Between 2002 and 2012 the population of the town Astopia fell by 4%.

If the population in 2012 was 252,960 find the population of the town in 2002.

The total amount of meat bought is 46/4 or 11.5 kg (or 11.5 kilograms).

To express the total amount of meat bought in fraction and decimal form, we need to add up the quantities of each type of meat.

The given quantities are:

2 kg and a quarter of beef

5 kg and two quarters (or a half) of chicken

4 kg and three quarters of pork loin

To find the total amount of meat bought, we can add these quantities:

2 kg and 1/4 + 5 kg and 1/2 + 4 kg and 3/4

To add these mixed numbers and fractions, we need to find a common denominator. The common denominator here is 4.

2 kg and 1/4 can be converted to 9/4 by multiplying 2 by 4 and adding 1.

5 kg and 1/2 can be converted to 9/2 by multiplying 5 by 2 and adding 1.

4 kg and 3/4 remains the same.

Now we can add the fractions:

9/4 + 9/2 + 4 + 3/4

To add the fractions, we need to find a common denominator, which is 4.

9/4 + 9/2 + 16/4 + 3/4

Now we can add the numerators and keep the denominator:

(9 + 18 + 16 + 3) / 4

The numerator becomes 46, so the total amount of meat bought is 46/4.

To express this as a decimal, we can divide the numerator by the denominator:

46 ÷ 4 = 11.5

Therefore, the total amount of meat bought is 46/4 or 11.5 kg (or 11.5 kilograms).

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Answer:

see explanation

Step-by-step explanation:

to find the first 5 terms substitute n = 1, 2, 3, 4, 5 into the explicit formula

36

a₁ = - [tex]3^{1-1}[/tex] = - [tex]3^{0}[/tex] = - 1 [ [tex]a^{0}[/tex] = 1 ]

a₂ = - [tex]3^{2-1}[/tex] = - [tex]3^{1}[/tex] = - 3

a₃ = - [tex]3^{3-1}[/tex] = - 3² = - 9

a₄ = - [tex]3^{4-1}[/tex] = - 3³ = - 27

a₅ = - [tex]3^{5-1}[/tex] = - [tex]3^{4}[/tex] = - 81

the first 5 terms are - 1, - 3, - 9, - 27, - 81

to find a₈ substitute n = 8 into the explicit formula

a₈ = - [tex]3^{8-1}[/tex] = - [tex]3^{7}[/tex] = - 2187

37

to find the first 5 terms substitute n = 1, 2, 3, 4, 5 into the explicit formula

a₁ = 2 × [tex](\frac{1}{2}) ^{1-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{0}[/tex] = 2 × 1 = 2

a₂ = 2 × [tex](\frac{1}{2}) ^{2-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{1}[/tex] = 2 × [tex]\frac{1}{2}[/tex] = 1

a₃ = 2 × [tex](\frac{1}{2}) ^{3-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{2}[/tex] = 2 × [tex]\frac{1}{4}[/tex] = [tex]\frac{1}{2}[/tex]

a₄ = 2 × [tex](\frac{1}{2}) ^{4-1}[/tex] = 2 × ([tex]\frac{1}{2}[/tex] )³ = 2 × [tex]\frac{1}{8}[/tex] = [tex]\frac{1}{4}[/tex]

a₅ = 2 × [tex](\frac{1}{2}) ^{5-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{4}[/tex] = 2 × [tex]\frac{1}{16}[/tex] = [tex]\frac{1}{8}[/tex]

the first 5 terms are 2 , 1 , [tex]\frac{1}{2}[/tex] , [tex]\frac{1}{4}[/tex] , [tex]\frac{1}{8}[/tex]

to find a₈ substitute n = 8 into the explicit formula

a₈ = 2 × [tex](\frac{1}{2}) ^{8-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{7}[/tex] = 2 × [tex]\frac{1}{128}[/tex] = [tex]\frac{1}{64}[/tex]

Answer:

