circumscribed circles
pls help

The rectangular form of the parametric equation is given as follows:

y = y = 2x² - 12x + 14, where x is in the interval [-1, 3].

The parametric equation in the context of this problem is defined as follows:

The interval for t is given as follows:

[-4,0].

Hence the bounds of the interval for x are given as follows:

From the first equation, we can write t as a function of x as follows:

t = x - 3.

Replacing into the second equation, the rectangular form of the equation is given as follows:

y = 2(x - 3)² - 4

y = 2(x² - 6x + 9) - 4

y = 2x² - 12x + 14.

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Answer: y = "2x^2 - 12x + 14", where x is on the interval ["-1, 3"].

Step-by-step explanation: I took the test.

Answer:

1. One solution

2. No solution

3. Infinite solutions

4. Two solutions

5. Three solutions

Step-by-step explanation:

You're welcome  ♡

Answerer Latoya got a better deal because she paid less than Herman by 25 cents

Step-by-step explanation:

Latoya 18 divided by 8= $2.25

Herman 15 divided by 6= $2.50

So Latoya saves $0.25 cents

The best estimate of the probability of receiving a prize would be the ratio of the number of prizes awarded to the total number of people who played the game.

a) For 10 people, 2 prizes were awarded, so the estimate of the probability of receiving a prize is 2/10 or 0.2 (20%).

b) For 50 people, 6 prizes were awarded, so the estimate of the probability of receiving a prize is 6/50 or 0.12 (12%).

c) For 100 people, 18 prizes were awarded, so the estimate of the probability of receiving a prize is 18/100 or 0.18 (18%).

Therefore, the best estimate of the probability of receiving a prize would be c) 100 people and 18 prizes awarded, which gives an estimate of 0.18 or 18%.

The first derivative of y is: y' = [4tan(πt)sec(πt) - πsec²(πt)]/(π³) and

The second derivative of y is: y'' = [4sec(πt)(π²sec²(πt) - 3πtan(πt))]/(π⁴)

The first derivative of a function represents the rate of change or slope of the function. The second derivative represents the rate of change of the first derivative or the curvature of the function.

To find the first derivative of y, we use the quotient rule:

y = [tan(πt)/π²] + [4 sec(πt)/π²]

y' = [π²(sec²(πt)) - 2tan(πt)sec(πt)]/(π⁴) + [4πsec(πt)tan(πt)]/(π⁴)

Simplifying this expression, we get:

y' = [(πsec(πt))(4tan(πt) - πsec(πt))]/(π⁴)

y' = [4πtan(πt)sec(πt) - π²(sec²(πt))]/(π⁴)

To find the second derivative of y, we again use the quotient rule:

y'' = [(π³(sec³(πt)) - 12πtan(πt)sec²(πt)) - 2π(4tan(πt)sec(πt) - πsec²(πt))(πsec(πt))]/(π⁶)

Simplifying this expression, we get:

y'' = [(4πsec(πt))(π²(sec²(πt) - 3tan(πt)))]/(π⁶)

y'' = [4sec(πt)(π²sec²(πt) - 3πtan(πt))]/(π⁴)

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Achieving a certain level οf incοme οr net wοrth: A finance prοfessiοnal might set a gοal οf achieving a certain level οf incοme οr net wοrth, which can prοvide a sense οf financial security and stability.

Stοcks, alsο knοwn as the shares οr equities, represent οwnership in  cοrpοratiοn. When yοu buy stοck, yοu are buying  small piece οf οwnership in  cοmpany. The value οf  stοck is determined by  supply and demand fοr  shares in  market, as well as by the cοmpany's financial perfοrmance, industry trends, and οther factοrs.

I can suggest sοme pοssible financial gοals that sοmeοne in a career related tο finance might have:

Saving fοr retirement: A financial prοfessiοnal might set a gοal οf saving a certain amοunt οf mοney each year fοr their retirement.

Investing in stοcks οr οther assets: A finance prοfessiοnal might set a gοal οf investing in stοcks οr οther assets in οrder tο grοw their wealth οver time.

Building an emergency fund: A finance prοfessiοnal might set a gοal οf building an emergency fund tο cοver unexpected expenses οr incοme disruptiοns.

