Let R be the region in the first quadrant bounded by the graphs of y = sqrt of x and y = x/3
a. Find the area of R
b. Find the volume of the solid generated when R is rotated about the vertical line x = -1
c. The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are squares. Find the volume of the solid.

Therefore, the correlation coefficient is -2.19.

To calculate the correlation coefficient, we first need to find the mean and standard deviation of both sets of data.

Weight:

Mean = (3175 + 3450 + 3225 + 3985 + 2440 + 2500 + 2290) / 7 = 2995

Standard Deviation = 633.72

Fuel Consumption:

Mean = (27 + 29 + 27 + 24 + 37 + 34 + 37) / 7 = 30

Standard Deviation = 5.73

Next, we need to find the covariance between the two sets of data.

Covariance = Σ[(weight - mean weight) * (fuel consumption - mean fuel consumption)] / (n - 1)

= [(3175-2995)(27-30) + (3450-2995)(29-30) + (3225-2995)(27-30) + (3985-2995)(24-30) + (2440-2995)(37-30) + (2500-2995)(34-30) + (2290-2995)*(37-30)] / 6

= -84317.62

Finally, we can calculate the correlation coefficient using the formula:

Correlation Coefficient = Covariance / (Standard Deviation of Weight * Standard Deviation of Fuel Consumption)

= -84317.62 / (633.72 * 5.73)

= -2.19

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The radius of the spherical container should be approximately 2.87941 inches to hold a volume of 100 cubic inches.

A sphere is simply a three-dimensional geometric object that is perfectly symmetrical in all directions.

Given the formula for volume of sphere in the question:
V = 4.18879r³

If a spherical container has a volume of 100 cubic inches, the we can calculate its radius using the above formula.

V = 4.18879r³

r³ = V/4.18879

r = ∛( V/4.18879 )

Plug in the volume of the container:

r = ∛( 100in³ / 4.18879 )

r = 2.87941 in

Therefore, the radius is approximately 2.87941 inches.

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The charity should use the range as a measure of variability to accurately represent the data. Range is the difference between the largest and smallest values in a dataset. So, the range as a measure of variability is appropriate for this dataset because it gives a sense of how spread out donations are from the smallest to the largest.

In this case, the largest donation was $55 and the smallest was $10, so the range is $55 - $10 = $45. Using the range as a measure of variability is appropriate for this dataset because it gives a sense of how spread out the donations are from the smallest to the largest.

It is also easy to calculate and understand, which makes it useful for reporting to donors and stakeholders.

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We can then perform row reduction to determine the rank of the matrix. After performing row reduction, we can see that the rank of the matrix is 2, which is less than the dimension of R^3, which is 3. Therefore, the two vectors in W do not span R^3, and so W is not a basis for R^3.

First, let's check for linear independence. We can set up the following equation:

c1[-2,3,0] + c2[6,-1,5] = [0,0,0]

This gives us the system of equations:

-2c1 + 6c2 = 0

3c1 - c2 = 0

5c2 = 0

The last equation tells us that c2 must be 0, which means the first equation simplifies to c1 = 0. Substituting these values into the second equation gives us 0 = 0, which is always true. This tells us that the only solution to the system is c1 = c2 = 0, meaning that W is linearly independent.

Next, let's check if W spans R^3. We can do this by checking if any vector in R^3 can be written as a linear combination of the two vectors in W. That is, we need to solve the equation:

c1[-2,3,0] + c2[6,-1,5] = [x,y,z]

This gives us the system of equations:

-2c1 + 6c2 = x

3c1 - c2 = y

5c2 = z

Solving for c1, c2, and z, we get:

c2 = z/5

c1 = (y + c2)/3 = (y + z/5)/3

x = -2c1 + 6c2 = -(2/3)(y + z/5) + 6z/5 = (-2/3)y + (22/15)z

This means that any vector [x,y,z] in R^3 can be written as a linear combination of the vectors in W. Therefore, W spans R^3.