The explicit formula for a geometric sequence is:

a_n = a_1 * r^(n - 1)

where:

In equation 36,

we can see that the first term is -3 and the common ratio is -3. Therefore, we can write the explicit formula for this sequence as:

a_n = -3 * (-3)^(n - 1)

Using this formula, we can find the first five terms and the 8th term in the sequence:

a_1 = -3

a_2 = -3 * (-3) = 9

a_3 = -3 * (-3)^2 = -27

a_4 = -3 * (-3)^3 = 81

a_5 = -3 * (-3)^4 = -243

a_8 = -3 * (-3)^7 = 6561

In equation 37,

we can see that the first term is 2 and the common ratio is 1/2. Therefore, we can write the explicit formula for this sequence as:

a_n = 2 * (1/2)^(n - 1)

Using this formula, we can find the first five terms and the 8th term in the sequence:

a_1 = 2

a_2 = 2 * (1/2) = 1

a_3 = 2 * (1/2)^2 = 1/2= 0.5

a_4 = 2 * (1/2)^3 =1/4= 0.25

a_5 = 2 * (1/2)^4 = 1/8=0.125

a_8 = 2 * (1/2)^7 = 1/64=0.015625

Answer:  j(x) = (x-1)(x+2)

The x-1 part is because of the root x = 1

The root x = -2 leads to the factor x+2

The algebra 2 is a vast field that deals with various mathematical concepts, such as equations and inequalities.

Algebra 2 is a branch of mathematics that mainly deals with equations and inequalities. An equation is a mathematical statement that implies that two expressions are equal.

Similarly, an inequality implies that two expressions are not equal but are related by a certain operation.There are different types of equations that one may encounter in Algebra 2.

A linear equation is an equation that can be represented by a straight line on the coordinate plane. A quadratic equation is one that can be written in the form ax² + bx + c = 0.

There are also exponential, logarithmic, and trigonometric equations that one may come across.Inequalities are statements that two expressions are not equal. Inequalities can be represented graphically on a coordinate plane, just like equations.

There are different types of inequalities such as linear, quadratic, exponential, and logarithmic inequalities.In conclusion,  These concepts help in solving real-world problems by providing a framework to analyze them mathematically.

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There are 24 different orders in which Greg, Albert, Joseph, and Finn can take their turns being 'it' in the game of hide-and-seek.

To determine the number of different orders in which the children can take their turns being 'it' in a game of hide-and-seek, we can use the concept of permutations.

Since there are four children (Greg, Albert, Joseph, and Finn), we need to find the number of permutations of these four children. A permutation represents an arrangement of objects in a specific order.

The formula for calculating permutations is given by:

P(n, r) = n! / (n - r)!

Where n is the total number of objects (children) and r is the number of objects (children) to be arranged.

In this case, we have n = 4 (four children) and r = 4 (all four children will take their turns as 'it'). Plugging these values into the formula, we get:

P(4, 4) = 4! / (4 - 4)!

= 4! / 0!

= 4! / 1

= 4 × 3 × 2 × 1 / 1

= 24

Therefore, there are 24 different orders in which Greg, Albert, Joseph, and Finn can take their turns being 'it' in the game of hide-and-seek.

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We subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend. Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

To calculate how much Miranda saves in one year by buying only one bottle of wine per week instead of two, we can follow these steps:

Determine the number of bottles of wine she buys in a year:

Miranda used to buy 2 bottles of wine per weekend, so in a week, she bought 2 bottles.

Since there are 52 weeks in a year, the number of bottles she bought in a year is 2 bottles/week * 52 weeks = 104 bottles.

Calculate the total cost of wine for the year:

Since each bottle costs $15, the total cost of wine for the year when buying 2 bottles per week would be 104 bottles * $15/bottle = $1560.

Calculate the cost of wine for one year when buying only one bottle per week:

With Miranda's decision to buy one bottle per week, the total number of bottles she buys in a year is 1 bottle/week * 52 weeks = 52 bottles.

