So, the correct option is: The median is the best measure of center because there are outliers present.
What is line plot?A line plot is a type of graph that is used to display data along a number line. In a line plot, each data point is represented by a dot or mark placed above the corresponding value on the number line. Line plots are commonly used to represent small to moderate-sized data sets, especially those with discrete values or categories. The line plot typically consists of a horizontal line, called the axis, representing the numerical values of the data, with marks or dots above the line indicating the frequency or count of each value in the data set. Each mark or dot is placed directly above the value on the number line that it represents.
Here,
Since the data in the graph is skewed to the right, there are outliers present. Outliers are data points that are significantly different from the other values in the data set. Therefore, the median is the most appropriate measure of center to represent the data in the graph because it is less affected by outliers compared to the mean.
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merrell claims that he randomly assigned rats to treatment groups. does the data shown in the dotplots above support his claim? why or why not? g
The data shown in the dot plots above doesn't support Merrell's claim that he randomly assigned rats to treatment groups. It doesn't support it because the dot plots are not similarly distributed.
There are a few discrepancies in the data that are not possible if rats were randomly assigned to treatment groups.
Difference between the two dot plots:
There is a noticeable difference between the two dot plots.
The dot plot on the left shows that most rats in Group 1 weigh between 200 and 240 grams.
On the other hand, in Group 2, most rats weigh between 240 and 280 grams.
The two groups' data are not similarly distributed, and there is an overlap in the weight of rats in the two groups. Furthermore, the data suggests that heavier rats were placed in Group 2 while lighter rats were placed in Group 1.
This difference implies that rats were not randomly assigned to treatment groups.
Other information should be considered:
The information shown in the dot plots alone is not enough to conclude that the rats were not randomly assigned to treatment groups.
We must investigate further and gather more information to confirm our assumptions about the rats' treatment groups. Randomization guarantees that each rat has an equal probability of being assigned to a treatment group.
The two groups must be equivalent in every aspect except for the treatment they get.
Therefore, it's essential to confirm that there was no selection bias in the rat selection process.
This means that rats were picked randomly from a large population, and no specific rat characteristics influenced the selection process.
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The following table gives the probability distribution of a random variable X
X 1 2 3 4
P(x) ¼ C ½ 1/8
a) Find the value of the constant C
b) Find
The value of the constant C is 3/4.
Probability is a branch of mathematics that deals with the likelihood or chance of an event occurring. It is a measure of the likelihood that a specific event will occur, expressed as a value between 0 and 1
The sum of the probabilities for all possible values of X should equal 1
P(1) + P(2) + P(3) + P(4) = 1
Substituting the given values, we get:
C/4 + 1/2 + 1/8 + C/4 = 1
Simplifying the equation, we get:
C/2 + 5/8 = 1
Move 5/8 to right hand side of the equation
C/2 = 3/8
C = (3/8) x 2
C = 3/4
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About 32,000 people in a region with an area 201 square miles. Find the population density in a people per square mile.
The population density of the people living in the given square area would be = 159.20 persons pre square miles.
What is population density?Population density can be defined as the amount of people that can be found in a unit area of a land.
Population density is found by dividing the total population by the total land area.
The total population of the region = 32,000 people
The total land area = 201 miles².
Therefore the population desity = 32,000/201 = 159.20 persons pre square miles.
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Vectors u and v are shown on the graph. vector u with initial point at the origin and terminal point at 0 comma negative 5, vector v with initial point at the origin and terminal point at negative 7 comma 0 Which of the following vectors represents u + v? a vector that points to the right 7 units and up 5 units b vector that points to the left 7 units and up 5 units c vector that points to the left 7 units and down 5 units d vector that points to the right 7 units and down 5 units
The resultant vector is 7 units to the left and 5 units down. As a result, graphs option (c) a vector pointing to the left 7 units and down 5 units is the correct answer.
What is graphs?Mathematicians use graphs to visually display or chart facts or values in order to express them coherently. A graph point usually represents a connection between two or more items. A graph, a non-linear data structure, is made up of nodes (or vertices) and edges. Glue the nodes, also known as vertices, together. This graph includes V=1, 2, 3, 5, and E=1, 2, 1, 3, 2, 4, and (2.5). (3.5). (4.5). Statistical graphs (bar graphs, pie graphs, line graphs, and so on) are graphical representations of exponential development. a logarithmic graph shaped like a triangle
We can add vectors geometrically by setting the starting point of the second vector at the terminal point of the first vector and drawing a vector from the first vector's initial point to the second vector's terminal point.
