100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!

The vector function that represents the curve of intersection is: r(t) = [x(t), y(t), z(t)] =[tex][t, 4t^2, 5t^2 + 32t^4][/tex]

To find a vector function that represents the curve of intersection between the paraboloid and the cylinder, we need to express the coordinates (x, y, z) in terms of a parameter t.

Let's start by expressing the cylinder equation in terms of x and y:

y =  [tex]4x^2[/tex]:

We can rewrite this as:

y - 4x^2 = 0

Now, we'll substitute this expression for y in the equation of the paraboloid:

z =[tex]5x^2 + 2y^2[/tex]

Replacing y with [tex]4x^2[/tex]:

[tex]z = 5x^2 + 2(4x^2)^2\\z = 5x^2 + 32x^4[/tex]

Now we have the equations for x and z in terms of t:

x = t

z = 5t^2 + 32t^4

To obtain the y-coordinate, we substitute the x value into the equation of the cylinder:

y =  [tex]4x^2[/tex]:

y =[tex]4t^2[/tex]

Therefore, the vector function that represents the curve of intersection is: r(t) = [x(t), y(t), z(t)] =[tex][t, 4t^2, 5t^2 + 32t^4][/tex]

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The equation of an ellipse is  $80x^2 + 160x + 591 = 0$. with a slope of $2

In the given equation $9x^2 + 16y^2 = 144$,

The coefficients of $x^2$ and $y^2$ are different, indicating that the ellipse is not a perfect circle.

If the line passing through the points $(0,5)$ and $(1,7)$.We can determine the slope by plugging the points

Slope = (y₂ -y₁)/(x₂-x₁)

Slope = (7-5)/1-0)

Slope = 2/1 = $2

Using the slope intercept form we determine the equation as .

$y = 2x + 5$.

By way of substitution into  the equation of the line into the equation of the ellipse, we have

$9x^2 + 16(2x + 5)^2 = 144$.

Expand to get

$80x^2 + 160x + 591 = 0$.

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The mathematics SAT score representing the top 1% of all scores is approximately 780.

a. The random variable in this case is the mathematics SAT score.

b. To find the probability that a person has a mathematics SAT score over 700, we need to calculate the z-score first.

The z-score is calculated as [tex]\frac{(X - \mu )}{\sigma}[/tex],

where X is the value we're interested in, μ is the mean, and σ is the standard deviation.

In this case, X = 700, μ = 514, σ = 117.

Using the formula, the z-score is [tex]\frac{(700 - 514)}{117 } = 1.59[/tex].

To find the probability associated with this z-score, we can consult a standard normal distribution table or use a calculator.

The probability is approximately 0.0564 or 5.64%.

c. To find the probability that a person has a mathematics SAT score of less than 400, we again calculate the z-score using the same formula.

X = 400, μ = 514, and σ = 117.

The z-score is [tex]\frac{(400 - 514) }{117 } = -0.9744[/tex].

Looking up the probability associated with this z-score, we find approximately 0.1635 or 16.35%.

d. To find the probability that a person has a mathematics SAT score between 500 and 650, we need to calculate the z-scores for both values.

Using the formula, the z-score for 500 is [tex]\frac{(500 - 514)}{117 } = -0.1197[/tex],

and the z-score for 650 is [tex]\frac{(650 - 514)}{117 } = 1.1624[/tex].

We can then find the area under the normal curve between these two z-scores using a standard normal distribution table or calculator.

Let's assume the probability is approximately 0.3967 or 39.67%.

e. To find the mathematics SAT score that represents the top 1% of all scores, we need to find the z-score corresponding to the top 1% of the standard normal distribution.

This z-score is approximately 2.33.

We can then use the z-score formula to calculate the corresponding SAT score.

Rearranging the formula,

[tex]X = (z \times \sigma ) + \mu[/tex],

where X is the SAT score, z is the z-score, μ is the mean, and σ is the standard deviation.

Substituting the values,

[tex]X = (2.33 \times 117) + 514 = 779.61[/tex].

Rounded to the nearest whole number, the mathematics SAT score representing the top 1% of all scores is approximately 780.

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The length of the fence around the enclosure not including the gates is:2l + 2w + 8 m

To find the length of the fence around the enclosure, we need to first find the perimeter of the rectangle and then subtract the combined length of the two gates from it.