Since W is both linearly independent and spans R^3, it is a basis for R^3. Therefore, the correct answer is A. W is a basis.

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To determine if a distribution is normal, you can employ several methods:

Histogram: Plot the data on a histogram and visually inspect its shape. A normal distribution typically has a symmetric bell-shaped curve, with the majority of the data points concentrated around the mean.

Quantile-Quantile (Q-Q) Plot: A Q-Q plot compares the quantiles of your data to the quantiles of a theoretical normal distribution. If the points on the plot roughly follow a straight line, it suggests that the data is normally distributed.

Skewness and Kurtosis: Calculate the skewness and kurtosis of the data. For a normal distribution, the skewness should be close to zero and the kurtosis should be close to three. Deviations from these values indicate departures from normality.

Shapiro-Wilk Test: Conduct a statistical test, such as the Shapiro-Wilk test, which tests the null hypothesis that the data comes from a normal distribution. If the p-value is above a predetermined significance level (e.g., 0.05), you fail to reject the null hypothesis and conclude that the data is normally distributed.

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Assuming that the size of the stack is constant, we would need to know the number of cans that can be stacked in order . Since the number of cans is evenly divisible by the number of cans per stack, there would be zero cans left over from the stack.

Assuming that the size of the stack is constant, we would need to know the number of cans that can be stacked in order to determine the answer. However, let's assume that a single stack can hold 10 cans. If Selene has 90 cans, then we can divide the total number of cans by the number of cans per stack to determine how many stacks are needed.

90 cans / 10 cans per stack = 9 stacks

Therefore, Selene would need 9 stacks to hold all 90 cans. However, there would be some cans left over from the last stack.

9 stacks x 10 cans per stack = 90 cans

90 cans - 90 cans = 0 cans

Since the number of cans is evenly divisible by the number of cans per stack, there would be zero cans left over from the stack.

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[tex]\begin{array}{cllll} -6x+5y&=&34\\ -6x-10y&=&4 \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{ \textit{using elimination method} }{\begin{array}{cllll} \text{\LARGE -1}(-6x+5y&=&34)\\ -6x-10y&=&4 \end{array}}\implies \begin{array}{clclll} 6x ~~ -5y&=&-34\\ -6x-10y&=&4\\\cline{1-3} 0 ~~ -15y&=&-30 \end{array} \\\\\\ -15y=-30\implies y=\cfrac{-30}{-15}\implies y=2 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{substituting on the 1st equation}}{-6x+5(2)=34\implies} -6x+10=34\implies x=\cfrac{24}{-6}\implies x=-4 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill (-4~~,~~2)~\hfill[/tex]

The answer is 2.6825  cm, which is not one of the given options. Therefore, the correct answer is E. none of these.

The bulk modulus, B is defined as:

B = (P / ΔV / V), where P is the pressure applied, ΔV is the change in volume, and V is the original volume.

For a cube, the change in volume, ΔV is related to the change in length, ΔL by:

[tex]\sf \Delta V = \Delta L^3[/tex].

Given that the cube has 2.0 cm sides, the original volume, V is:

[tex]\sf V = (2.0 \ cm)^3 = 8.0 \ cm^3[/tex].

The pressure applied is [tex]\sf P = 2.0 \times 10^5[/tex] Pa and the bulk modulus is [tex]\sf B = 4.7 \times 10^5 \ N/m^2[/tex].

We can rearrange the bulk modulus formula to solve for ΔL as:

[tex]\sf \Delta L = \huge \text(\dfrac{P}{B} \huge \text) \times \dfrac{V}{3}[/tex].