Next we align the starting point of vector v with the terminal point of vector u and draw a vector from the initial point of vector u to the terminal point of vector v:
The resultant vector is 7 units to the left and 5 units down. As a result, option (c) a vector pointing to the left 7 units and down 5 units is the correct answer.
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Cory writes the polynomial x7 + 3x5 + 3x + 1. Melissa writes the polynomial x7 + 5x + 10. Is there a difference between the degree of the sum and the degree of the difference of the polynomials?
Adding their polynomials together or subtracting one polynomial from the other both result in a polynomial with degree 5.
Adding their polynomials together results in a polynomial with degree 14, but subtracting one polynomial from the other results in a polynomial with degree 5.
Adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.
Answer:
The last option.
Adding their polynomials together results in a polynomial with degree 7, but subtracting one polynomial from the other results in a polynomial with degree 5.
Step-by-step explanation:
Both polynomials are of degree 7.
The 7th degree terms are both x^7.
When you add the polynomials, the x^7 term of one polynomial will be added to the x^7 term of the other polynomial resulting in 2x^7. The sum of the polynomials is also of degree 7.
When you subtract the polynomials, the x^7 term of one polynomial subtracted from the other x^7 term of the other polynomial will eliminate the x^7 terms. The difference of the polynomials will have a 3x^5 or a -3x^5 term resulting in a 5ht degree polynomial.
Answer: The last option.
You have a bag of 10 marbles. Three of them are red, five are blue, and two are green. What is the probability of randomly picking a green or blue marble out of the bag?
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Question
You have a bag of 10 marbles. Three of them are red, five are blue, and two are green. What is the probability of randomly picking a green or blue marble out of the bag?
You have a bag of 10 marbles. Three of them are red, five are blue, and two are green. The probability of randomly picking a green or blue marble out of the bag is 7/10, 0.7 or 70%.
Given that,
there is a bag containing 10 marbles, out of which 3 are red, 5 are blue and 2 are green, and
we are supposed to find the probability of picking a green or blue marble out of the bag.
Probability is defined as the likelihood of an event occurring out of all the possible outcomes.
It is represented in the form of a fraction, decimal or percentage.
So, in order to find the probability of picking a green or blue marble out of the bag, we need to first find the total number of marbles in the bag, then the number of marbles that are green or blue, and then divide the number of green or blue marbles by the total number of marbles.
Let us calculate the probability of picking a green or blue marble from the bag.
Total number of marbles in the bag = 10
Number of green or blue marbles in the bag
= 5 + 2 = 7
Now,
the probability of picking a green or blue marble out of the bag
= Number of green or blue marbles in the bag / Total number of marbles in the bag= 7 / 10
The probability of picking a green or blue marble out of the bag is 7/10.
This can also be expressed as a decimal value of 0.7 or as a percentage of 70%.
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For each of the following quadratic equations written in standard form, y = ax² + ba+c, enter the values of a, b, and c in order, separated by commas. a) y = 3x² - 8x +1 b) y = 4x² - 3 c) y = 8x² + 2x
Answer:
Step-by-step explanation:
[tex]y=ax^2+bx+c[/tex]
a) [tex]a=3,b=-8,c=1[/tex]
b) [tex]a=4,b=0,c=-3[/tex]
c) [tex]a=8,b=2,c=0[/tex]
please help me solve this i’ll mark brainliest
Answer:
Step-by-step explanation:
From the parallel lines, we know that corresponding angles are congruent, so:
(2y+78) = X
We also know that vertically opposite angles are equal:
X=5y
Substitute for X:
2y+78=5y
Solve:
2x-5x=-78
-3x=-78
x=26
Therefore, the measure of angle WXZ is 26.
Solve this quadratic equation.
x⁴ - x³ - 4x² - x + 1 = 0 is the solution of quadratic equation.
What in mathematics is a quadratic equation?
A quadratic equation is a second-order polynomial equation in one variable using the formula x = ax2 + bx + c = 0 and a 0. Given that it is a second-order polynomial equation, the algebraic fundamental theorem ensures that it has at least one solution.