Therefore, the cost of wine for one year when buying only one bottle per week would be 52 bottles * $15/bottle = $780.

Calculate the amount saved in one year:

To determine the amount saved, we subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend.

Amount saved = $1560 - $780 = $780.

Therefore, Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

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The Equation of circle for Circle B is (x - 2) ² + (y + 4)²  = 38.4.

For circle A:

Radius of circle A = √36 = 6

Area of circle A = π(6)^2 = 36π

Let's assume the radius of circle B is r.

Circumference of circle A = 2π(6) = 12π

Circumference of circle B = 3.2 x Circumference of circle A

= 3.2 x 12π

= 38.4π

Since the ratio of the areas is equal to the ratio of the circumferences, we have:

Area of circle B / Area of circle A = (r^2π) / (36π) = 38.4π / 36π

Simplifying, we get:

r² / 36 = 38.4 / 36

Cross-multiplying, we have:

36 x r² = 38.4 x 36

Dividing both sides by 36:

r^2 = 38.4

Taking the square root of both sides:

r = √38.4

Now we have the radius of circle B. Let's substitute this value into the equation for circle B:

(x - 2) ² + (y + 4)² = (√38.4)²

Simplifying:

(x - 2) ² + (y + 4)²  = 38.4

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Answer:

100

Step-by-step explanation:

#20 naira - 20 * 100

= 2000kobo

2000/20

100

QED✅✅

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It should be noted that the number of sample size based on the information will be 300.

In order to calculate the sample size, we can use the following formula:

n = z² * p * (1-p) / E²

n = 1.96² * 0.5 * (1-0.5) / 0.05²

= 300

A questionnaire is a method of gathering data that makes use of written questions to be answered by the respondents. It is a popular method of data collection because it is relatively easy to administer and can be used to collect a wide variety of information.

In the first column, the population is the entire National Capital Region (NCR). The sample is the city of Manila. In the second column, the population is all STEM students. The sample is all academic track students. In the third column, the population is all the tablespoons of sugar in the jar. The sample is one tablespoon of sugar. In the fourth column, the population is all the vowels in the word "juice". The sample is the vowel "i".

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Sin 0 is approximately 0.4709.

To find sin 0 given that cos 0 = 15/17, we can use the Pythagorean identity:

[tex]sin^2 0 + cos^2 0 = 1[/tex]

Rearranging the equation, we have:

[tex]sin^2 0 = 1 - cos^2 0[/tex]

Since we know that cos 0 = 15/17, we can substitute this value into the equation:

[tex]sin^2 0 = 1 - (15/17)^2[/tex]

Calculating this, we find:

[tex]sin^2 0 = 1 - (225/289)\\sin^2 0 = (289/289) - (225/289)\\sin^2 0 = 64/289[/tex]

Taking the square root of both sides, we get:

sin 0 = √(64/289)

Now, we need to determine the sign of sin 0. Since cos 0 is positive (15/17), sin 0 will also be positive in the first and second quadrants.

sin 0 = √(64/289)

sin 0 ≈ √0.2213

sin 0 ≈ 0.4709 (rounded to four decimal places)

Therefore, sin 0 is approximately 0.4709.

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Answer:

To find the points where the rate of change of x is half the rate of change of y, we need to solve the equations:

   dy/dx = k

   dy/dt = k

where k is a constant representing the rate of change.

Step-by-step explanation:

The percent decrease in total principal and interest paid between a 30-year term mortgage and a 15-year mortgage with a principal balance of $484,500.00 and a 6.5% APR is 31.0%. (Option A)

To calculate the  percent decrease, we can use the following formula.

Percent decrease = ( (30-year factor -   15-year factor) / 30-year factor) x 100

Substituting the values from the chart.  