Let's assume the length of the rectangle is 'l' and the width is 'w'.

From the given data, we know that each gate is 4 m wide.

Therefore, the width of the rectangle is:

Width = w + (4 m + 4 m) = w + 8 m

The perimeter of the rectangle is:

P = 2l + 2(w + 8 m) = 2l + 2w + 16 m

Now, we need to subtract the combined length of the two gates from the perimeter:

P - 2 × 4 m = 2l + 2w + 16 m - 8 m = 2l + 2w + 8 m

So, the length of the fence around the enclosure not including the gates is:2l + 2w + 8 m

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The area of the rectangle is 315/8 square feet.

To find the area of a rectangle, you need to multiply its length by its width. In this case, the length is given as [tex]$7 \frac{1}{2}$[/tex] ft and the width is given as [tex]$5 \frac{1}{4}$[/tex] ft. To work with mixed numbers, it's helpful to convert them to improper fractions.

[tex]$7 \frac{1}{2}$[/tex] ft can be written as an improper fraction as [tex]\((2 \cdot 7 + 1) / 2 = \frac{15}{2}\)[/tex] ft.

[tex]$5 \frac{1}{4}$[/tex] ft can be written as an improper fraction as [tex]\((4 \cdot 5 + 1) / 4 = \frac{21}{4}\)[/tex] ft.

Now, we can calculate the area by multiplying the length and width:

Area = [tex]\(\left(\frac{15}{2} \text{ ft}\right) \times \left(\frac{21}{4} \text{ ft}\right) = \frac{15 \times 21}{2 \times 4} \text{ ft}^2\)[/tex]

To multiply fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together:

Area = [tex]\(\frac{15 \times 21}{2 \times 4} \text{ ft}^2 = \frac{315}{8} \text{ ft}^2\)[/tex]

Therefore, the area of the rectangle is 315/8 square feet.

Note: The complete question is:

Find the area of the rectangle shown.

[tex]$7 \frac{1}{2}$[/tex] ft is the length of the rectangle and [tex]$5 \frac{1}{4}$[/tex] ft is the breadth of the rectangle.

The figure has been attached.

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The linear function that has the same y-intercept as the given graph is the equation y = -2x + 10, corresponding to option 3.

To determine the linear function with the same y-intercept as the graph, we need to find the equation of the line passing through the points (3, 4) and (5, 0).

First, let's find the slope of the line using the formula:

slope (m) = (change in y) / (change in x)

m = (0 - 4) / (5 - 3)

m = -4 / 2

m = -2

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line:

y - y1 = m(x - x1)

Using the point (3, 4) as our reference point, we have:

y - 4 = -2(x - 3)

Expanding the equation:

y - 4 = -2x + 6

Simplifying:

y = -2x + 10

Now, let's check the given options to find the linear function with the same y-intercept:

Option 1: The table with x-values (-3, -1, 1, 3) and y-values (-4, 2, 8, 14)

The y-intercept is not the same as the given line. So, this option is not correct.

Option 2: The table with x-values (-4, -2, 2, 4) and y-values (-26, -18, -2, 6)

The y-intercept is not the same as the given line. So, this option is not correct.

Option 3: The table with x-values (-5, -3, 3, 5) and y-values (-15, -11, 1, 5)

The y-intercept is the same as the given line (10). So, this option is correct.

Option 4: The table with x-values (-6, -4, 4, 6) and y-values (-26, -14, 34, 46)

The y-intercept is not the same as the given line. So, this option is not correct.

Therefore, the linear function that has the same y-intercept as the given graph is the equation y = -2x + 10, corresponding to option 3.

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The cardinal number of the set is 5.

The cardinal number of the given set A = {13, 19, 25, 31, 39} is 5.

Cardinal number is the number of elements present in a set, so counting the number of elements of the set will give us the cardinal number of the set.

In this case, we have the set A with 5 elements: 13, 19, 25, 31, and 39.

Thus, the cardinal number of the set is 5.

Hence, the answer is 5.

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The length of the major axis is: 2√{18 - 6√5}

The foci of an ellipse are usually denoted by the coordinates:

(c, 0) and (-c, 0)

where c is the distance from the center of the ellipse to each focus.

