divisible by 1 and itself. Since 17, 19, 46, and 55 are all prime numbers, they have no other factors besides 1 and themselves, which means they cannot be divided evenly by any other number.

Answer:

x = 10

Step-by-step explanation:

To find the value of x:

1) add up like-terms

4 + 2x+1x-4=30

4 + 3x - 4 = 30

2) Add up the number on the left side

0 + 3x = 30

3x = 30

3) isolate the x

3x/3 = 30/3

x = 10

Check the picture below.

Answer:

  D)  $27

Step-by-step explanation:

You want an estimate of a reasonable amount to pay for 10 gallons of gas at $229 per gallon, and 2 quarts of oil at $1.89 per quart.

For estimating purposes, we like to round numbers to 1 or 2 significant figures.

The cost of 10 gallons of gas is about 10 × $2.30 = $23.

The coast of 2 quarts of oil is about 2 × $2 = $4.

So, the total cost of the gas and oil is about $23 +4 = $27.

You expect the total to be near $27.

__

Additional comment

Part of estimating is estimating the error in your estimate. Here the error is $0.01 per gallon of gas, or $0.10 for 10 gallons. The error is $0.11 per quart of oil, so  $0.22 for 2 quarts. The final estimate of $27 is $0.10 +0.22 = $0.32 too high. $27 is different from the next answer choice by $2, so is still the best answer choice.

Answer:

[tex]\huge\boxed{\sf h = 130\ m}[/tex]

Step-by-step explanation:

Time = t = 5.2 s

Velocity = v = 25 m/s

Height = h = ?

h = vt

h = 25(5.2)

[tex]\rule[225]{225}{2}[/tex]

Answer:

The valley is 130 m deep.

Step-by-step explanation:

Formula we use,

→ Height = Time × Speed

Now the height will be,

→ Time × Speed

→ 5.2 × 25

→ 130 metres

Thus, valley is 130 m deep.

The length of the hypotenuse AC is √(h² + 4).

Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

We have,

Right triangle ACD.

AC² = CD² + AD²

AC² = h² + 2²

AC² = h² + 4

AC = √(h² + 4)

Thus,

AC is √(h² + 4).

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Answer: C: $7,000.00

Step-by-step explanation:

The credit limit which gives an acceptable debt ratio is $7000.

A credit limit is the most money you are permitted to spend on a credit card or line of credit by a lender. Knowing your limit, however, does not imply that going for it is a wise decision.

Given:

Owed Amount = $ 3450.0

So, Debt Ratio= (Total liabilities) ÷ (Total amount of worth possessed)

A debt ratio of approximately 0.5 is considered to be reasonable.

Money Owed by you = $ 3450.00

0.5 = 3450/credit limit

credit limit = 3450 × 10/5

credit limit = 6900

So, the Credit limit which gives Acceptable Debt of $ 3450.00 when debt ratio is approximately more or less than 0.5 is $7000.

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The equation that is quadratic in all the equations is 6(2x+4)²=(2x+4)+2

A quadratic equations defined as any equation that can be rearranged in standard form as ax²+bx+c=0 or -b±[tex]\frac{\sqrt{} b^{2}-4ac }{2a}[/tex]

From the list of equations the one that carries the standard form for a quadratic equation is

6(2x+4)²=(2x+4+)+2

Expand the brackets to have

6[(2x+4)(2x+4)=(2x+4)+2

6[4x²+8x+8x+16]=(2x+4)+2

Collecting like terms and opening the brackets further;

6[4x²+16x+16]=2x+4+2

=24x²+96x+96=2x+6

Collecting like terms we have

24x²+96x-2x+96-6=0

⇒24x²94x+90=0

In conclusion, the equation carries 2 as its highest power therefore, it is a quadratic equation.

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The value of angle a is 45, the value of angle b is 135, the value of angle c is 135 and d is 135.

Straight lines with little depth or width are present. You will learn about a number of lines, including transversal, intersecting, and perpendicular lines. A figure called an angle is one in which two rays originate from the same point. In this area, you might also encounter contrasting and related viewpoints.

Given that the line cutting the two parallel lines. The value of the angles will be calculated as,

Angle a will be,

a + 135 = 180

a = 180 - 135

a = 45

Angle b = 135

Angle c = 135

Angle  d = 135

Therefore, the value of angle a is 45, the value of angle b is 135, the value of angle c is 135 and d is 135.

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