Paying οff debt: A finance prοfessiοnal might set a gοal οf paying οff debt, such as student lοans οr credit card debt, in οrder tο imprοve their financial stability.

Saving fοr a dοwn payment οn a hοme: A finance prοfessiοnal might set a gοal οf saving fοr a dοwn payment οn a hοme in οrder tο purchase a prοperty and build equity.

Achieving a certain level οf incοme οr net wοrth: A finance prοfessiοnal might set a gοal οf achieving a certain level οf incοme οr net wοrth, which can prοvide a sense οf financial security and stability.

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The length of the line segment PR from the given parameters is: 8x - 8

The line segment addition postulate is a postulate that states that if we are given two points on a line segment, X and Z, a third point Y lies on the line segment XZ if and only if the distances between the points meet the requirements of the equation XY + YZ = XZ

We are given the parameters:

PQ = 3x - 2

QR = 5x + 6

Thus:

PR = PQ + QR

PR = 3x - 2 + 5x - 6

PR = 8x - 8

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The shaded area to the right of 4 indicates all values larger than 4 being inequality included in the solution set, while the closed circle at 4 represents the number 4 being included in the solution set.

A relationship between two terms or values which is not equal is known as an inequality in mathematics. Inequality follows imbalances, therefore. In mathematics, an inequality makes a link among two quantities that are not equal. Egality and inequality are different. Use the not similar symbol most frequently when two components are not equal (). Various inequalities are employed to contrast values of all sizes. By changing the two parts until only the variables are left, one can solve many straightforward inequalities. However a variety of factors support inequality: Negative values are split or added on either side. Switch between the left & right.

The graph below shows the set of solutions to the inequality where x + 2 is larger than or equal to 6:

From negative 9 to positive 9 is a number line. At positive 4, a closed circle appears. Positive numbers four through nine are shaded on the number line.

This means that the inequality will be satisfied by any value of x larger than or equal to 4. The graph includes the value 4 in the solution set since it has a closed circle there, and because it is shaded to the right of 4, it also includes all values higher than 4.

Here is an illustration of the graph:

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

     -9   -8   -7   -6   -5   -4   -3   -2   -1    0

                                           o        

                                         ---->      

                                           4        

                                         ----]      

                                           >        

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

      -9   -8   -7   -6   -5   -4   -3   -2   -1    0

The shaded area to the right of 4 indicates all values larger than 4 being included in the solution set, while the closed circle at 4 represents the number 4 being included in the solution set.

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Answer:

[tex]5(4+c)[/tex]

Answer:

9000000 Milligrams

Step-by-step explanation:

We know that,

1 kilogram = 10^6 Miligram

So, 9  Kilpgrams = 9×10^6 = 9000000 Milligrams

Answer:

Step-by-step explanation:

We can use the two given data points to find the slope of the line that represents the population growth over time.

The population at time t=3 is 1285, which means that the population has increased by P1 = 1285 - P0, where P0 is the initial population at time t=0 (which is not given). Similarly, the population at time t=8 is 2460, which means that the population has increased by P2 = 2460 - P0.

The time elapsed between t=3 and t=8 is 8-3 = 5 years.

The slope of the line that represents the population growth over time is given by:

slope = (P2 - P1) / (t2 - t1)

where t1 and t2 are the corresponding times.

Substituting the given values, we get:

slope = (2460 - 1285) / (8 - 3)

slope = 235

This means that the population is increasing by 235 people per year.

To find the initial population at time t=0, we can use either of the two given data points. Let's use the population at time t=3:

P1 = 1285 - P0

235 * 3 = 705

1285 - 705 = 580

Therefore, the initial population at time t=0 (i.e., in the year 1990) was 580 people.

We can now use the slope to find the population at any given time t after 1990.

Let P(t) be the population at time t after 1990.

Then, P(t) = P0 + slope * (t - 0)

P(t) = 580 + 235t

For example, the population in the year 2000 (i.e., when t=10) would be:

P(10) = 580 + 235(10) = 3330

Therefore, the population in the year 2000 would be 3330 people.

Answer:

Nonexamples of base and exponent in math would be any mathematical expressions that do not involve raising a base to an exponent. For example, 2 + 3 or √4 do not involve base and exponent.

An example of coefficient and exponential form is 5x², where 5 is the coefficient and x² is the exponential form. A nonexample could be x²/5, which does not have a coefficient in front of the exponential form.