We are given the coordinates of the foci as:

(2, -3 + √5) and (4,-8).

Let us now find the distance between the two foci:

d = √{(2 - 4)² + (-3 + √5 + 8)²}

d = √{4 + (√{5} - 3)² }

d =  √{18 - 6√5}

The ellipse is tangent to the x-axis, and as such the major axis of the ellipse will be parallel to the x-axis. The vertices lie on the x-axis.

The length of the major axis is given by2a,

where a is the distance from the center of the ellipse to either vertex.

The distance between the two vertices is 2a = 2c, which is twice the distance between the center and either focus.

Therefore, the length of the major axis is:

2a = 2d = 2√{18 - 6√5}

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Sample standard deviation = 51.03

Sample variance = 2604.33

Mean (average):

[tex]= (107.5 + 108.2 + 133.6 + 151.4 + 155.8 + 149 + 106.11 + 16.1 + 181 + 165.7 + 130.6 + 126.5 + 109.5 + 139.6 + 139.4 + 140.8 + 143.2 + 132.7 + 171 + 115.7) / 20\\= 1894.28 / 20\\= 94.714[/tex]

Squared differences from the mean:

[tex](107.5 - 94.714)^2 = 161.39\\(108.2 - 94.714)^2 = 182.14\\(133.6 - 94.714)^2 = 1507.99\\(151.4 - 94.714)^2 = 3217.15\\(155.8 - 94.714)^2 = 3750.18\\(149 - 94.714)^2 = 2922.18\\[/tex]

[tex](106.11 - 94.714)^2 = 130.46\\(16.1 - 94.714)^2 = 6659.49\\(181 - 94.714)^2 = 7334.04\\(165.7 - 94.714)^2 = 5041.45\\\\(130.6 - 94.714)^2 = 1281.47\\(126.5 - 94.714)^2 = 1007.35\\(109.5 - 94.714)^2 = 217.75\\(139.6 - 94.714)^2 = 1997.15\\(139.4 - 94.714)^2 = 1990.18\\(140.8 - 94.714)^2 = 2108.42\\(143.2 - 94.714)^2 = 2353.86\\(132.7 - 94.714)^2 = 1425.68\\(171 - 94.714)^2 = 5852.45\\(115.7 - 94.714)^2 = 441.51[/tex]

Sum of squared differences:

= 161.39 + 182.14 + 1507.99 + 3217.15 + 3750.18 + 2922.18 + 130.46 + 6659.49 + 7334.04 + 5041.45 + 1281.47 + 1007.35 + 217.75 + 1997.15 + 1990.18 + 2108.42 + 2353.86 + 1425.68 + 5852.45 + 441.51

= 49482.29.

Sample variance:

s^2 = Sum / (n-1)

= 49482.29 / (20-1)

= 2604.33105263

= 2604.33

Sample standard deviation:

s = sqrt(s^2)

s = sqrt(2604.33)

s = 51.0326366162

s = 51.03

Fulls:

An author is doing research on an upcoming book on skyscrapers in America. The author gathers data for 20 skyscrapers throughout America and decides to focus on the height data, given in meters, which are reproduced below.

Height of skyscraper (meters)

107.5

108.2

133.6

151.4

155.8

149

106.1

116.1

181

165.7

130.6

126.5

109.5

139.6

139.4

140.8

143.2

132.7

171

115.7

Calculate the sample standard deviation and the sample variance.

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The GPA of a student who studies for 15 hours a week is 3.125 GPA.

In Mathematics, the standard form of the equation of a regression line is represented or modeled by the following mathematical expression;

y = bx + c

Where:

Since the student would study for 15 hours a week, the predicted GPA can be calculated as follows;

y = 0.149x + 0.89

y = 0.149(15) + 0.89

y = 3.125 GPA.

In conclusion, we can logically deduce that a student who studies for 15 hours a week would most likely have a 3.125 GPA.

