Find the area inside the loop of the following limacon: r=7−14sinθ

Step-by-step explanation:

FCP fundamental counting principle :

in total there are

3×3×2 = 18

different combinations

listing method means listing all different triplets of such outfit combinations.

white shirt = ws

red blouse = rb

floral shirt = fs

short = sh

jeans = j

jogger pants = jp

shoes = s

sandals = sa

{

(ws, sh, s),

(ws, sh, sa),

(ws, j, s),

(ws, j, sa),

(ws, jp, s),

(ws, jp, sa),

(rb, sh, s),

(rb, sh, sa),

(rb, j, s),

(rb, j, sa),

(rb, jp, s),

(rb, jp, sa),

(fs, sh, s),

(fs, sh, sa),

(fs, j, s),

(fs, j, sa),

(fs, jp, s),

(fs, jp, sa)

}

tree diagram (I cannot draw here) :

OOTD

/ | \

ws rb fs

/ | \ / | \ / | \

sh j jp sh j jp sh j jp

/ | / | / | / | / \ / | / | / | / |

s sa s sa s sa s sa s sa s sa s sa s sa s sa

as you can see, we get a expected 18 different triplets in our listed set, and we get 18 different end leaves in and therefore 3-level paths through our tree.

For the given function,

Vertical asymptote = -5/4

Horizontal asymptote = -5/4

Domain =  (-∞,-5/4) ∪ (-5/4,∞)

Range =  (-∞,-5/4) ∪ (-5/4,∞)

What is a function?

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

The given function is

[tex]f(x) = \frac{-5x+2}{4x+5}[/tex]

The given function has a vertical asymptote at x = -5/4. This is because the function tends to ∞ as x tends to -5/4.

The horizontal asymptote of the function is when x tends to ∞ .

[tex]\lim_{x \to \infty} \frac{-5x+2}{4x+5} = \frac{-5+2/x}{4+5/x} = \frac{-5}{4}[/tex]

The horizontal asymptote of function is at y = -5/4.

Since at x = -5/4, the function becomes undefined.

The domain is all real numbers except x = -5/4

Domain =  (-∞,-5/4) ∪ (-5/4,∞)

Since the function is not defined at y = -5/4 as well, the range is also

(-∞,-5/4) ∪ (-5/4,∞) .

Therefore for the given function,

Vertical asymptote = -5/4

Horizontal asymptote = -5/4

Domain =  (-∞,-5/4) ∪ (-5/4,∞)

Range =  (-∞,-5/4) ∪ (-5/4,∞)

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The mixture of different shade of green that is bluer will be; ⇒ 2 cups of yellow and 4 cups of blue.

Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

Given that;

A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue.

now,

Since, A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue.

Hence, The mixture of different shade of green that is bluer is find as;

Let Cups of yellow = 2

And, Cups of blue = x

So, we can formulate;

⇒ 2/x < 2/3.5

⇒ x > 3.5

So, Assume x = 4

Hence, The mixture of different shade of green that is bluer will be;

⇒ 2 cups of yellow and 4 cups of blue.

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Answer:

Step-by-step explanation:

The Laplace inverse is not a standard function on most calculators. However, if you have a graphing calculator that supports Laplace transforms, you can use it to find the Laplace inverse by performing the inverse Laplace transform on a function.

Here's the general procedure for finding the Laplace inverse on a graphing calculator:

Enter the Laplace transform of the function you want to find the inverse of.

Select the inverse Laplace transform function from the calculator's menu.

Enter the desired complex variable, usually represented by "s" in Laplace transform notation.

The calculator will return the inverse Laplace transform of the function, which represents the original function in the time domain.

Note: The exact steps for finding the Laplace inverse on a graphing calculator will vary depending on the model and brand of the calculator. It's best to consult the manual or a reference guide for your specific calculator to determine how to perform the inverse Laplace transform.

The percent error of the measurement, to the nearest tenth of a percent, is 5.6.

A ratio or value that may be stated as a fraction of 100 is called a percentage. And it is represented by the symbol '%'.

Given:

John measured a line to be 1.7 inches long.

The actual length of the line is 1.8 inches.

