Dividing the sum above by the total number of values in the data set.
How do you calculate the data’s mean absolute deviation?Take the absolute value of each integer in the data collection and remove the mean. Then add the absolute values together. Divide the total number of values in the data set by the sum above to get the mean absolute deviation. Finally, round up to the tenth.
A collection of data’s Mean Absolute Deviation (MAD) is the average distance between each data item and the mean. The mean absolute deviation is the “average” of each point’s “positive distances” from the mean. The bigger the MAD, the greater the variability in the data (the data is more spread out).
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A pet store had 4 cats to feed.If they only had one-ninth of a bag of cat food and each cat got the same amount,what fraction of the bag would each cat get in all.
Answer:
0.02777777778 or 1/36
Step-by-step explanation:
1/9 divided by 4 = 0.02777777778 or 1/36
Without calculation, find one eigenvalue and two linearly independent eigenvectors of A ={2 2 2 }
one eigenvalue and two linearly independent eigenvectors of A ={2 2 2 } are [ 1, 2 ]
1st step:
Lambda I
here I is identity matrix.
As our matrix is 2X2 , so we will take I= [ 1 0, 0 1]
consider lambda is represented as $
so $I matrix will be [$ 0, 0 $]
2nd Step:
A - Lambda I
i.e .[2 2, 2] - [0, 0 $]
=> [2-$ 2, 2$]
3rd Step:
Det [A-$I]
i.e. [(2-$)(4-$) - (-2)(-4)] = 0
=>8-2$-4$+$²-8
=>$²-6$
4th Step:
Det [A-$I]=0
i.e. $²-6$ =0
$($-6)=0
i.e. $ =0, & $ =6 (eigen values)
5th Step
put these eigen values in [A-$I] matrix
i.e if we put $ =0
we get [2 -2, -4 4]
consider this matrix as B
then
B X⁻= 0⁻ (⁻ is a bar sign notation)
[2 -2, -4 4] [x₁ x₂] =[0 0]
by row reduction
-2R₁+ R₂ -> R₂
so
2X₁-2X₂=0
X₁=X₂
ie eigen vector will be [0 0]
now consider
i.e if we put $ =6
we get [-4 -2, -4 -2]
consider this matrix as B
then
B X⁻= 0⁻ (⁻ is a bar sign notation)
[-4 -2, -4 -2] [x₁ x₂] =[0 0]
by row reduction
R₁- R₂ -> R₂
so
-4X₁-2X₂=0
2X₁=X₂
if we put X₁=1
x₂=2
So eigen vector will be
[1 2]
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Let f be the continuous function defined on (−1,8) whose graph, consisting of two line segments. Let g and h be the function defined by g(x)=√x^2−x−3 and h(x)=5e^x−9sinx. The function is defined by m(x)=f(x)/2g(x). Find m′(5).
The function is defined by m(x)=f(x)/2g(x) is m′(5))=[tex]\sqrt[3]{3/2}[/tex] on graph.
How to defined the function?A function is a relation that is "well-behaved," by which we mean that, given a starting point, we know the exact one ending spot to go to; given an x-value, we get only and precisely one corresponding y-value. As a result, even though all functions are relations [since they pair information], not all relations are functions. Family members who behave well are a subset of all your relations, just as well-behaved functions are a subset of all mathematical relations.)
f(x)=2x+3,XE[-1,2]
-2/3x+25/3, XE[2,8]
g(x)=√x^2−x+3 XE
m′(5))=[tex]\sqrt[3]{3/2}[/tex]
The thing to notice here is that f(x) is expressed differentially in different domain.
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Find the marginal probability mass function pX(x). Round the answers to two decimal places1) Px(0)
2) Px(1)
3) Px(2)
Answer Px1 and px2 is the same
Step-by-step explanation:
use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = x3, y = 0, x = 1, x = 3
The volume of the cylindrical shell will be 20 cubic unit using integration.