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Complete Question:

Using your equation from step 2d, estimate the GPA of a student who studies for 15 hours a week. Justify your answer. The equation for the predicted GPA is y=0.149x+0.89

Answer:

to the nearest thousand, 5829 is 6000

Step-by-step explanation:

We have to round 5829 to the nearest thousand,

We have,

Thousand : Hundred : Ten : Unit

     5          :       8        :    2   :   9

Now, we have an 8 in the hundred position, since 8 > 5, we have to round up ( if the number at the hundred position was less than 5, we would have left the number to the left as is)

Now, we have 5 in the thousand position, due to rounding, we get,

5+1 = 6

so, to the nearest thousand, 5829 is 6000

Therefore, 22 + 4x(3 - 1) simplifies to 30.
To simplify the expression 22 + 4x(3 - 1), we can apply the order of operations (PEMDAS/BODMAS) which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let's simplify the expression step by step:

Evaluate the expression within the parentheses:

3 - 1 = 2

Now the expression becomes: 22 + 4x2

Perform the multiplication:

4 x 2 = 8

Now the expression becomes: 22 + 8

Finally, perform the addition:

22 + 8 = 30

Therefore, 22 + 4x(3 - 1) simplifies to 30.

Answer:

x=9

Step-by-step explanation:

7x9+13x9=180

7x Is equivalent to that angle that is under 13x, which means that if we add both of those together we should get 180.

The domain of the function graphed in this problem is given as follows:

C. {x | x  ≠ -2, 3}.

The function is this problem is defined for all real values except x = -2 and x = 3, hence the domain is given as follows:

C. {x | x  ≠ -2, 3}.

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In a dilation, the size of the object can be changed while keeping its shape, which is the case in this scenario.

A transformation of rectangle ABCD results in rectangle A’B’C’D. We need to identify which transformation maps the pre-image to the image.Two-dimensional transformations can be classified into four main categories; translations, reflections, rotations, and dilations. We will evaluate each of these to determine which of them maps the pre-image to the image.

TranslationsA translation is a transformation that moves an object in a straight line without changing its shape or orientation. It involves moving an object by a fixed distance in a particular direction. The image is congruent to the pre-image. This transformation does not result in a rectangle ABCD being transformed to A’B’C’D.

It results in a rectangle that has the same size and shape as the pre-image being moved to a different location.ReflectionsA reflection is a transformation that flips an object across a line without changing its shape or orientation. A reflection through a line known as the x-axis, or the y-axis results in a mirror image of the object.

This transformation does not result in a rectangle ABCD being transformed to A’B’C’D.RotationsA rotation is a transformation that turns an object about a fixed point. The fixed point is known as the center of rotation. The angle through which the object is rotated is known as the angle of rotation. A rotation does not change the size or shape of an object, but it changes its orientation. This transformation does not result in a rectangle ABCD being transformed to A’B’C’D.

DilationsA dilation is a transformation that changes the size of an object. The size change is described by a scale factor. The scale factor is greater than 1 for an enlargement and between 0 and 1 for a reduction. The center of dilation is the point about which the object is enlarged or reduced. This transformation does not result in a rectangle ABCD being transformed to A’B’C’D.After evaluating each of the four types of transformations, we can conclude that the transformation that maps the pre-image rectangle ABCD to the image rectangle A’B’C’D is a dilation.

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Reason: The two line segments CF and AH point in the same direction, and they never intersect. Therefore, we consider them parallel.

Skew lines are ones that point in different directions. An example of a pair of skew lines would be AB and DE. Skew lines never intersect, but we don't consider them parallel.

After 8 hours, the hire costs will be the same for Connie and her friends from Ace and David and his brother from Best.

To find the number of hours when the hire costs will be the same for Connie and her friends from Ace and David and his brother from Best, we need to set up an equation.

Let x represent the number of hours.

For Ace Bicycle Shop, the cost is given by C_Ace[tex](x) = 4 + 1x[/tex], where 4 is the flat fee and 1x represents the hourly charge.

For Best Bicycle Shop, the cost is given by C_Best[tex](x) = 8 + 0.5x[/tex], where 8 is the flat fee and 0.5x represents the hourly charge.

To find the number of hours when the costs are the same, we equate the two equations:

[tex]4 + 1x = 8 + 0.5x[/tex]

Subtracting 0.5x from both sides, we get:

[tex]0.5x = 4[/tex]

Dividing both sides by 0.5, we find:

[tex]x = 8[/tex]

Therefore, after 8 hours, the hire costs will be the same for Connie and her friends from Ace and David and his brother from Best.