An example of expanded form is 5x + 2, where the expression is fully written out. The standard form of the same expression would be 2 + 5x. A nonexample of expanded form is 2(x + 3), where the expression is not fully written out.

Examples of exponent laws include the product rule (a^m * a^n = a^(m+n)), quotient rule (a^m / a^n = a^(m-n)), and power rule ((a^m)^n = a^(mn)). A nonexample of an exponent law could be the sum of powers rule, which does not exist in traditional exponent laws.

The solutions for all the parts are given below.

Exponentiation is one of the mathematics operations.

Let mᵃ, where m and a are the real numbers.

And m is multiplied by 'a' times to itself.

So, a is the exponent of m.

1) Non-examples of a base and exponent in math would be: "2 + 3" or "5 - 7" as these are expressions and not a base raised to an exponent.

2) An example of a coefficient and exponential form would be "2x^3" where 2 is the coefficient and x^3 is the exponential form. A nonexample would be "3x + 2" as it is an expression and not in exponential form.

3) An example of an expanded form would be 235 = 2 x 100 + 3 x 10 + 5 x 1, where each digit is multiplied by its corresponding place value. An example of standard form would be 235 written as is, without being broken down into its place values. A nonexample of expanded form would be 235 written as 2 + 3 + 5, as this is not a multiplication of the digits with their place values.

4). Examples of exponent laws include:

Product law: aᵇ x aⁿ = a⁽ᵇ ⁺ⁿ⁾

Quotient law: [tex]a^m / a^n = a^{(m-n)}\\[/tex]

Power law: [tex](a^m)^n = a^{(m \times n)}[/tex]

Negative exponent law: [tex]a^{(-n)} = 1/a^n[/tex]

Zero exponent law: [tex]a^0 = 1[/tex].

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The perimeter of triangle N is 72 inches.

Similar triangles are two triangles that have the same shape, but may differ in size. In other words, their corresponding angles are congruent and their corresponding sides are proportional. This means that if you were to scale one triangle up or down, it would still maintain the same shape as the other triangle.

Here given the triangles M and N are similar, the ratio of sides N:M is 4:3

We can say that M and N with constant term x is, (by ratio proportion rule)

M = 3x and N = 4x and

Given the lengths of triangle M are 12 inches, 18 inches and 24 inches

for triangle M = 3x, (Using above ratio proportion rule)

3x = 12 inches                    3x = 18 inches                    3x = 24 inches

x = 4                                    x = 6                                    x = 8

Similarly, for similar triangle N = 4x, (Using above ratio proportion rule) put all values of x = 4, 6, 8

for x =4                                for x =6                                for x =8

4x = 4×4                               4x = 4×6                              4x = 4×8  

16 inches                             24 inches                            32 inches

Length of triangle N is 16 inches, 24 inches and 32 inches

Perimeter of triangle N = 16+24+32 = 72 inches

Therefore, the Perimeter of triangle N is 72 inches.

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Based on the information given in the histogram, option B. The distribution of the data is asymmetrical, so the mean and the median are likely outside the "1,000-1,099 photocopies" category is the correct statement.

What is histogram?

The histogram shows that the distribution of the data is asymmetrical, with a longer tail towards the right. This means that there are more schools that made a higher number of photocopies than those that made a lower number of photocopies. Therefore, the mean and median are likely to be higher than the midpoint of the "1,000-1,099 photocopies" category, which has the highest frequency.

In a skewed distribution, the mean is typically affected by the outliers, while the median is a better measure of central tendency. Therefore, the median is more likely to be within the "1,000-1,099 photocopies" category, while the mean is likely to be higher.

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Answer:

10

Step-by-step explanation:

5×6=20+y

30=20+y

30-20=y

10=y

therefore y=10

After solving the given problem, we found that the difference between the area of the dilated square and the original square is 32 square centimeters.

The area of the original square with a side length of 2 centimeters can be calculated as:

A = s²

A = 2²

A = 4 square centimeters

When this square is dilated by a scale factor of 3, the new side length will be:

s' = 3s

s' = 3(2)

s' = 6 centimeters

The area of the dilated square can be calculated as:

A' = s'²

A' = 6²

A' = 36 square centimeters

The difference between the area of the dilated square and the original square is:

A' - A = 36 - 4

A' - A = 32 square centimeters

Therefore, the difference between the area of the dilated square and the original square is 32 square centimeters.