= (($ 6.65 - $8.71) / $6.65) x 100

= ( -$ 2.06 / $6.65) * 100

= -0.309 *  100

Percent decrease ≈ - 30.9 %

Since the question asks for the percent decrease as a positive value, we take the absolute value of the result which is

Percent decrease ≈ 31%

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Answer: I think it's H, if it isn't and you get a second try it should be N

Step-by-step explanation: Both H and N start with 0/-3 except F which means F is wrong. Maybe you should wait for a better answer...but it shouldn't be F

The missing part indicated in the figures are ? = 12, TX = 9 and x = 20

Figure a

The missing part can be calculated using the following equation

?/(11 - 5) = 22/11

Evaluate the difference

?/6 = 22/11

So, we have

? = 6 * 22/11

Evaluate the expression

? = 12

Figure b

The missing part can be calculated using the following equation

TX/3 = 6/2

So, we have

TX = 3 * 6/2

Evaluate

TX = 9

Figure c

The value of x can be calculated using the following equation

1/4x + 6 = 2x - 29

So, we have

x + 24 = 8x - 116

Evaluate

-7x = -140

Divide

x = 20

Hence, the value of x is 20

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When viewed through the magnifying glass, the termite appears to be approximately 1.58 mm in length.

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The solution to the gaps in the angle proof are:

Multiplication Property of equality

m∠D = ¹/₂ * [arc EFG] and m∠F = ¹/₂ * [arc GDE]

substitution property

The inscribed angle Theorem states that the inscribed angle measures half of the arc of which it is composed.

Thus, we can say that:

m∠D = ¹/₂ * [arc EFG]

and

m∠F = ¹/₂ * [arc GDE]

Therefore:

arc EFG + arc GDE = 360°-------> full circle

Applying multiplication property of equality, we have:

¹/₂ * arc EFG + ¹/₂ * arc GDE = 180°

Applying substitution property of equality, we have:

m∠D = ¹/₂ * [arc EFG]

m∠F = ¹/₂ * [arc GDE]

¹/₂ * arc EFG + ¹/₂ * arc GDE = 180° ----> m∠D + m∠F = 180°

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The correct statement is that they have different x- and y-intercepts, but we cannot determine their end behavior based on the given information.

Based on the information provided, we can compare the two functions f and g as follows:

Function f:

- It passes through the points (-10, -1) and (2, 8).

- It intercepts the x-axis at -2 units and the y-axis at 2 units.

Function g:

- It is represented by the table with x-values -2, -1, 0, 1, 2, and corresponding y-values 0, 2, 8, 26.

Comparing the two functions based on their intercepts and end behavior:

A. They have the same y-intercept and the same end behavior as x approaches ∞: This statement is incorrect because the y-intercepts of f and g are different. Function f intercepts the y-axis at 2 units, while function g intercepts the y-axis at 0 units. Additionally, we do not have information about the end behavior of either function.

B. They have the same x-intercept and the same end behavior as x approaches ∞: This statement is incorrect because the x-intercepts of f and g are different. Function f intercepts the x-axis at -2 units, while function g does not intercept the x-axis.

C. They have different x- and y-intercepts but the same end behavior as x approaches ∞: This statement is partially correct. Function f and g have different x- and y-intercepts, but we don't have information about their end behavior.

D. They have the same x- and y-intercepts: This statement is incorrect as the x- and y-intercepts of f and g are different.

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first off, let's take a peek at the picture above

hmmm the hyperbola is opening sideways, that means it has a horizontal traverse axis, it also means that the positive fraction will be the one with the "x" variable in it.

now, the length of the horizontal traverse axis is 4 units, from vertex to vertex, that means the "a" component of the hyperbola is half that or 2 units, and 2² = 4, with a center at the origin.

[tex]\textit{hyperbolas, horizontal traverse axis } \\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(x- 0)^2}{ 2^2}-\cfrac{(y- 0)^2}{ (\sqrt{3})^2}=1\implies {\Large \begin{array}{llll} \cfrac{x^2}{4}-\cfrac{y^2}{3}=1 \end{array}}[/tex]

It would cost approximately $119.42 to install the molding around the circular room, assuming the cost per foot is $4.22 and the diameter of the room is 9 feet.