The answer could be simple or complicated. The equation of the second degree is a quadratic equation. This indicates that it contains a phrase that is squared, at least once.
x² + 1/x² - x - 1/x - 4 = 0
x⁴ + 1 -x³ - x - 4x²/x² = 0
x⁴ - x³ - 4x² - x + 1 = 0
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Suppose you select a number at random from the sample space {1, 2, 3, 4, 5, 6, 7, 8}. probability.
P(the number is positive)
P(the number is even)
Suppose you select a number at random from the sample space {1, 2, 3, 4, 5, 6, 7, 8}. probability.
P(the number is positive)
P(the number is even)
Answer:
The probability of selecting a number at random from the sample space {1, 2, 3, 4, 5, 6, 7, 8} that is positive is:
There are 8 possible outcomes in the sample space, and 4 of them are positive (1, 2, 3, 4).
Therefore, P(the number is positive) = 4/8 = 1/2 = 0.5
The probability of selecting a number at random from the sample space {1, 2, 3, 4, 5, 6, 7, 8} that is even is:
There are 8 possible outcomes in the sample space, and 4 of them are even (2, 4, 6, 8).
Therefore, P(the number is even) = 4/8 = 1/2 = 0.5
So the probability of selecting a number at random from the sample space {1, 2, 3, 4, 5, 6, 7, 8} that is positive is 0.5 and the probability of selecting a number at random from the same sample space that is even is also 0.5.
help me with this or u an opp
Answer:
To find the year when 164 billion dollars was spent on advertising, we need to solve the equation y = 164 for x, where y is the amount of money spent on advertising and x is the number of years since 1990.
Substituting y = 164 into the given equation, we have:
164 = 0.937 + 2.2x + 130
Simplifying the equation, we get:
2.2x = 164 - 0.937 - 130
2.2x = 33.063
x = 15.03
So, 164 billion dollars was spent on advertising 15.03 years after 1990. To find the year, we add 15.03 to 1990:
1990 + 15.03 = 2005.03
Since we are looking for a year within the range of 1990-2000, we round down to the nearest whole number:
2005.03 rounded down to the nearest whole number is 2005.
Therefore, 164 billion dollars was spent on advertising in the year 2005, which is outside the given range. Therefore, the answer is none of the options provided.
Mr. Clark has these two spinners. A student wins a prize by spinning a 1 or an X. Nathan spins both spinners one time. How many possible outcomes will include a 1 or an X?
There are two possible outcomes that include a 1 or an X.
How to determine the number of possible outcomes that include a 1 or an X.
First we need to count the number of outcomes on both spinners that have a 1 or an X.
Spinner 1 has 3 possible outcomes: 1, 2, or 3.
Spinner 2 has 4 possible outcomes: X, Y, Z, or W.
To find out how many possible outcomes will include a 1 or an X, we can use the principle of inclusion-exclusion. We start by adding the number of outcomes on each spinner that have a 1 or an X:
Number of outcomes on spinner 1 that have a 1: 1Number of outcomes on spinner 2 that have an X: 1Total number of outcomes that have a 1 or an X: 1 + 1 = 2However, we need to subtract the number of outcomes that have both a 1 and an X, as we have counted these twice:
Number of outcomes that have both a 1 and an X: 0
Therefore, the total number of possible outcomes that include a 1 or an X is:
2 - 0 = 2
So, there are two possible outcomes that include a 1 or an X.
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Bao has 39 m of fencing to build a three-sides fence around a rectangular plot of land that sits in a riverbank. (The fourth side of the enclosure would be the river.) The area of the land is 169 square meters. List each set of possible dimensions (length and width) of the field.
The pοssible dimensiοns (length and width) οf the rectangular plοt are:
Length = 21 meters, width = 8.05 meters
Length = 8.05 meters, width = 21 meters
Let the length οf the rectangular plοt be L and the width be W. Then we knοw that the perimeter οf the rectangular plοt, which is equal tο the length οf fencing Baο has, is given by:
2L + W = 39
Alsο, the area οf the rectangular plοt is given by:
L × W = 169
We can use these twο equatiοns tο sοlve fοr L and W. First, we can sοlve the equatiοn 2L + W = 39 fοr W:
W = 39 - 2L
Substituting this into the equation for the area, we get:
L × (39 - 2L) = 169
Expanding and rearranging this equation, we get a quadratic equation in terms of L:
[tex]-2L^2 + 39L - 169 = 0[/tex]
We can solve for L using the quadratic formula:
[tex]L = [ -39 \± sqrt(39^2 - 4(-2)(-169)) ] / (2(-2))\\L = [ -39 \± sqrt(1681) ] / 4\\L = [ -39 \± 41 ] / 4[/tex]
L = 1/2 or L = 21
If L = 1/2, then W = 38, which gives a negative area and is therefore not possible.