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The closest option to the total distance driven is 270.

We kept an eye on them from afar. Compared to earlier, she perceives a distance between her and her brother. They had been close friends once, but now there was a great distance between them. He desires to disassociate himself from his old boss.

We must combine the distances for each hour in order to calculate the total distance travelled:

Distance total = 62 + 70 + 68 + 72

272 miles roundtrip.

Thus, option 270 comes the closest to the total mileage driven.

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The probability that exactly 2 of the workers interviewed are union members is approximately 0.044.

To solve this problem, we can use the binomial probability formula, which is:

[tex]P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)[/tex]

where:

n is the sample size (number of workers being interviewed)

k is the number of successes we are interested in (in this case, 2 of the workers being union members)

p is the probability of success (in this case, the probability that a worker is a union member)

(n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items (this can be calculated using the formula n! / (k! * (n - k)!))

Plugging in the values, we get:

[tex]P(X = 2) = (4 choose 2) * (0.96)^2 * (1 - 0.96)^(4 - 2)[/tex]

[tex]= (6) * (0.96)^2 * (0.04)^2[/tex]

= 0.04403136

Therefore, the probability that exactly 2 of the workers interviewed are union members is approximately 0.044.

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The sequence of transformations is:

Let's only follow the coordinates of one of the vertices to identify the transformations, we clearly have a reflection over a vertical line and a reflection over a horizontal line, so first let's apply thes two.

Vertex A starts at (1, -4)

First a reflection over the x-axis will change the sign of the y-component, then we get:

A₁ = (1, 4)

Then a reflection over the y-axis changes the sign of the x-component to:

A₂ = (-1, 4)

Finally we have a translation, we can see that:

A' = (2, 4).

Then we have a translation (a, b) such that:

(-1 + a, 4 + b) = (2, 4)

So we can see that a = 3 and b = 0, then we have a translation of 3 units to the right.

That is the sequence of transformations.

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Answer: (a) The probability of being male or being a nursing major is the sum of the probabilities of being male and being a nursing major, minus the probability of being both male and a nursing major (since this intersection is counted twice):

P(being male or being nursing major) = P(male) + P(nursing major) - P(male and nursing major)

From the table, we have:

P(male) = 1110/3538 ≈ 0.314

P(nursing major) = 1016/3538 ≈ 0.287

P(male and nursing major) = 94/3538 ≈ 0.027

Therefore:

P(being male or being nursing major) ≈ 0.314 + 0.287 - 0.027 ≈ 0.574

Rounded to the nearest thousandth as needed, the probability of being male or being a nursing major is approximately 0.574.

(b) The probability of being female or not a nursing major is the sum of the probabilities of being female and not a nursing major:

P(being female or not being a nursing major) = P(female) + P(not nursing major)

From the table, we have:

P(female) = 2428/3538 ≈ 0.686

P(not nursing major) = 1728/3538 ≈ 0.489

Therefore:

P(being female or not being a nursing major) ≈ 0.686 + 0.489 ≈ 1.175

This probability is greater than 1, which is not possible. Therefore, we need to subtract the probability of being both female and a nursing major (which was counted twice):

P(being female or not being a nursing major) = P(female) + P(not nursing major) - P(female and nursing major)

From the table, we have:

P(female and nursing major) = 700/3538 ≈ 0.198

Therefore:

P(being female or not being a nursing major) ≈ 0.686 + 0.489 - 0.198 ≈ 0.977

Rounded to the nearest thousandth as needed, the probability of being female or not a nursing major is approximately 0.977.

(c) The probability of not being female or a nursing major is the complement of the probability of being female or a nursing major:

P(not being female or being a nursing major) = 1 - P(being female or being nursing major)

From part (a), we have:

P(being male or being nursing major) ≈ 0.574

Therefore:

P(not being female or being a nursing major) ≈ 1 - 0.574 ≈ 0.426

Rounded to the nearest thousandth as needed, the probability of not being female or a nursing major is approximately 0.426.

(d) The events "being male" and "being a nursing major" are not mutually exclusive, because there are 94 males majoring in nursing (as shown in the table). Mutually exclusive events cannot occur at the same time, but being male and a nursing major is a possible combination. Therefore, the correct answer is OB. Yes, because there are 94 males majoring in nursing.