The amount of error = 1.8 - 1.7

The amount of error = 0.1

Let 1 - n be the required percentage of error.

Then applying the percentage formula,

1.8 x n /100 = 1.7

Applying cross multiplication,

we get,

1.8n = 170

n = 170/1.8

n = 94.4

So, 1 - n = 100 - 94.4 = 5.6

Therefore, the percentage is 5.6.

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If A, B, and C are nxn invertible matrices, the equation   C^-1(A+X)B^-1=I(n) has a solution which is X=CB-A. Since invertible matrices are those whose inverse matrix exists.

For A to be the matrix  here A⁻¹ is the inverse of it, so AA⁻¹=I, here I is the identity matrix ,since the matrix equation given to us is: C⁻¹(A+X)B⁻¹=In, now multiplying B on the right side and  C on the left side we get :CC⁻¹(A+X)B⁻¹B=CIₙB

=>Iₙ(A+X)Iₙ=CIₙB

=>A+X=CB, according to the property  of the identity matrix.Now adding -A on both sides we get,

=>A+X+(-A)=CB+(-A)

=>X=CB-A, so here the solution for the given matrix equation is: X=CB-A

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The person with the largest container is Julie and the person with the smallest container is Jeff.

The volume of each container has to be determined.

Volume of a cylinder = πr²h

Where:

Volume of Ms. Liu's container:

3.14 x 3² x 12 = 339.12 in³

Volume of Ms. Julie's container:

Volume = 3.14 x 3² x 24 = 678.24 in³

Volume of Jeff's container:

Volume = 3.14 x 1² x 24 = 75.36 in³

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Answer:

8 cm

Step-by-step explanation:

15 - 7 = 8

The length of the right side equals the length of the left side.  The left side has a length of 15 and the part that we know for the right side is 7.

Helping in the name of Jesus.

Answer to question 1 is   [tex]\text{y} \le -\frac{2}{5}\text{x}+100[/tex]

=========================================================

Explanation:

The first task is to find the equation of the line through the given points in the table.

Pick any two points you want. I'll pick (0,100) and (50,80)

Determine the slope:

[tex](x_1,y_1) = (0,100) \text{ and } (x_2,y_2) = (50,80)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{80 - 100}{50 - 0}\\\\m = \frac{-20}{50}\\\\m = -\frac{2}{5}\\\\[/tex]

The slope is -2/5.

The y intercept is b = 100 because of the point (0,100)

We go from [tex]\text{y} = m\text{x}+b[/tex] to [tex]\text{y} = -\frac{2}{5}\text{x}+100[/tex]

Each point from the table is found on this line. For instance, let's check the point (140,44)

[tex]\text{y} = -\frac{2}{5}\text{x}+100\\\\44 = -\frac{2}{5}*140+100\\\\44 = -56+100\\\\44 = 44\\\\[/tex]

This confirms (140,44)

I'll let you check the other points.

------------------

Any point on the line mentioned represents a scenario where César spends all of the $50.

If he goes above the line, then he'll spend more than $50. If he goes below, he spends less than $50.

This means we want to shade the region below the boundary line [tex]\text{y} = -\frac{2}{5}\text{x}+100[/tex]

We'll replace the equal sign with a "less than or equal" sign

We arrive at [tex]\text{y} \le -\frac{2}{5}\text{x}+100[/tex] as the final answer to question 1.

Answer: Answer to question 1 is   =========================================================Explanation:The first task is to find the equation of the line through the given points in the table. Pick any two points you want. I'll pick (0,100) and (50,80) Determine the slope:The slope is -2/5.The y intercept is b = 100 because of the point (0,100)We go from  to Each point from the table is found on this line. For instance, let's check the point (140,44)This confirms (140,44)I'll let you check the other points.------------------Any point on the line mentioned represents a scenario where César spends all of the $50.If he goes above the line, then he'll spend more than $50. If he goes below, he spends less than $50.This means we want to shade the region below the boundary line We'll replace the equal sign with a "less than or equal" signWe arrive at  as the final answer to question 1.

Step-by-step explanation:

The measure of angle C in radians is 3.6 radians.

The arc length of a circle is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc.

Given that, the circle has a radius of 7 and angle C intercepts the arc with a length of 28π.