Substituting the values, we get:

[tex]\sf \Delta L = (2.0 \times 10^5 \ \dfrac{Pa}{4.7} \times 10^5\ N/m^2) \times \dfrac{8.0 \ cm^3}{3} = 0.6825 \ cm[/tex]

Therefore, the final length of any side of the cube is:

[tex]\sf L = 2.0 \ cm + \Delta L = 2.0 \ cm + 0.6825 \ cm = 2.6825 \ cm[/tex]

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Complete Question:

A cube with 2.0-cm sides is made of material with a bulk modulus of4.7 × 10^5 N/m2. When it is subjected to a pressure of 2.0 × 10^5 Pa the length of its any of its sides is:

A. 0.85 cm

B. 1.15 cm

C. 1.66 cm

D. 2.0 cm

E. none of these

Answer:

[tex]( { {(x + 4)}^{2}) }^{3} = {(x + 4)}^{6} [/tex]

The standard deviation of the total number of pages the person will read is approximately 179.3.

To find the standard deviation of the total number of pages the person will read, we need to consider that the number of pages in the romance novel and the number of pages in the detective novel are independent random variables.

The variance of the sum of two independent random variables is equal to the sum of their variances. Therefore, the variance of the total number of pages is the sum of the variances of the romance novel and the detective novel.

For the romance novel:

Mean = 364

Standard Deviation = 47

For the detective novel:

Mean = 404

Standard Deviation = 173

Variance of the total number of pages = Variance of romance novel + Variance of detective novel

Variance = (47^2) + (173^2)

Standard Deviation = √Variance = √(47^2 + 173^2) ≈ 179.3 (rounded to one decimal place)

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Answer: 33.6

Step-by-step explanation:

Answer:

33.6

Step-by-step explanation:

First, let's multiply 8.4 by 100.

This gives us the number 840.

Next, divide 840 by 25.

That gives us 33.6, our final answer.

Since you cannot have a fraction of a sample, you should round up to the nearest whole number. Therefore, a sample size of 542 is required to be 99.9% confident that your estimate is within 5% of the true population proportion.

To estimate a population proportion with a high degree of confidence (99.9%) and a margin of error (5%), you need to determine an appropriate sample size. Since you have no reasonable estimate for the population proportion, it's common to use 0.5 as a conservative estimate to ensure the largest possible sample size.
For this calculation, you can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where n is the sample size, Z is the Z-score associated with the desired confidence level (99.9%), p is the estimated population proportion (0.5), and E is the margin of error (0.05).
For a 99.9% confidence level, the Z-score is 3.291. Plugging the values into the formula:
n = (3.291^2 * 0.5 * (1-0.5)) / 0.05^2
n ≈ 541.16

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The result of the lines after the translation is given as follows:

They do not intersect.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The vector notation is given as follows:

(g,h).

The meaning is given as follows:

The vector for this problem is given as follows:

(2,1).

Meaning that the lines are shifted 2 units right and one unit down.

Both lines, that do not intersect, are translated, hence the lines continue not intersecting.

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Step-by-step explanation:

To find the area of the whole mosaic, we need to multiply the length by the width. In this case, the mosaic is a 7 x 7 array of square tiles, and each tile is 1 ft long.

Area = Length × Width

Area = 7 ft × 7 ft

Area = 49 square feet

So, the area of the whole mosaic is 49 square feet.

Rounding to the nearest cent, the expected profit for one round of the coin game is approximately -$0.38.

To calculate the expected profit for one round of the coin game, we need to consider the probabilities of winning and losing, as well as the corresponding amounts won or lost.

There are 2 possibilities for winning: either getting all three heads or all three tails. Each of these outcomes has a probability of (1/2) * (1/2) * (1/2) = 1/8.

The amount won in each winning case is $15. Therefore, the expected profit from winning is:

(1/8) * $15 = $1.875

The probability of losing is 1 - probability of winning. In this case, it is 1 - (1/8 + 1/8) = 6/8 = 3/4.

The amount lost in each losing case is $3. Therefore, the expected loss from losing is:

(3/4) * (-$3) = -$2.25

Finally, we can calculate the expected profit by subtracting the expected loss from the expected profit from winning:

$1.875 - $2.25 = -$0.375

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The difference is that tropical rainforests are near the equator while temperate rainforests are found further north near coastal areas. The Option A is correct.