Step-by-step explanation:

17.  Number of hours, h     10 + 2h    Total Cost

         1                             10 + 2 x 1          12

         2                            10 + 2 x 2         14

         3                            10 + 2 x 3         16

         4                             10 + 2 x 4        18

This is true because as shown in the first example, you have to sub in the number of hours into the "h" value, and multiply it times two. After that, you add ten.

The three accοmpanying questiοns, assuming that the firm is prοfit-maximizing and dοes nοt shut dοwn in the shοrt run

1)Firm's Tοtal Revenue=  $78000

2)Firm's Tοtal Cοst =$128700

3)Firm's Lοss = - $50700

Wοrd prοblems are οften described verbally as instances where a prοblem exists and οne οr mοre questiοns are pοsed, the sοlutiοns tο which can be fοund by applying mathematical οperatiοns tο the numerical infοrmatiοn prοvided in the prοblem statement. Determining whether twο prοvided statements are equal with respect tο a cοllectiοn οf rewritings is knοwn as a wοrd prοblem in cοmputatiοnal mathematics.

Here in the given graph,

Cοst per unit = $300

Equilibrium quantity = $260 Then,

Firm's Tοtal Revenue = P * Q

=> $300*260 = $78000 (Equilibrium where, MR = MC)

Firm's Tοtal Cοst = Cοst per Unit * equilibrium quantity

=> $495*260 = $128,700

Firm's Lοss = TR - TC

=> $78000 - $128,700 = - $50700

Hence the answers are,

1)Firm's Tοtal Revenue=  $78000

2)Firm's Tοtal Cοst =$128700

3)Firm's Lοss = - $50700

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The quadratic equation for the height of the rock, h = -16·t² + 79·t + 50, indicates;

a. The times at which the rock will be 125 feet above the beach are about; 3.66 seconds and 1.28 seconds

b. The maximum height reached is about 147.5 feet and it will take about 2.5 seconds for the rock to reach maximum height

A quadratic equation is an equation of the form, f(x) = a·x² + b·x + c, where; a ≠ 0, and a, b, and c are numbers.

a. The function for the height of the rocket is presented as follows;

h(t) = -16·t² + 79·t + 50

The time at which the height is 125 feet can be found as follows;

h(t) = 125 feet

h(t) = 125 = -16·t² + 79·t + 50

-16·t² + 79·t + 50 - 125 = 125 - 125 = 0

-16·t² + 79·t - 75 = 0

The quadratic formula indicates that we get;

t = (-79 ± √(79² - 4 × (-16) ×(-75)))/(2 × (-16)) = (79 ± √(1441))/32

t = (79 ± √(1441))/32

t = (79 + √(1441))/32 ≈ 3.66

t = (79 - √(1441))/32 ≈ 1.28

b. The time at which the rock is at the maximum height can be obtained by using the formula for finding the input value at the maximum value of a quadratic function, which is; at the maximum height, x = -b/2·a

Therefore, at the maximum height, we get;

The time, t = -79/(2 × (-16)) = 2.46875

The time it takes the rock to reach the maximum height is 2.46875 seconds ≈ 2.5 seconds

The maximum height is therefore;

h(2.46875) = -16×2.46875² + 79×2.46875 + 50 = 147.515625 ≈ 147.5

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 3\%\to \frac{3}{100}\dotfill &0.03\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.03}{1}\right)^{1\cdot 4}\implies A=3000(1.03)^4 \implies A \approx 3376.53[/tex]

The equations of the other two sides are AD: y = -3x + 3 and DC: y = 1/2x - 1/2.

What is point-slope form?

The point-slope form is a way to write the equation of a straight line in algebra. It is used when you know the coordinates of a point on the line and the slope of the line. The point-slope form of a linear equation is:

y - y₁ = m(x - x₁)

where m is the slope of the line, and (x1, y1) are the coordinates of a point on the line.

Since AB and BC are two sides of a parallelogram, they are parallel to each other. Thus, we can use the slope formula to find the slopes of these sides.