We know that, formula to find the arc length of circle is Arc length= θ/360 ×2πr

28π= θ/360×2π×7

4= θ/360×7

θ=1440/7

θ=205.7

θ=3.6 radians

Therefore, the measure of angle C in radians is 3.6 radians.

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The statement which is true is more males were surveyed than females, the correct option is A.

Suppose two dimensions are there, viz X and Y. Some values of X are there as X_1, X_2, ... , X_n and some values of Y are there as Y_1, Y_2, ... , Y_n

List them in title of the rows and left to the columns. There will be n \times k table of values will be formed(excluding titles and totals), such that:

Value(ith row, jth column) = Frequency for intersection of  X_i  and Y_j  (assuming X values are going in rows, and Y values are listed in columns).

Then totals for rows, columns, and whole table are written on bottom and right margin of the final table.

For n = 2, and k = 2, the table would look like:

[tex]\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&n(X_1 \cap Y_1)&n(X_1\cap Y_2)&n(X_1)\\X_2&n(X_2 \cap Y_1)&n(X_2 \cap Y_2)&n(X_2)\\\rm Total & n(Y_1) & n(Y_2) & S \end{array}[/tex]

where S denotes total of totals, also called total frequency.

n is showing the frequency of the bracketed quantity, and intersection sign in between is showing occurrence of both the categories together.

We are given that;

            Males  Females  Total

Soccer      |  31   |    50   |  81

Basketball  |  21   |    50   |  71

Total       |  52   |   100   | 152

Now,

A. More males were surveyed than females.

True. From the table, we can see that 52 males were surveyed, while only 100 females were surveyed.

B. More females like soccer.

False. From the table, we can see that 31 males and 50 females like soccer, so a total of 81 people like soccer. This is the same as the number of females surveyed, so we cannot say that more females like soccer.

C. An equal number of people like soccer and basketball.

False. From the table, we can see that 81 people like soccer, while 71 people like basketball.

D. Twice as many females than males like basketball.

False. From the table, we can see that 21 males and 50 females like basketball. Twice the number of females who like basketball is 100, which is the total number of females surveyed, but this is not twice the number of males who like basketball.

Therefore, by the two-way table the answer will be more males were surveyed than females.  

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a. Here's a table of values for the number of rounds and the number of teams participating:

Round Number of Teams

1             16

2              8

3                  4

4             2

5       1 (Champion)

.b. domain: [1 5]

Range: {1 16]

C. Rate of decay = 8

D. Decay factor = 0.5

b. The reasonable domain of this function would be the number of rounds, and the range would be the number of teams participating in each round.

The domain would be all positive integers from 1 to the maximum number of rounds,

The range would be all positive integers from 1 to the starting number of teams (16 in this case).

c. To find the rate of decay, we can calculate the ratio of the change in the number of teams to the change in the number of rounds.

Using Round 1 to Round 2,

= (16 - 8) / (2 - 1)

= 8.

d. The decay factor is the ratio of the number of teams in a given round to the number of teams in the previous round.

Using Round 1 to Round 2, the decay factor is

= 8 / 16

= 0.5

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The  function that models y in terms of x is y=0.804x

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

m=4.02-0/5-0

m=4.02/5 = 0.804

Now let us find y intercept

0=0.804(0)+b

b=0

Hence, the  function that models y in terms of x is y=0.804x

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To get Desmos to show both square roots, you can use the formula y = √x + √(-x).  Desmos can then graph this equation, showing both the positive and negative square roots.

To get Desmos to show both square roots, you can use the formula y = √x + √(-x).This formula can be used to graph two square roots. The first square root is √x, which is the positive square root of x. The second square root is √(-x), which is the negative square root of x. To calculate the roots, you would take the square root of both x and -x. For example, if x is 4, the two roots would be √4 = 2 and √(-4) = 2i, where i is the imaginary unit. Desmos can then graph this equation, showing both the positive and negative square roots.

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Answer: 0 or No answer

Step-by-step explanation:

2t-2t is 0 so there is no t and 5-5 is 0 so theres no number.

The required equivalent expression is 3 · z³ to the given expression 3/4 x ( z^3 x 4). which is the correct answer .