The tropical rainforests are located near equator and spans across regions such as the Amazon Basin in South America, the Congo Basin in Africa and parts of Southeast Asia. These rainforests are characterized by high temperatures and humidity throughout the year with no distinct seasons.

The temperate rainforests are found further north, primarily in coastal areas of temperate regions. Examples include the Pacific Northwest in North America which experience milder temperatures compared to tropical rainforests and have distinct seasonal variations.

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The function csc(theta), which stands for cosecant, is defined as the reciprocal of the sine function: csc(theta) = 1/sin(theta).

In trigonometry, the sine function is undefined when the denominator is zero, which occurs at angles where the sine function equals zero. The sine function equals zero at angles that are multiples of pi (180 degrees), such as 0, pi, 2pi, etc.

Since csc(theta) is the reciprocal of sin(theta), csc(theta) is undefined at angles where sin(theta) equals zero. Therefore, csc(theta) is undefined at theta = 0, pi, 2pi, etc., or in general, at any angle where the sine function equals zero.

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To find the volume of the cube, we need to know the length of one side of the cube. However, since it is not provided in the question, we cannot calculate the exact volume.

We can use a formula to relate the volume of the cube to its surface area. The formula is:

Surface Area = 6 * (side)^2

Given that the surface area of the cube is 294 cm², we can set up the equation:

294 = 6 * (side)^2

Divide both sides of the equation by 6:

49 = (side)^2

Take the square root of both sides:

side = √49

side = 7 cm

Now that we know the length of one side of the cube is 7 cm, we can calculate the volume using the formula:

Volume = side^3

Volume = 7^3

Volume = 343 cm³

Therefore, the volume of the cube is 343 cm³.

If Jason follows the given condition in the given data question then answer will be Jason had 4 nickels.

Let's assume the number of nickels Jason had is represented by the variable 'n.' Since the problem states that there were 3 times as many dimes as nickels, the number of dimes can be represented as 3n. Additionally, the problem states that there were twice as many quarters as dimes, so the number of quarters can be represented as 2(3n) or 6n.

Now, we can create an equation based on the total value of the coins: 5n (nickels) + 10(3n) (dimes) + 25(6n) (quarters) = 37 (total value in cents).

Simplifying the equation, we have 5n + 30n + 150n = 37.

Combining like terms, we get 185n = 37.

Dividing both sides by 185, we find n = 0.2.

Since we can't have a fraction of a nickel, we round down to the nearest whole number.

Therefore, Jason had 4 nickels.

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The dilation of △JKP is centered at P(3, 2) and has a scale factor of 2.

A coordinate rule to represent the dilation is (x, y)  →  (2(x - 3) + 3, 2(y - 2) + 2).

In Mathematics and Geometry, a dilation is a type of transformation which typically changes the size of a geometric object, but not its shape.

Next, we would determine scale factor that was used to dilate △JKP as follows;

Scale factor = side length of image/side length of pre-image

Scale factor = J'K'/JK

Scale factor = 4/2

Scale factor = 2

Now, we can write the coordinate rule that represent the dilation by using a scale factor of 2 centered at the point P (3, 2) by using this mathematical equation:

(x, y)  →  (k(x - a) + a, k(y - b) + b)

(x, y)  →  (2(x - 3) + 3, 2(y - 2) + 2)

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Missing information:

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

Answer:

$19,000

Step-by-step explanation:

Let's start by finding the sum of the prices of all 80 mobile phones:

80 × 30,000 = 2,400,000

Next, we can find the sum of the prices of the remaining 78 mobile phones by multiplying the new average by 78:

78 × 29,500 = 2,301,000

We know that the highest price is $80,000. Let's assume that the lowest price is x dollars. We can set up an equation to solve for x:

2,400,000 - 80,000 - x = 2,301,000

Simplifying the equation:

2,320,000 - x = 2,301,000

Subtracting 2,301,000 from both sides:

19,000 = x

Therefore, the lowest price of the mobile phone is $19,000.