Slope of AB:

m = (y₂ - y₁)/(x₂ - x₁)

= (6 - 3)/(6 - 0)

= 3/6

= 1/2

Slope of BC:

m = (y₂ - y₁)/(x₂ - x₁)

= (3 - 6)/(7 - 6)

= -3/1

= -3

Now that we know the slopes of AB and BC, we can use the point-slope form to find the equations of the other two sides.

Equation of AD:

We know that AD is parallel to BC and passes through point A. So, we can use the point-slope form with the slope of BC and point A.

y - y₁ = m(x - x₁)

y - 3 = -3(x - 0)

y - 3 = -3x

y = -3x + 3

Equation of DC:

We know that DC is parallel to AB and passes through point C. So, we can use the point-slope form with the slope of AB and point C.

y - y₁ = m(x - x₁)

y - 3 = 1/2(x - 7)

y - 3 = 1/2x - 7/2

y = 1/2x - 1/2

Therefore, the equations of the other two sides are:

AD: y = -3x + 3

DC: y = 1/2x - 1/2

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1) The nth term if the sequences are (a) 9a-7  (b)  13-6a  

A sequence is a list of numbers or objects in a special order.1 It is an enumerated collection of objects in which repetitions are allowed and order matters.

The given sequences are

1) (a)  2,11,20,...

the first term a= 2, the common difference d = 11-2 =9

The nth term is given as

Tn = a + (n-1)d

Tn = 2 + (n-1)*9

Opening the brackets to get

2 + 9n -9

Rearrange to have

9n -7

(b)  the sequence is 7,1,-5,...

a = 7 ,

d= 1-7 = -6

Tn = a + (n-1)d

Tn = 7 + (n - 1)-6

Tn = 7 + -6n +6

Tn = 7+6 -6n

Tn = 13 -6n

2) Remember for every Fibonacci sequence

1st+2nd=3rd

2nd+3rd=4th

3rd+4th=5th

4th+5th=6th

Remember for every arithmetic Progression

Tn = a + (n-1)d

For every Geometric Progression,

Tn = arⁿ⁻¹

First five terms        Next three terms      Name of sequence(A,G,F,Q)

  10,6,2,-2,-6                 -10,-14,-18                    arithmetic

  2,8,32,126512        2048, 8192,32768          Geometric

  20,13,33,46,79           125,204,329                 Fibonacci

  200,100,50,25,12.5   6.25, 3.125, 1.5625      Geometric

46,39,32,25,18,              11,4 , -3                         Arithmetic

-2,,2,0,2,2,4                  6,10,16                            Fibonacci

3,6,10,15, 21                 28,35,42                        Arithmetic

25,15,0,-20,-45,           -80,-125,-175                Arithmetic

2,5,7,12,19                    31,50,81                       Fibonacci

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Check the picture below.

Answer:

Step-by-step explanation:

The two given points are (8, 0) and (8, -8).

Using the distance formula, the distance between the two points can be calculated as:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where x1 = 8, y1 = 0, x2 = 8, and y2 = -8

d = sqrt((8 - 8)^2 + (-8 - 0)^2)

d = sqrt(0 + 64)

d = sqrt(64)

d = 8

Therefore, the distance between the two points is 8 units.

The smaller acute angle of the right triangle is x-3 = 12.6 degrees.

What is Right Triangle?

A triangle in which one angle is 90 degree is called a Right Angled Triangle.

In a right triangle, if one acute angle is x-3 degrees, then the other acute angle is 90-(x-3) = 93-x degrees.

Using the given equation, we have:

sin(x-3) = cos(6x+9)

Taking the sine of both sides:

sin(x-3) = sin(90 - (6x+9)) = sin(81-6x)

Now we have:

sin(x-3) = sin(81-6x)

Since the two angles have the same sine, they must differ by a multiple of 360 degrees. Thus:

x-3 = 81-6x + 360n or x-3 = 6x-81 + 360n

where n is an integer.

Simplifying each equation:

7x = 84 + 360n or 5x = 78 + 360n

Dividing both sides by 7 or 5, respectively:

x = 12 + 51.43n or x = 15.6 + 72n

The first equation gives us values of x that are too large for an acute angle, so we use the second equation:

x = 15.6 + 72n

The smallest such x that satisfies 0 < x < 90 is when n=0, which gives:

x = 15.6 degrees

Therefore, the smaller acute angle of the right triangle is x-3 = 12.6 degrees.

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