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Given a jar of candy that has a certain number of pieces of candy. The given data also tells that a total of 16 pieces are excluded from each color except yellow, where 18 pieces are excluded. The objective is to determine the number of pieces of each color and the total number of candy pieces.Except for 16 pieces, all the candy is red.

It means that there are 16 red candies in the jar that are not included in this count. Let us assume that there are "r" red candies in the jar.Therefore, the total number of red candies is:r + 16.

Except for 16 pieces, all candy is blue.Similarly, we can assume that there are "b" blue candies in the jar.Therefore, the total number of blue candies is:b + 16.

Except for 16 pieces, all candy is green.Likewise, we can assume that there are "g" green candies in the jar.So, the total number of green candies is:g + 16.

Except for 18 pieces, all candy is yellow.We can assume that there are "y" yellow candies in the jar.Therefore, the total number of yellow candies is:y + 18.

The total number of candy pieces in the jar can be obtained by adding all the individual pieces of each color. That is:Total candy pieces

= (r + 16) + (b + 16) + (g + 16) + (y + 18)

Total candy pieces

= r + b + g + y + 66

Now, we will solve this system of equations using the values obtained above

:r + b + g = y + 50y = r + b + g + 18

We need to substitute the second equation in the first equation,

r + b + g = r + b + g + 18 + 50r + b + g = r + b + g + 68

This simplifies to 50 = 68, which is not possible.

Therefore, the given information is inconsistent.

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Answer:

x-intercept(s):  (6,0)

y-intercept(s):  (0,−9)

Step-by-step explanation:

To find the x-intercept, substitute in 0 for y and solve for x. To find the y-intercept, substitute in 0 for x and solve for y.

Hello!

We will go tu use the pythagorean theorem!

So:

BA² = BC² + AC²

AC² = BA² - BC²

AC² = 52² - 20²

AC² = 2304

AC = √2304

AC = 48

Answer:

Step-by-step explanation:

considering A). hemisphere diameter = 48 yd

volume of hemisphere = (2/3)*pi*(r)^3

the radius of hemisphere = diameter/2 = 48/2 = 24 yd

hence, the volume of hemisphere = (2/3)*pi*(24)^3  yd^3

taking pi = 3.14

volume = 28938.24 yd^3 approximately volume = 28940 yd^3

considering B). Circumference of a great circle = 26 m

circumference of sphere = 2*pi*r

therefore r = 4.14 m

volume of sphere = (4/3)*pi*(r)^3

volume of sphere = 297.077 m^3

Answer:

24. A. 28952.9 yd²

B. 297.2 m³

Step-by-step explanation:

24.

A.

diameter=48 yd

radius (r)=diameter/2=48/2=24yd

We have

Volume of Hemisphere= ⅔*πr³=⅔*π*r³

Substituting value

Volume of Hemisphere=⅔*π*24³=28952.9 yd³

B.

Circumference of great circle=2πr=26m

2πr=26

r=26/(2π)

r=4.14

Now

Volume of sphere = 4/3*πr³

Volume of sphere=4/3*π*4.14³=297.2 m³

Using this points we can use to plot the graph:

(x, f(x))

(0, 20)

(1, 16)

(2, 12.8)

(3, 10.24)

(4, 8.192)

(5, 6.5536)

To graph the function [tex]f(x) = 20 \times 0.8^x,[/tex] we can plot some points and connect them to create the graph.

Here are some points we can use to plot the graph:

(x, f(x))

(0, 20)

(1, 16)

(2, 12.8)

(3, 10.24)

(4, 8.192)

(5, 6.5536)

Using these points, we can plot them on a coordinate system and connect them with a smooth curve to obtain the graph of the function [tex]f(x) = 20 \times 0.8^x,[/tex]

Note: The x-values used in this example are just for illustration purposes. Depending on the range and precision desired, more points can be plotted to get a more accurate representation of the graph.

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The fraction of the plot of land not occupied by the flowers is 0.0625 or 1/16.

Let's calculate the fraction of the plot of land that is not occupied by the flowers.

Given that initially, 1/4 of the land is allocated for orchids, we have 1 - 1/4 = 3/4 of the land remaining.