Point A is between the 7th and 8th tick marks. Once you zoom in, you can see that the decimal would be 7.4. So the number could be represented by the expression 7.4 times 10 to the 11th power.

In Geometry and Mathematics, a number line refers to a type of graph that is usually composed of a graduated straight line, which typically comprises both negative and positive numerical values (numbers) that are located at equal intervals along its length.

In this scenario and exercise, we would determine the numerical value (number) that is represented by point A by solving as follows;

10¹² × 7/10 = 7 × 10¹¹

10¹² × 4/10 = 4 × 10¹¹

By adding the numbers, we have:

7 × 10¹¹ + 4/10 × 10¹¹ = 7.4 × 10¹¹

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Missing information:

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

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{2}\\ o=\stackrel{opposite}{9} \end{cases} \\\\\\ c=\sqrt{ 2^2 + 9^2}\implies c=\sqrt{ 4 + 81 } \implies c=\sqrt{ 85 }\implies c\approx 9.2[/tex]

Answer:163.362817987

Step-by-step explanation:

Circumference = (rx2)π

= (26x2)π

= 52π

= 163.362817987cm

a. The probability he hits the target 3 times is 0.2013. b. The probability he hits his target at least twice is 0.9984. c. The probability he hits the target at most 4 times is 0.1074. d. The expected number of shots that hit the target is 8.

a. The probability he hits the target 3 times = (10C3) * (0.8)^3 * (0.2)^7 = 0.2013

b. To calculate the probability he hits his target at least twice, we need to calculate the probability of hitting it twice, hitting it thrice, hitting it four times, ..., hitting it ten times.

So, P(hitting at least twice) = P(hitting twice) + P(hitting thrice) + ... + P(hitting ten times).

Therefore,

P(hitting at least twice) = (10C2) * (0.8)^2 * (0.2)^8 + (10C3) * (0.8)^3 * (0.2)^7 + ... + (10C10) * (0.8)^10 * (0.2)^0 = 0.9984

c. The probability he hits the target at most 4 times = P(hitting 0 times) + P(hitting 1 time) + P(hitting 2 times) + P(hitting 3 times) + P(hitting 4 times)

= (10C0) * (0.8)^0 * (0.2)^10 + (10C1) * (0.8)^1 * (0.2)^9 + (10C2) * (0.8)^2 * (0.2)^8 + (10C3) * (0.8)^3 * (0.2)^7 + (10C4) * (0.8)^4 * (0.2)^6

= 0.1074

d. The expected number of shots that hit the target can be calculated as:

Expected number = 10 * 0.8 = 8

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2. The conversions are as follows: a. (11/9)π, b. -80°/9, and c. 3.5 remains in degrees.

3. The trigonometric functions are: a. - (√2)/2, b. - (√2)/2, and c. -1

4. The possible values of θ between 0° and 360° are 210° and 330°.

5. The possible values of θ between 0° and 2π are 45° and 315°.

2. Converting angles:

(a) 220° to radians:

To convert degrees to radians, multiply the degree measure by π/180.

220° x (π/180) = (22/18)π = (11/9)π

(b) -4π/9 to degrees:

To convert radians to degrees, multiply the radian measure by 180/π.

-4π/9 * (180/π) = -80°/9

(c) 3.5 remains in degrees.