Indicate integrationIntegration is the process of calculating an integral. Integrals are used by mathematicians to compute a range of useful quantities, such as areas, volumes, displacement, etc. Usually, when we talk about integrals, we mean definite integrals. Indefinite integrals are used for antiderivatives. Integration is one of the two primary calculus topics in mathematics, along with differentiation (which measure the rate of change of any function with respect to its variables).
Fundamental IntegralAn integral that has both the upper and lower boundaries is said to be definite. Only a real line may be used for X to lay. A Riemann Integral is another name for the Definite Integral.
n example of an Integral is:
f(x)dx, from a to b.
Indefinite integral
∫f(x)dx = F(x) + C
Where C is any constant and the function f(x) is called the integrand.
Now,
As we can see in image
Volume = ∫x³.dx from 1 to 3 Using integration formula
=x⁴/4
=81/4-1/4
=80/4
volume =20 cubic unit
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PLS ASWNSER NO MATTER WHAT AWNSER CORRECTLY THIS IS A REVIEW AND IS GRADED OUT OF 100% I need 80% to get an A- PLS HELP
The scale factor of the dilation is given by the equation s = 2.5
What is Dilation?Resizing an item uses a transition called Dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. Dilation transformations ensure that the shape will stay the same and that corresponding angles will be congruent
Given data ,
Let the triangle be represented as RST
Let the dilated triangle be represented as R'S'T'
Now , the coordinates of ΔRST is
The coordinate of R = R ( 0 , 4 )
The coordinate of S = S ( 2 , 4 )
The coordinate of T = T ( 2 , 2 )
And , the coordinates of ΔR'S'T' is
The coordinate of R' = R' ( 0 , 10 )
The coordinate of S = S ( 5 , 10 )
The coordinate of T = T ( 5 , 5 )
So , the dilation scale factor is given by coordinate of R' / coordinate of R
Substituting the values in the equation , we get
Scale factor s = 10/4
Scale factor s = 5/2
Scale factor s = 2.5
Hence , the dilation scale factor is 2.5
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Point R is at (2, 1.2) and Point T is at (2, 2.5) on a coordinate grid. The distance between the two points is . (Input numbers and decimal point only, such as 8.2.)
the number of visitors that the website receives on a given day d the list of visitor counts for the 42 days that were tracked b the average number of visitors per day for the 42 days that the administrator tracked f the average number of visitors per day that have come to the website since it was created a all the days that the website has been in existence c the 42 randomly selected days that the administrator tracked
Population, Statistic, sample, Variable, data, parameter are the correct matching vocabulary words.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
The correct matching of the vocabulary words and the examples given of the web administrator's process are:
Population - All the days that the website has been in existence.
Statistic - The average number of visitors per day for the 42 days that the administrator tracked.
Sample - The 42 randomly selected days that the administrator tracked.
Variable - The number of visitors that the website receives on a given day.
Data - The list of visitor counts for the 42 days that were tracked.
Parameter - The average number of visitors per day that have come to the website since it was created.
Hence, Population, Statistic, sample, Variable, data, parameter are the correct matching vocabulary words.
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Compute the Instantaneous rate of change of the function f(x) = 3x2 + 12 at x = 4 ?
The answer to these questions is "7," but before we get there, let's define the instantaneous rate and its further computation.
Compute the Instantaneous rate of change ?A rate at a specific time is called an immediate rate.The negative ball of the reactant concentration curve with respect to time t is calculated by calculating the pitch of the product amounting with respect to time t.
Given:
at X = 3
y = 21 according to the graph.
In light of this, given that X = 3 and the instantaneous rate of changes,
that is half.
Consequently, y/x = 21/3 = 7 is the instantaneous rate of change.
The final response is "7," as a result.
The answer to these questions is "7," but before we get there, let's define the instantaneous rate and its further computation.
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A real estate agent has 17 properties that she shows. She feels that there is a 40% chance of selling any one property during a week. The chance of selling any one property is independent of selling another property. Compute the probability of selling no more than 3 properties in one week
A real estate agent has 17 properties that she shows, the probability of selling no more than 3 properties in one week is 0.0464 or 4.64%.
