Catherine walks her dog 3/4 mile everyday. how far does she walk each day?

Final answer:

To find the value of k so that the line containing the points (-5,k) and (2,10) is parallel to the line containing the points (5,5) and (1,-4), we need to find the slope of the second line and use it to find the value of k.

Explanation:

To find the value of k so that the line containing the points (-5, k) and (2, 10) is parallel to the line containing the points (5, 5) and (1, -4), we need to find the slope of the second line and use it to find the value of k. Firstly, let's find the slope of the second line:

The formula we apply is the slope

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

slope = -9 / -4 = 9/4;

slope = (-4 - 5) / (1 - 5)

The lines now have the same slope since they are parallel.

So, the slope of the line passing through (-5, k) and (2, 10) should also be 9/4. We can find the slope using the formula:

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

Substituting the coordinates (-5, k) and (2, 10):

9/4 = (10 - k) / (2 - (-5))

By simplifying the equation, we get:

9 = 7(10 - k)

Now, let's solve for k:

90 - 9k = 63

-9k = 63 - 90

-9k = -27

k = 3

The slope of a line perpendicular to the given line is -5/2.

The equation of the line given is 5y = 10 + 2x. To find the slope of this line, we need to rearrange the equation into the slope-intercept form, y = mx + b, where m represents the slope. Dividing both sides of the equation by 5, we get y = 2/5x + 2. Comparing this equation to the slope-intercept form, we can see that the slope is 2/5.

To find the slope of a line perpendicular to this line, we use the fact that the product of the slopes of two perpendicular lines is -1. Therefore, the slope of the perpendicular line is the negative reciprocal of the slope of the given line. The negative reciprocal of 2/5 is -5/2.

Therefore, the slope of a line that is perpendicular to the line with the equation 5y = 10 + 2x is -5/2.

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To sketch a triangle with an acute angle θ and cot(θ) = 2, we can choose an angle in the first quadrant, such as 30 degrees, and draw a right triangle with sides 2 and 1. The hypotenuse can be found using the Pythagorean theorem.

To sketch a triangle with an acute angle θ and cot(θ) = 2, we can use the fact that cotangent is the reciprocal of the tangent function, so tan(θ) = 1/2. Since tangents are positive in the first and third quadrants, we can consider an angle in the first quadrant. Let's choose the angle as 30 degrees.

Now, we can draw a right triangle with one of the acute angles as 30 degrees. Since tan(θ) = opposite/adjacent, we can choose the side opposite the angle as 1 and the side adjacent as 2. Therefore, the triangle has sides 2, 1, and the hypotenuse can be found using the Pythagorean theorem.

The triangle has an acute angle θ and cot(θ) = 2. It has sides of length 2, and 1, and the hypotenuse can be found using the Pythagorean theorem.

The expected recorded score when the total number of points earned on the two parts is recorded is 20.4.

The expected recorded score when the maximum of the two scores is recorded is 25.7.

We have,

(a)

If the score recorded in the grade book is the total number of points earned on the two parts (x + y):

We will calculate the expected value (E) using the joint probability mass function (pmf) values given in the table.

E(x + y) = Σ[(x + y) * p(x, y)]

Using the table values:

E(x + y) = (0 * 0.02) + (5 * 0.06) + (10 * 0.02) + (15 * 0.10) + (20 * 0.04) + (25 * 0.13) + (30 * 0.20) + (35 * 0.10) + (40 * 0.01) + (45 * 0.15) + (50 * 0.16) + (55 * 0.01)

E(x + y) = 20.4

(b)

If the maximum of the two scores is recorded:

We need to find the maximum value between x and y for each combination of scores in the table and calculate the expected value using the joint pmf.

E(max(x, y)) = Σ[max(x, y) * p(x, y)]

Using the table values:

E(max(x, y)) = (0 * 0.02) + (5 * 0.06) + (10 * 0.15) + (15 * 0.20) + (20 * 0.20) + (25 * 0.15) + (30 * 0.15) + (35 * 0.15) + (40 * 0.16) + (45 * 0.16) + (50 * 0.16) + (55 * 0.01)

E(max(x, y)) = 25.7

Therefore,

The expected recorded score when the total number of points earned on the two parts is recorded is 20.4.

The expected recorded score when the maximum of the two scores is recorded is 25.7.

Learn more about probability mass functions here:

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The expected recorded score in the grade book (E(X+Y)) is calculated by multiplying each possible total score by its probability and summing these products. The expected maximum score is calculated similarly, but considers the probabilities of each possible maximum score. Application of these principles requires access to full probability mass function data.

The student's question relates to the calculation of expected values in the context of a two-part quiz, with scores registered as the total (x+y) or the maximum of the two scores. Given a joint probability mass function (pmf) for x and y, we can calculate expected scores in both cases.