3. Evaluating trigonometric functions:

(a) sin(-225°):

Since sine is an odd function, sin(-θ) = -sin(θ).

sin(-225°) = -sin(225°)

To find the exact value of sin(225°), use the reference angle of 45° in the second quadrant:

sin(225°) = -sin(180° + 45°) = -sin(45°)

sin(45°) = (√2)/2

Therefore, sin(-225°) = - (√2)/2

(b) cos(5π/4):

To find the exact value of cos(5π/4), use the reference angle of π/4 in the third quadrant:

cos(5π/4) = -cos(π/4)

cos(π/4) = (√2)/2

Therefore, cos(5π/4) = - (√2)/2

(c) tan(3π/4):

To find the exact value of tan(3π/4), use the reference angle of π/4 in the second quadrant:

tan(3π/4) = -tan(π/4)

tan(π/4) = 1

Therefore, tan(3π/4) = -1

4. Finding possible values of θ for sin θ = - ¹/₂:

Given that sin θ = - ¹/₂ is true for θ in the third and fourth quadrants.

In the third quadrant, the reference angle is 30°, so θ = 180° + 30° = 210°.

In the fourth quadrant, the reference angle is also 30°, so θ = 360° - 30° = 330°.

Therefore, the possible values of θ between 0° and 360° are 210° and 330°.

5. Finding possible values of θ for cos θ = (√2)/2:

Given that cos θ = (√2)/2 is true for θ in the first and fourth quadrants.

In the first quadrant, the reference angle is 45°, so θ = 45°.

In the fourth quadrant, the reference angle is also 45°, so θ = 360° - 45° = 315°.

Therefore, the possible values of θ between 0° and 2π are 45° and 315°.

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Among the given alternatives, the following option accurately describes the expected frequencies for a chi-square test: They can contain fractions or decimal values.

The answer is B.

A chi-square test is a statistical method used to measure the relationship between two categorical variables. The null hypothesis for a chi-square test is that there is no relationship between the variables.

The expected frequency (EF) is the number of times an outcome is expected to occur during a trial based on probabilities. In a chi-square test, the expected frequencies are calculated based on probabilities and sample sizes.

The expected frequency can be a fraction or a decimal, and it is determined by multiplying the row total and column total of each cell by the overall total and dividing it by the total number of observations.

The chi-square test statistic is calculated by comparing the observed frequencies to the expected frequencies. The test statistic is then compared to a critical value in a chi-square distribution table to determine whether to accept or reject the null hypothesis.

Hence, the answer of the question is B.

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Using the given mean and standard deviation, we can calculate the probability of a randomly selected high jumper having a personal best of no more than 230 cm. By using the normal distribution and the cumulative distribution function (CDF), we can find this probability.

To find the probability, we can standardize the value of 230 cm using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get z = (230 - 221) / 11 = 0.818. This standardized value represents the number of standard deviations that 230 cm is away from the mean.

Next, we use a standard normal distribution table or a calculator to find the cumulative probability associated with the standardized value of 0.818. This probability represents the area under the normal curve to the left of 0.818. Using technology, we can find this probability to be approximately 0.7910.

Therefore, the probability that a randomly selected high jumper has a personal best of no more than 230 cm is 0.7910, or 79.10%.

This means that approximately 79.10% of the high jumpers in the nearby state have a personal best height of 230 cm or lower.

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The line that passes through the points (1, 7) and (10, 1) has a slope of -2/3.

The slope of a line is a measure of its steepness and is defined as the change in the y-coordinate over the change in the x-coordinate between any two points on the line.

The slope of the line that passes through the pair of points (1, 7) and (10, 1) can use the slope formula.

The slope formula states that the slope of a line passing through two points, (x1, y1) and (x2, y2), is given by:

slope = (y2 - y1) / (x2 - x1)

(x1, y1) = (1, 7) and (x2, y2) = (10, 1).

Substituting the values into the formula, we get:

slope = (1 - 7) / (10 - 1)

= -6 / 9

= -2/3

The slope of the line as follows:

For every one unit increase in the x-coordinate the y-coordinate decreases by 2/3 units.

Alternatively for every one unit increase in the x-coordinate the line drops by 2/3 units.

The slope of a line is useful in many applications such as calculating the rate of change of a quantity over time measuring the angle between two lines and determining whether lines are parallel or perpendicular.

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