If L = 21, then W = 169/21, which simplifies to W = 8.05 (rounded to two decimal places).
Therefore, the possible dimensions (length and width) of the rectangular plot are:
Length = 21 meters, width = 8.05 meters
Length = 8.05 meters, width = 21 meters
Note that these are the only possible dimensions, as the sum of the length and width must be less than or equal to 19.5 meters (half of the available fencing).
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which formula represents the volume of a cyclinder
Answer:
[tex]\pi r^2h[/tex]
Step-by-step explanation:
A cylinder is a surface or solid bounded by two parallel planes and generated by a straight line moving parallel to the given planes and tracing a curve bounded by the planes and lying in a plane perpendicular or oblique to the given planes
The formula for calculating the volume of a cylinder is similar to calculating the volume of a cube. The volume of a cube is equal to the Area of the Base (length times width) times the height of the cube.
[ V = (L x W) x H ] or [ V = A x H ]
The volume of a cylinder is the Area of the Base times the Height of the Cylinder
[ V = A x H ].
The Base of a cylinder is a circle. The formula for calculating the Area of a Circle is the constant Pi (∏) times the square of the Radius
[ A = ∏ x R² ]
and the formula for calculating the Radius of a Circle is the Diameter divided by 2
[ R = 1/2D ].
Therefore the formula for calculating the volume of the cylinder is
[ V = (∏ x (1/2D)² ].
Pls help I need this today
Answer:
In a 45° 45° 90° triangle, the two legs are congruent and the hypotenuse is √2 times the length of each leg.
Let x be the length of each leg.
Then, using the Pythagorean theorem, we have:
x^2 + x^2 = 24^2
2x^2 = 576
x^2 = 288
x = √288 = 12√2
Therefore, the length of each leg is 12√2.
Answer:
each leg is 16.97
Step-by-step explanation:
[tex]\frac{24}{\sqrt{2} }[/tex] = 16.97
HELP ASAP!!! Solve for x if the figure is a rectangle. (Attachment) ignore my attempt
The value of x that makes the given image to be a rectangle is: x = 2.5
How to find the length of the diagonal of a rectangle?The length of the diagonals of a rectangle are equal and congruent to each other. It is also pertinent to note that the diagonals of a rectangle bisect each other at the center. Thus:
From the given parameters, we see that half of the diagonals have been labelled and as such we have:
8x - 3 = 4x + 7
Rearrange to have like terms on same side and so we have:
8x - 4x = 7 + 3
4x = 10
x = 10/4
x = 2.5
Thus, the value of x from the calculation above of the rectangle is 2.5
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if the number of degrees of freedom for a chi-square distribution is 25, what is the standard deviation? round to four decimal places. standard deviation
The standard deviation for a chi square distribution for given degree of freedom is 7.0711.
The chi-square distribution is parameterized by the number of degrees of freedom (df). As the number of degrees of freedom increases, the shape of the distribution becomes more symmetrical and approaches a normal distribution.
The standard deviation (σ) of a chi-square distribution with k degrees of freedom is given by the formula:
σ = sqrt(2k)
Therefore, for a chi-square distribution with 25 degrees of freedom:
σ = sqrt(2(25)) = sqrt(50) ≈ 7.0711
Rounding this to four decimal places gives a standard deviation of approximately 7.0711.
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Which ONE of the following statements best describes the use of spaces in folder and file names in ArcGIS?
Using spaces in folder names in ArcGIS is permitted but strongly discouraged
The statement that best describes the use of spaces in folders and files names in ArcGIS is "Using spaces in folder and file names in ArcGIS is permitted but strongly discouraged."
ArcGIS is a geographic information system software that is used for creating, editing, analyzing, and sharing spatial data. The software is widely used by GIS professionals, environmental scientists, geographers, and other experts to manage and analyze spatial data.