Algebraic expressions are mathematical statements with a minimum of two terms containing variables or numbers.

her, we have,

We have been given expression as:

3/4 x ( z^3 x 4)

To determine the equivalent expression to the given expression

3/4 x ( z^3 x 4)

⇒ 3* z^3

Here the variable z has an exponent is 3 which is positive,

Rearrange the terms of variables z  in the above expression

⇒3 · z³

Therefore, the required equivalent expression is 3 · z³ to the given expression 3/4 x ( z^3 x 4).

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There will be 360 different ways without using the same airline or shipping line.

A combination in mathematics is a selection of items from a set with distinct members, where the order of selection is irrelevant.

Let's first consider the number of ways the man can choose his outbound journey. He has 6 airlines to choose from, and once he has made his choice of airline, he has 4 shipping lines to choose from.

Therefore, there are 6 × 4 = 24 ways he can choose his outbound journey.

For his return journey, he cannot use the same airline or shipping line that he used for his outbound journey.

This means he now has 5 airlines and 3 shipping lines to choose from. Therefore, there are 5 × 3 = 15 ways he can choose his return journey.

The total number of ways he can make his journey is:

24 × 15 = 360

Therefore, he can make his journey in 360 different ways without using the same airline or shipping line.

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The graph of the equation is added as an attachment

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

y = 3x

The above expression is a linear equation

A linear equation is represented as

y = mx + c

Where

Slope = m and y-intercept = c

This means that

y = 3x has a slope of 3 and a y-intercept of 0

Next, we plot the graph

See attachment for the graph of the equation

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Answer:

x = 3

Step-by-step explanation:

5x + 1 = 2x + 10

1.) Subtract 2x from both sides to get 2x on the other side

5x - 2x + 1 = 10

2.) Subtract 1 from both sides to get 1 to the other side

5x - 2x = 10 - 1

3.) Combine like terms

3x = 9

4.) Divide 3 on each side to get x by itself

x = 3

Answer:

$1350.

Step-by-step explanation:

If Justin files a claim for $2100, he will pay the first $750 toward the claim, and his insurance company will pay the remaining costs.

So, Justin will pay $750.

And, the insurance company will pay $2100 - $750 = $1350.

Kelly has to pay a tax of $22,562.50 which is approximately around of 20% her taxable income.

The term "taxable income" describes the basis on which an income tax system levies tax. In other words, the amount of income that was taxed by the government.

In most cases, it covers some or all sources of income and is then diminished by costs and other deductions. Every gross income received and utilized to determine your tax liability is referred to as taxable income.

It can be stated simply as your adjusted gross income less any deductions. This covers all pay, gratuities, salaries, and bonuses received from employers. Investments and ill-gotten gains are also mentioned.

The given taxable income is over $43,650. So use the information on line 3 to determine the tax.

Tax =$5,975+25% of the amount over 43,650

=5,975+0.25(110,000 - 43,650)

=5.975+16587.50

=22,562.50

Kelly has to pay a tax of $22,562.50 which is approximately around of 20% her taxable income.

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Answer: D, 1.01

1.01 is in between 0.62 and 1.05

Answer: D-1.01

Step-by-step explanation:

The inequality that can be used to determine how many hours Matilda will need to babysit to buy the dress is 40 + 9x >= 112 , here the x needs to be greater than or equal to 8 i.e. x >=8.

Given, Matilda needs at least $112 to buy a new dress.

Money she has =$40

Money she earns for each hour of babysitting = $9

Let, x be the number of hours she will work to earn the minimum amount.

Therefore, the inequality that can be used to determine how many hours Matilda will need to babysit to buy the dress = 40 + 9x >= 112

                                                                               x >=8 hour

So, The inequality that can be used to determine how many hours Matilda will need to babysit to buy the dress is 40 + 9x >= 112 . She needs to work for at least 8 hours to earn the minimum required amount.

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The next value R in the sequence is 6250.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

2, 10, 50, 250, 1250, R

Now,

We see that the next number in the sequence is a multiple of 5 of the previous number.

i.e

2 x 5 = 10

10 x 5 = 50

50 x 5 = 250

250 x 5 = 1250

So,

R = 1250 x 5

R = 6250

Thus,

The next value R in the sequence is 6250.