To calculate the cost of installing molding around a circular room, we need to find the circumference of the room. The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter.

In this case, the diameter is given as 9. We can substitute this value into the formula to find the circumference:

C = π * 9

C ≈ 28.27 feet

Now that we know the circumference of the room is approximately 28.27 feet, we can calculate the cost of installing the molding. The cost per foot is given as $4.22.

Cost = Cost per foot * Circumference

Cost = $4.22 * 28.27

Cost ≈ $119.42

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The only system of inequalities that has no solution is x<2 and x>7.(option-b)

The system of inequalities that has no solution is x<2 and x>7.

If x is less than 2, it cannot be greater than 7 at the same time. These two inequalities are contradictory, and there is no number that could satisfy both of them simultaneously.

The system of inequalities x>2 and x<7 forms an open interval between 2 and 7. Any value of x within this interval can satisfy both inequalities, so this system has a solution.

The system of inequalities x<2 and x<7 forms an inequality that is satisfied by any value of x that is less than 7, so this system also has a solution.

Lastly, the system of inequalities x>2 and x>7 forms an inequality that is not satisfied by any value of x, as there is no number that is simultaneously greater than 2 and greater than 7.(option-b)

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Answer:

495

Step-by-step explanation:

using the definition

n[tex]C_{r}[/tex] = [tex]\frac{n!}{r!(n-r)!}[/tex]

where n! = n(n - 1)(n - 2) ... × 3 × 2 × 1

then

12[tex]C_{4}[/tex]

= [tex]\frac{12!}{4!(12-4)!}[/tex]

= [tex]\frac{12!}{4!(8!)}[/tex]

cancel 8! on numerator/ denominator

= [tex]\frac{12(11)(10)(9)}{4!}[/tex]

= [tex]\frac{11880}{4(3)(2)(1)}[/tex]

= [tex]\frac{11880}{24}[/tex]

= 495

The length of side X in simple radical form with a rational denominator is 25/√3.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the side lengths are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side):

Adjacent side = 5/√3

Hypotenuse, x = 5 × 5/√3

Hypotenuse, x = 25/√3.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The type of transformation shown is vertical translation.

Since, Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Given a polygon EFGH.

It is transformed to another polygon with the same size E'F'G'H'.

Here the polygon EFGH is just moved downwards as it is and mark is as E'F'G'H'.

If it is rotation or reflection, the points will change it's correspondent place.

Translation is a type of transformation where the original figure is shifted from a place to another place without affecting it's size.

Therefore, here translation is done.

Since the shifting is done vertically, it is vertical translation.

Hence the transformation is vertical translation.

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The area of triangle ADE is,

⇒ A = 48 square units

We have to given that,

In the diagram , DE is a midsegment of triangle ABC.

And, The area of triangle ABC is 96 square units

Now, We know that,

Since DE is a midsegment of triangle ABC, it is parallel to AB and half the length of AB. Therefore, DE is half the length of AB.

Hence, the area of triangle ADE is half the area of triangle ABC,

That is,

A = 96 / 2

A = 48 square units

Thus, The area of triangle ADE is,

⇒ A = 48 square units

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The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.

We have,

To evaluate the limit of the expression as x approaches 0, we can simplify the expression first.

The expression is given as lim(x approaches 0) (cosec(2x) - x cosec³(x)).

Using trigonometric identities, we can rewrite cosec(2x) as 1/sin(2x) and cosec³(x) as 1/(sin(x))³.

Substituting these into the expression,

We get lim(x approaches 0) (1/sin(2x) - x (1/(sin(x))³)).

Now, let's evaluate the limit term by term:

lim(x approaches 0) (1/sin(2x)) = 1/sin(0) = 1/0 (which is undefined).

lim(x approaches 0) (x (1/(sin(x))³))

= 0 (1/(sin(0))³)

= 0 x 1/0 (which is also undefined).

Since both terms of the expression are undefined as x approaches 0, we cannot determine the limit of the expression.