Generally, the expected value E(X) of a discrete random variable X is calculated by multiplying each possible value of X by its probability, and then summing these products. This is represented by the formula E(X) = Σ xP(x). Assuming the same logic for the expected total score on the quiz, E(X+Y) would be calculated by summing the products of each possible total score and its probability.

To find the expected maximum score, we'd perform a similar calculation, but consider the probabilities of each possible maximum score instead.

Without explicit probabilities in the question, no exact solutions can be provided. However, the processes outlined here would allow the student to solve such problems with access to the full pmf data.

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The angular speed of Earth is [tex]\fbox{\begin\\\ \bf 7.268\times 10^{-5}\ \text{radian per second}\\\end{minispace}}[/tex].

Further explanation:

It is given that the radius of Earth is [tex]6.37\times10^{6}\ \text{meters}[/tex] and it rotates once in every  [tex]24[/tex] hours.

The angular speed of earth is nearly constant and Earth takes one day to complete one rotation that means the time interval for earth to complete one revolution is [tex]1\ \text{day}[/tex].

The total angular displacement covered by the earth to complete one revolution is [tex]2\pi\text{radian}[/tex].

One day is equivalent to [tex]24\ \text{hours}[/tex], [tex]1\ \text{hour}[/tex] is equivalent to [tex]60\ \text{minutes}[/tex] and [tex]1\ \text{minute}[/tex] is equivalent to [tex]60\ \text{seconds}[/tex].

The total number of seconds in [tex]1\ \text{day}[/tex] is calculated as follows:

[tex]\fbox{\begin\\\ \begin{aligned}1\ \text{day}&=24\ \text{hours}\\&=24\times60\ \text{minutes}\\&=1440\ \text{minutes}\\&=1440\times60\ \text{seconds}\\&=86400\ \text{seconds}\end{aligned}\\\end{minispace}}[/tex]    

Therefore the time taken to complete one revolution is [tex]86400\ \text{seconds}[/tex].

The angular velocity is defined as the rate of change of angular displacement. So the formula for angular velocity is, as follows:

[tex]\fbox{\begin\\\ \omega=\dfrac{\theta}{t}\\\end{minispace}}[/tex]

In the above equation [tex]\omega[/tex] represents angular velocity, [tex]\theta[/tex] represents angular displacement and [tex]t[/tex] represents time.

To obtain the value of angular velocity of Earth substitute [tex]2\pi[/tex] for [tex]\theta[/tex] and [tex]86400[/tex]  for [tex]t[/tex] in the above equation.

[tex]\begin{aligned}\omega&=\dfrac{2\pi}{86400}\\&=\dfrac{2\times3.14}{86400}\\&=\dfrac{6.28}{86400}\\&=7.268\times10^{-5}\end{aligned}[/tex]

Therefore, the value of [tex]\omega[/tex] is [tex]7.268\times10^{-5}[/tex].

Thus, the angular speed of the Earth is [tex]\fbox{\begin\\\ \bf 7.268\times 10^{-5}\ \text{radian per second}\\\end{minispace}}[/tex].

Learn more:

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Angular motion

Keyword: Angular velocity, angular displacement, time, one revolution, radian/second, Earth, 86400 seconds, 2pi, speed, minutes, omega, theta.

62 +62 = 124

360 - 124 = 176

176/2 = 88

BED = 88 degrees

The difference in value is due to the decimal place values. A digit in the hundredths place indeed has ten times the value of the same digit in the thousands place. This happens due to the way decimal place values work.

The digit in the hundredths place and the digit in the thousands place refer to decimal place values. When we mention that a digit in the hundredths place has ten times the value of the same digit in the thousands place, it's because of the way decimal place values function. For example, if we take a number 0.1203, the digit 2 is in the hundredths place. Now suppose you have another number, 0.003, where 3 is in the thousands place. Even if the digit remains same i.e., 2 or 3, their value will continue to change based on their place. Hence, one should know the value of the same digit can change depending on its position in the number.

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Answer:

y=(x–5)^2

Step-by-step explanation:

The standard deviation for the number of times a 3 will be rolled in 360 dice rolls is approximately 17.32, calculated using the binomial distribution's standard deviation formula.

The question asks for the standard deviation of the number of times a 3 will be rolled when a die is rolled 360 times. Since each face of a six-sided die has an equal probability of landing upright, the chance of rolling a 3 is 1/6. To calculate the standard deviation, we first find the expected value (mean) of the number of times a 3 will come up, which is 360 rolls multiplied by the probability of 1/6, resulting in an expected value of 60. Then we use the binomial distribution formula, where the standard deviation (σ) is given by the square root of the product of the number of trials (n), the probability of success (p), and the probability of failure (q), which is σ = √(npq). In this case, it is σ = √(360*(1/6)*(5/6)) which simplifies to σ = √(300) or approximately 17.32.