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The completed two-column table used to prove that the quadrilateral WXYZ is a parallelogram is presented as follows;

Statement     [tex]{}[/tex]                                              Reason

1. m∠y = 49, m∠z = 131°, m∠w = 49  [tex]{}[/tex]       Given

2. m∠y + m∠z + m∠w + m∠x = 360 [tex]{}[/tex]    1. ∠ sum property of a quadrilateral

3. 49 + 131 + 49 + m∠x = 360[tex]{}[/tex]                Substitution Property of Equality

4. 229 + m∠x = 360   [tex]{}[/tex]                           2. Summation of values

5. m∠x = 131                                           3. Subtraction Property of Equality

6. ∠x ≅ ∠z and ∠w ≅ ∠y [tex]{}[/tex]                     Angles with equal measures are ≅

7. WXYZ is a parallelogram                 4. The opposite ∠s of a ║ are ≅

A parallelogram is a quadrilateral that has two pairs of parallel facing sides.

The details of the reasons used in the two-column table method to prove that the quadrilateral WXYZ is  parallelogram are illustrated as follows;

∠ sum property of a quadrilateral

The angle (∠) sum property of a quadrilateral, indicates that the addition of the four interior angles of a quadrilateral has a result of 360°

Summation of a set of values

The sum of the values 49 + 131 + 49 is 229

Subtraction Property of Equality

The subtraction property of equality states that the subtraction of the same amount from both sides of an equation produces two new quantities or expressions on both sides of the equation that remain equivalent.

Angles with the same measures are ≅

Angles that are congruent, ≅, are angles that have the same measure

The opposite angles of a ║ are ≅

The angles located in the facing corners of a parallelogram, ║, are congruent.

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The empirical rule states that in a normal distribution, 68% of the observations lie within 1 standard deviation from the mean, 95% of the observations lie within 2 standard deviations from the mean, and , 99.7% of the observations lie within 3 standard deviations from the mean.

a) -  Mean = U = 330 grams

standard deviation= [tex]\sigma=40 \: grams[/tex]

As per the empirical rule, 68% of the observations lie within 1 standard deviation from the mean.

Let [tex]X_1[/tex] and [tex]X_2[/tex] be the 2 random value, between which 68% of the observations lie.

Mathematically,

[tex]X1=U-\sigma[/tex]

[tex]X1[/tex] = 330 - 40

[tex]X1[/tex] =  290

Similarly,

[tex]X1=U-\sigma[/tex]

[tex]X1[/tex] = 330 + 40

[tex]X1[/tex] = 370

Thus, about 68 % of organs will be between 290 and 370 grams in weight.

b) - In this, we need to find the percentage of organs weighing between 250 grams and 410 grams.

We can rewrite 250 as follows:

250= 330− 80 ⇒ 30−2(40) ⇒ 330−2(σ)

The above value (250) lies within 2 standard deviations from the mean.

Similarly,

We can rewrite 410 as follows:

410 = 330+80 ⇒ 330+2(40) ⇒ 330+2(σ)

The above value (410) lies within 2 standard deviations from the mean.\

As per the empirical rule, 95% of the observations lie within 2 standard deviations from the mean.

Thus, 95% of organs weighs between 250 grams and 410 grams.

c)-As we have calculated in part(b) that 95% of the organs lie within 250 and 410 grams.

So, area outside these values = Total area under curve - 95%

= 100 - 95 ⇒ 5%

So, 5% of the area lies outside the 250 and 410 grams range.

From the property of normal curve, we know that the area under the curve is divided into 2 equal parts.So, we divide the area- 5% into 2 equal parts which gives 2.5%.

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In the research effort where a researcher want to survey the people at the local pool about the food served in the snack shack by deciding to walk around the kiddy pool and survey parents, this is an example of random sample because every parent in the pool have equal chance of being chosen for the study.

A random sample is drawn at random from the entire population to represent the entire data set, with each member having an equal chance of being chosen. Researchers can generate a random sample by employing techniques such as lotteries or random draws.

Because the data set is chosen using random selection, every member of the population has an equal chance of being chosen.

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