Therefore,

The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.

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Mikey Johnson shipped out 34 2/7 pounds of electrical supplies. The supplies are placed in individual packets that weigh 2 1/7 pounds each. Therefore, Mikey shipped out 16 packets of electrical supplies.

To solve the problem, we can use the following steps.Step 1: Find the weight of each packet.

We are given that the weight of each packet is 2 1/7 pounds.

To convert this mixed number into an improper fraction, we can multiply the whole number by the denominator and add the numerator.

This gives us: 2 1/7 = (2 × 7 + 1) / 7= 15 / 7 pounds.

Therefore, the weight of each packet is 15/7 pounds.

Now, divide the total weight by the weight of each packet.

We are given that the total weight of the supplies shipped out is 34 2/7 pounds.

To convert this mixed number into an improper fraction, we can multiply the whole number by the denominator and add the numerator.

This gives us: 34 2/7 = (34 × 7 + 2) / 7= 240 / 7 pounds.

Therefore, the total weight of the supplies is 240/7 pounds.

To find the number of packets that Mikey shipped out, we can divide the total weight by the weight of each packet.

This gives us: 240/7 ÷ 15/7 = 240/7 × 7/15= 16.

Therefore, Mikey shipped out 16 packets of electrical supplies.

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The correlation coefficient (r) is -0.96.

Given the data:

x: 0, 1, 2, 2, 3, 4, 5, 5, 6

y: 12, 9.5, 9, 8.5, 8.5, 6, 5, 5, 3.5

Step 1: Calculate the mean (average) of x and y.

Mean of x = (0 + 1 + 2 + 2 + 3 + 4 + 5 + 5 + 6) / 9 = 3

Mean of y  = (12 + 9.5 + 9 + 8.5 + 8.5 + 6 + 5 + 5 + 3.5) / 9 = 7

Step 2: Calculate the deviations from the mean for x and y.

Deviation of x (dx) = x - X

Deviation of y (dy) = y - Y

x: -3, -2, -1, -1, 0, 1, 2, 2, 3

y: 5, 2.5, 2, 1.5, 1.5, -1, -2, -2, -3.5

Step 3: Calculate the product of the deviations of x and y.

(dx * dy): -15, -5, -2, -1.5, 0, -1, -4, -4, -10.5

Step 4: Calculate the sum of the products of the deviations (Σ(dx * dy)).

Σ(dx * dy) = -15 - 5 - 2 - 1.5 + 0 - 1 - 4 - 4 - 10.5 = -43

Step 5: Calculate the standard deviations of x (σx) and y (σy).

Standard deviation of x (σx) = √(Σ(dx²) / (n-1))

Standard deviation of y (σy) = √(Σ(dy²) / (n-1))

Calculating dx²:

dx^2: 9, 4, 1, 1, 0, 1, 4, 4, 9

Σ(dx²) = 33

Calculating dy²:

dy²: 25, 6.25, 4, 2.25, 2.25, 1, 4, 4, 12.25

Σ(dy²) = 61

Calculating the standard deviations:

σx = √(33 / (9-1)) = √(33 / 8) ≈ 1.84

σy = √(61 / (9-1)) = √(61 / 8) ≈ 2.70

Step 6: Calculate the correlation coefficient (r).

r = Σ(dx dy) / (√(Σ(dx²)  Σ(dy²)))

r = -43 / (√(33  61))

r ≈ -43 / (√(2013))

r ≈ -43 / 44.87

r ≈ -0.96

Therefore, the correlation coefficient (r) is approximately -0.96.

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We can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.

Given the equation of the surface is xy + 15x + z^2 = 209.

Let's determine the shortest distance from the surface to the origin.

The shortest distance between the surface and the origin is given by the perpendicular distance, which can be calculated as follows:

Firstly, we need to determine the gradient of the surface, which is the vector normal to the surface.

For this purpose, we need to write the surface equation in the standard form, which is:xy + 15x + z^2 = 209 xy + 15x + (0)z^2 - 209 = 0 (the coefficients of x, y, and z are a, b, and c).