After planting the orchids, 2/3 of the remaining land is allocated for roses. Therefore, the fraction of land allocated for roses is (2/3) * (3/4) = 2/4 = 1/2.

Subtracting the land allocated for roses from the remaining land, we have 3/4 - 1/2 = 1/4 of the land remaining.

Finally, 0.75 of the remaining land is allocated for tulips. Therefore, the fraction of land allocated for tulips is 0.75 * (1/4) = 0.1875.

To find the fraction of the plot of land not occupied by the flowers, we subtract the fractions of land allocated for flowers from 1:

1 - (1/4 + 1/2 + 0.1875) = 1 - 0.9375 = 0.0625.

Therefore, the fraction of the plot of land not occupied by the flowers is 0.0625.

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Answer:

41.93 degrees

Step-by-step explanation:

Answer:(x^3/y^5)

Step-by-step explanation: Firstly write y to the negative fifth power as y^-5 and then write x to the negative third power as x^-3...After that write accordingly to the question that is...

y^-5/x^-3

which leads to gives us as:

(x^3/y^5)

Answer:

w = 120

x = 60

y = 120

z = 60

Step-by-step explanation:

w = 120 (vertically opposite angles)

sum of co interior angles is 180

⇒ w + x = 180 and x + y = 180

w + x = 180

⇒ 120 + x = 180

⇒ x = 180 - 120

⇒  x = 60

x + y = 180

⇒  60 + y = 180

⇒   y = 180 - 60

⇒   y = 120

z = x (corresponding angles)

z = 60

A residual plot is a scatterplot in which the residuals (vertical distances between the predicted and actual values) are plotted against the independent variable. A residual is defined as the difference between the predicted value (based on the regression equation) and the actual value.

Residual plots are a valuable tool for checking the adequacy of the model. It helps us check whether the assumptions of linearity, independence, equal variance, and normality are met or not.

The most basic way to create a residual plot is to plot the residuals against the fitted values. If the points in the residual plot are randomly scattered around the horizontal axis, then the assumption of linearity has been met.

If the points show a pattern, such as a curved line, then the assumption of linearity has been violated.To create a residual plot, follow these steps:

Step 1: Estimate the regression equation and obtain the predicted values (ŷ) and residuals (e). ŷ = b0 + b1X

Step 2: Plot the residuals on the vertical axis and the independent variable (X) on the horizontal axis

.Step 3: Look for patterns in the residual plot. If the points are randomly scattered around the horizontal axis, then the assumptions of linearity, independence, equal variance, and normality are met. If there is a pattern, such as a curved line, then the assumptions have been violated. A residual plot can be used to detect outliers, influential observations, and nonlinearity.

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The line parallel to the x-axis passing through the points (5,5) and (-5,5) has a slope of 0, indicating a horizontal line with a constant y-coordinate value of 5.

To determine the slope of the line that represents this relationship, we can use the formula for slope, which is given by:

slope = (change in y-coordinates) / (change in x-coordinates)

In this case, we are given two points on the line: (5,5) and (-5,5).

The change in y-coordinates is 5 - 5 = 0, as the y-coordinate remains constant.

The change in x-coordinates is -5 - 5 = -10.

Substituting these values into the slope formula, we get:

slope = 0 / -10 = 0

Therefore, the slope of the line that represents this relationship is 0.

A slope of 0 indicates that the line is parallel to the x-axis. This means that the line has a constant y-coordinate value for all x-coordinate values. In this case, the line passes through the point (5,5) and (-5,5), and it remains at y = 5 for all x-values.

Visually, a line with a slope of 0 would be a horizontal line on the coordinate plane. It does not have an upward or downward slope but remains parallel to the x-axis.

It's important to note that the slope of 0 indicates a relationship where the dependent variable (y) does not change with respect to the independent variable (x). In this case, no matter the value of x, the corresponding y-value remains constant at 5.

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A point on the scatter plot is defined as (19, 3), which means that an input of 19 is mapped to an output of 3.

The scatter plot is built inserting all the points of the table in a graph, which are in the following input-output format:

(Input, Output).

One point on the scatter plot for this problem has the coordinates given as follows:

(19, 3).

Hence the meaning is that an input of 19 is mapped to an output of 3.

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