In ArcGIS, it is possible to use spaces in folder and file names, but it is not recommended. This is because spaces can cause problems when the software tries to access files with spaces in their names. Therefore, it is better to use underscores or hyphens instead of spaces when naming folders and files in ArcGIS. This helps to avoid issues when working with data in ArcGIS.
The statement "Using spaces in folder names in ArcGIS is permitted but strongly discouraged" is the best description of the use of spaces in folder and file names in ArcGIS.
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Mr. Kha Lipat wants to earn 8% on his investment. How much money should he invest today in order to receive 400. 00 one year from now?
Mr. Kha Lipat should invest $5,000 today in order to receive $400.00 in interest one year from now at an 8% interest rate.
To calculate how much money Mr. Kha Lipat should invest today to receive $400.00 one year from now at an 8% interest rate, we can use the formula for calculating simple interest though compound intrest:
I = P * r * t
where I is the interest earned, P is the principal (the initial amount invested), r is the interest rate (as a decimal), and t is the time period (in years).
We know that Mr. Kha Lipat wants to earn $400.00 in interest, the interest rate is 8% or 0.08 (as a decimal), and the time period is 1 year. We can plug these values into the formula and solve for P:
I = P * r * t
400 = P * 0.08 * 1
400 = 0.08P
P = 400 / 0.08
P = 5000
Therefore, Mr. Kha Lipat should invest $5,000 today in order to receive $400.00 in interest one year from now at an 8% interest rate.
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1. a) Determine the equation of the line in slope y-intercept form that runs
through points (12,-7) and (-8,3).
b) Line is perpendicular to the line from part a) and has the same
y-intercept as 5x-6y + 54 = 0. State the equation of line in standard form.
Answer:
a)
[tex]y = - \frac{1}{2} x - 1[/tex]
b)
[tex]2x - y = - 9[/tex]
Step-by-step explanation:
a)
[tex]m = \frac{3 - ( - 7)}{ - 8 - 12} = \frac{10}{ - 20} = - \frac{1}{2} [/tex]
[tex] - 7 = - \frac{1}{2} (12) + b[/tex]
[tex] - 7 = - 6 + b[/tex]
[tex]b = - 1[/tex]
[tex]y = - \frac{1}{2} x - 1[/tex]
b) Slopes of perpendicular lines are negative reciprocals of each other, so if the original line has slope -1/2, the perpendicular line will have a slope of 2.
[tex]5x - 6y + 54 = 0[/tex]
[tex]5x - 6y = - 54[/tex]
We see that the y-intercept of this line is at (0, 9), so in slope-intercept form, we have
[tex]y = 2x + 9[/tex]
Putting this in standard form:
[tex] - 2x + y = 9[/tex]
[tex]2x - y = - 9[/tex]
Find the equation of L
[tex]y=\frac{7}{2.4} x-6[/tex] ; if estimated then : [tex]y=\frac{7}{2.5} x-6[/tex]
Explination:
Y-intercept is -6, (0,-6)
The rise over run it [tex]\frac{7}{2.4} x[/tex]. The denominator is 2.4 because it nearly 2.5.
3. as stephen dubner and steven levitt develop their essay, they use a great deal of quantification. note three particular examples. how does their use of numbers affect their argument?
Stephen Dubner and Steven Levitt are the authors of the essay, and as they develop their essay, they use a great deal of quantification.
The following are three specific examples that they used: They write, "If a martini is made with 4 ounces of gin, that’s 2.8 standard drinks. "They also said, "Let's consider the five most unsafe hours for driving, which are Saturday and Sunday mornings from 1 am to 6 am. In the middle of this time period, 3 am on Saturday morning, the likelihood of an accident is three times higher than at noon on a weekday. "In another instance, they note that "The average person who has been murdered has approximately 300 friends and relatives, which means that a homicide victim’s personal network is quite extensive. "The authors use these specific numbers to make their argument more convincing. Using quantification provides an air of authority to the author's claims. By providing specifics, the author can communicate more information than would otherwise be possible.
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1. The orthocenter is outside the triangle. 2. An altitude is the same line segment as an angle bisector. 3. If the perpendicular bisector of one side of a triangle intersects the opposite vertex, then the triangle is isosceles
The solution of the match is
(1) Orthocenter (C) The point at which the altitudes of a triangle
meet.