The gradient of the surface is given by the vector: ∇f = (a, b, c) = (y + 15, x, 2z) at the point P(x, y, z), which is a point on the surface.

Here, the normal vector is ∇f = (y + 15, x, 2z).

Now, let's consider a point A on the surface which is closest to the origin.

Let the coordinates of A be (a, b, c).

Therefore, the position vector of A is given by: OA = ai + bj + ck.

The direction of the position vector is in the direction of the normal vector, and therefore: OA is parallel to ∇f.

Thus, we can write: OA = λ∇f = λ(y + 15)i + λxj + 2λzkWhere λ is a scalar.

Since A lies on the surface, we have: a*b + 15a + c^2 = 209.

We also know that OA passes through the origin.

Therefore, the position vector of A is perpendicular to the direction vector OA.

This gives us: OA·OA = 0⟹ (ai + bj + ck)·(λ(y + 15)i + λxj + 2λzk) = 0

Simplifying this equation gives us:aλ(y + 15) + bλx + c(2λz) = 0

Also, we know that OA passes through the origin.

Therefore, the magnitude of OA is equal to the distance of A from the origin.

Hence, we can write: |OA| = √(a^2 + b^2 + c^2)

The value of λ can be obtained from the equation: aλ(y + 15) + bλx + c(2λz) = 0orλ = -2cz / (b + a(y + 15))

Substituting this value of λ in OA, we get: OA = λ(y + 15)i + λxj + 2λzk= -2cz/(b + a(y + 15)) (y + 15)i - 2cz/(b + a(y + 15)) xj - 4cz^2/(b + a(y + 15))k

Substituting this value of λ in |OA|, we get: |OA| = √[(2cz/(b + a(y + 15)))^2 + (2cz/(b + a(y + 15)))^2 + (4cz^2/(b + a(y + 15)))^2] = 2cz√[(y + 15)^2 + x^2 + 4z^2] / |b + a(y + 15)|

The distance of the point A from the origin is |OA|, which is minimized when the denominator is maximized. The denominator is given by |b + a(y + 15)|.

Thus, we have to maximize the denominator with respect to a and b. The condition for maximum value of the denominator is obtained by differentiating the denominator with respect to a and b separately and equating it to zero. The values of a and b obtained from these equations are substituted in the equation a*b + 15a + c^2 = 209 to obtain the coordinates of the point A, which is closest to the origin.

Hence, we can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.

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The value of [tex]\(r\)[/tex] is [tex]\(r = -2\)[/tex], according to the given cartesian points.

To find the value of [tex]\(r\)[/tex], we can use the slope-intercept form of a linear equation, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope of the line. Given that the line has a slope of [tex]\(-10\)[/tex] and passes through the cartesian points [tex]\((4, 8)\)[/tex]and \[tex]((5, r)\)[/tex], we can calculate the slope as follows:

[tex]\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{r - 8}}{{5 - 4}} = r - 8\][/tex]

Since the slope is [tex]\(-10\)[/tex], we can equate it to the calculated slope and solve for [tex]\(r\)[/tex]:

[tex]\[-10 = r - 8\][/tex]

Simplifying the equation, we have:

[tex]\[r - 8 = -10\][/tex]

Adding [tex]\(8\)[/tex] to both sides, we get:

[tex]\[r = -10 + 8\][/tex]

Therefore, the value of [tex]\(r\)[/tex] is [tex]\(r = -2\)[/tex].

In conclusion, the value of [tex]\(r\)[/tex] in the line with a slope of [tex]-10[/tex] passing through the points [tex](4, 8)[/tex] and [tex](5, \(r\))[/tex] is [tex]\(r = -2\)[/tex]. This satisfies the equation and represents the y-coordinate of the second point. This value of [tex]r[/tex] indicates that the second point lies on the line with a slope of -[tex]10[/tex] passing through ([tex]4,8[/tex]).

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