(2) Centroid (D) The point at which the medians of a triangle
meet.
(3) Circumcenter (A) The point of intersection of the perpendicular
bisectors of the sides of a triangle.
(4) Incentre (B) The point at which the three angle bisectors
of a triangle meet.
The orthocenter of a triangle is the point where the altitudes of the triangle intersect. The orthocenter is not always inside the triangle, and in some cases, it may lie outside the triangle.
The centroid of a triangle is the point where the three medians of the triangle intersect. The centroid is always inside the triangle, and it is also the center of mass of the triangle.
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. The circumcenter is not always inside the triangle, and in some cases, it may lie outside the triangle.
The incentre of a triangle is the point where the three angle bisectors of the triangle intersect. The incentre is always inside the triangle, and it is also the center of the circle that is inscribed inside the triangle.
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Complete Question:
Match the following.
(1) Orthocenter (A) The point of intersection of the perpendicular
bisectors of the sides of a triangle.
(2) Centroid (B) The point at which the three angle bisectors of
a triangle meet.
(3) Circumcenter (C) The point at which the altitudes of a triangle
meet.
(4) Incentre (D) The point at which the medians of a triangle
meet.
Brody is driving on a long road trip. He currently has 9 gallons of gas in his car. Each hour that he drives, his car uses up 2 gallons of gas. How much gas would be in the tank after driving for 2 hours? How much gas would be left after tt hours?
Answer: 5 Gallons of gas
Step-by-step explanation: Start with 9, and subtract that by how much he lost within those two hours (2t = 2(2) = 4, 9 - 4 = 5)
After driving for 2 hours, Brody would have 5 gallons of gas left in the tank. The formula to find gas left after tt hours is gas left = 9 - 2tt.
Explanation:To find out how much gas would be in Brody's tank after driving for 2 hours, we need to subtract the amount of gas used in 2 hours from the initial amount of gas in the tank. Each hour, Brody's car uses up 2 gallons of gas, so in 2 hours it would use up 2 x 2 = 4 gallons. Therefore, after driving for 2 hours, Brody would have 9 - 4 = 5 gallons of gas left in the tank.
To find out how much gas would be left after tt hours, we can use the formula: gas left = initial gas - gas used in tt hours. In this case, initial gas is 9 gallons, and gas used in tt hours is 2 x tt gallons. So the formula becomes: gas left = 9 - 2tt.
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simplify 2(2x+5)-(3x-2)
Solution: x + 12
Step-by-step explanation:
first simplify then distribute!
Answer:x+12
Step-by-step explanation:
we have to multiply number outside of brackets to each of the numbers inside it:
2(2x+5)=4x+10
-(3x-2)=-1(3x-2)=-3x+2
4x+10-3x+2=x+12
What is 35 to the nearest degree
I'm sorry, 35 is not a measure of angle and therefore cannot be rounded to the nearest degree. Are you asking for the sine, cosine, or tangent of 35 degrees?
Help I don’t have a clue ?
Answer: Yes
Step-by-step explanation:
Because she tested it more times, her results should (in theory) be more accurate, due to more tests leading to the results being more diverse. The results indicate that one side of the coin (the tails side) is more likely to land down, which leads the head side up.
a 1-kilogram mass is attached to a spring whose constant is 16 n/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. determine the initial conditions and equations of motion if the following is true.
The final equation of motion for the system is: x(t) = [(8v0 - 2x0)/6]*[tex]e^{(-2t)} + [(2x0 - 8v0)/6]*e^{(-8t).[/tex]
To determine the initial conditions and equations of motion, we can use the following steps:
Write down the equation of motion for the system. The equation of motion for a mass-spring-damper system is given by:
m * [tex]d^2x/dt^2[/tex] + c * dx/dt + k * x = 0
where m is the mass, c is the damping coefficient, k is the spring constant, x is the displacement from the equilibrium position, and t is time.
In this case, the mass is 1 kg, the spring constant is 16 N/m, and the damping coefficient is 10 times the instantaneous velocity, or 10 * dx/dt.
Substituting these values into the equation of motion, we get:
1 * [tex]d^2x/dt^2[/tex] + 10 * dx/dt + 16 * x = 0
Determine the initial conditions. The initial conditions refer to the position and velocity of the mass at time t = 0.
Let x(0) be the initial displacement of the mass from the equilibrium position, and let v(0) be the initial velocity of the mass. Then the initial conditions are:
x(0) = ?
v(0) = ?
These values are typically given in the problem statement.
Solve the differential equation. To solve the differential equation, we can use the characteristic equation method. First, we assume that the displacement of the mass is of the form:
x(t) = A[tex]e^{(rt)[/tex]
where A and r are constants to be determined.
Taking the first and second derivatives of x(t), we get:
dx/dt = Are^(rt)
d^2x/dt^2 = Ar^2*e^(rt)
Substituting these expressions into the equation of motion, we get:
A * ([tex]r^2[/tex] + 10r + 16) * [tex]e^{(rt)[/tex]= 0
Since [tex]e^{(rt)[/tex]is never zero, we can divide both sides by it and simplify to obtain the characteristic equation:
[tex]r^2 + 10r + 16 = 0[/tex]
Solving for r using the quadratic formula, we get:
r1 = -2
r2 = -8
Therefore, the general solution for the displacement of the mass is:
[tex]x(t) = Ae^{(-2t)} + Be^{(-8t)[/tex]
where A and B are constants determined by the initial conditions.
To find A and B, we use the initial conditions x(0) = x0 and v(0) = v0. Taking the derivative of x(t) and setting t = 0, we get:
v(0) = -2A - 8B
Using the second initial condition, we get:
x(0) = A + B = x0
Solving for A and B, we get:
A = (8v0 - 2x0)/6
B = (2x0 - 8v0)/6
Therefore, the final equation of motion for the system is:
[tex]x(t) = [(8v0 - 2x0)/6]*e^{(-2t)} + [(2x0 - 8v0)/6]*e^{(-8t)[/tex]
Complete question: A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. Determine the equations of motion if (a) the mass is initially released from rest from a point 1 meter below the equilibrium position, and then (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 12 m/s.
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The sum of two numbers is 18. Their difference is -8. Find the two numbers.
Part A
Write a system of equations that represents the situation.
x + y =
x-y =
Part B
Solve the system of equations. Express the coordinates as decimals if necessary.
The two numbers in the system of equation is 5 and 13.
What is system of equation?A group of multiple-variable equations that must all be solved concurrently make up a system of equations. Finding values for the variables that satisfy every equation in a system of equations is the objective. Each equation in the system reflects a connection between the variables. In order to find a singular solution or a group of solutions, systems of equations can be solved using a variety of methods, such as substitution or elimination. In many disciplines, including mathematics, physics, engineering, and economics, systems of equations can occur.
Let us suppose the two numbers as x and y.
x + y = 18
x - y = -8
The second equation can be written as:
x + 8 = y
Substitute the value of y in equation 1:
x + x + 8 = 18
2x = 10
x = 5
Substitute the value of x to get y:
x + y = 18
5 + y = 18
y = 13
Hence, the two numbers in the system of equation is 5 and 13.
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a farmer want to create a rectangular field with one side alongn a straight stream using 1100 feet of fencing, but there will be fencing only one 3 sides and not on the side bordered by the stream. what dimensions will maximize the area of the field?
To maximize the area of a rectangular field with one side along a stream using 1100 feet of fencing, the dimensions should be 183.33 feet by 550 feet.
Let's call the length of the field that runs parallel to the stream "x" and the width of the field that runs perpendicular to the stream "y".
The total amount of fencing that the farmer has available is 1100 feet, and we know that three sides of the rectangular field will be fenced. Since one side of the field is already bordered by the stream and does not require fencing, we can set up the following equation:
2x + y = 1100 - x
Simplifying this equation, we get:
3x + y = 1100
We want to maximize the area of the field, which is given by the formula:
A = xy
We can use the equation we derived earlier to solve for y in terms of x:
y = 1100 - 3x
Substituting this into the area formula, we get:
A = x(1100 - 3x)
Expanding the parentheses, we get:
A = 1100x - 3x^2
To find the maximum area, we need to find the value of x that maximizes this equation. We can do this by taking the derivative of A with respect to x and setting it equal to zero:
dA/dx = 1100 - 6x = 0
Solving for x, we get:
x = 183.33 feet
We can use this value of x to find the corresponding value of y:
y = 1100 - 3x = 550 feet
Therefore, the dimensions of the rectangular field that maximize the area are 183.33 feet by 550 feet.
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