There are only two possible outcomes for each attribute. Selling one home does not affect the likelihood of selling another. Therefore, to answer this query, we employ the binomial probability distribution.
Binomial probability distribution
The probability of precisely x successes on n repeated trials is known as a binomial probability, and X can only have two possible outcomes.
[tex]P(X=x) = C_{n} , x.P^x.(1-P)^{n-x}[/tex]
Given by the following formula.
[tex]C_{n} , x = \frac{n!}{x! (n-x)!}[/tex]
And p is the probability of X happening.
n = 17
P = 0.4
P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
In which,
[tex]P(X=x) = C_{n} , x.P^x.(1-P)^{n-x}[/tex]
[tex]P(X=x) = C_{n} , x.P^x.(1-P)^{n-x}\\P(X=0) = C_{17} ,0.(0.4)^0.(0.6)^{17} = 0.0002\\P(X=1) = C_{17} ,1.(0.4)^1.(0.6)^{16}=0.0019\\P(X=2) = C_{17} ,2.(0.4)^2.(0.6)^{15}=0.0102\\P(X=3) = C_{17} ,3.(0.4)^3.(0.6)^{14}=0.0341[/tex]
P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.0002 + 0.0019 + 0.0102 + 0.0341 = 0.0464
Therefore probability of selling no more than 3 properties in one week is 0.0464 or 4.64%.
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Use the graph of the function to determine the indicated limit, if it exists. lim f(x) Select the correct choice below and fill in any answer boxes in your choice. A. lim f(x) = B. The limit does not exist and is neither conor -
lim (x → a) f(x) = ∞ , The limit does not exist and is neither ∞ nor - ∞
What is function?The vertical line test is used to verify function. That is, a graph must have a unique y value for it to be referred to as a function for a certain (unique) value of x. If for a particular x, and y values are coming it means it is not a function. If it crosses y-axis twice it means for a particular x value , two different value of y will exist. And that is not possible for a function
From the graph, we know that limit at point a is:
lim (x→a) f(x) exists if and only if
lim (x →a⁺) f(x) = lim (x→a⁻) f(x) and ≠ ∞
∴ right hand side limit = left hand side limit
lim (x →a⁺) f(x) = - ∞
lim (x →a⁻) f(x) = - ∞
∴ lim (x → a) f(x) = ∞
thus, option B is the correct answer.
The limit does not exist and is neither ∞ nor - ∞
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Use the graph at the right Write directions to translate the original triangle to make the new triangle . original b. What are the coordinates of the vertices (corners) of the new shape ? new On your graph, reflect the original triangle across the y -axis. What are the coordinates of the new triangle?
The directions used to translate the original triangle to a new triangle are translation to the right on the x-axis by 2 units and then it is down on the y-axis by 3 units.
What are transformations?The transformation, or f: X X, is the name given to a function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation. A function can be moved in one way or another using translation, rotation, reflection, and dilation. A function can also be scaled using rotation around a point.
The directions used to translate the original triangle to a new triangle are:
First the graph is translated to the right on the x-axis by 2 units and then it is translated down on the y-axis by 3 units.
The new coordinates of the triangle are:
A' = (-1, -2)
B' = (-2, 0)
C' = (0, 0)
Hence, the directions used to translate the original triangle to a new triangle are translation to the right on the x-axis by 2 units and then it is down on the y-axis by 3 units.
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Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (6, 36).
The tangent line to the parabola y = x² at the point (6, 36) is (0.141, 1.68)
How to find the unit vector parallel to the tangent lineThe equation of the tangent line at the point (6, 36) can be found using the derivative of the function and the point-slope form of a line:
f(x) = x²
f'(x) = 2x
y = f'(x) * (x - 6) + 36
Substituting f'(x) = 2x,
y = 2x * (x - 6) + 36
Now, we can find a unit vector that is parallel to this line by finding a non-zero multiple of the direction vector of the line, which is (1, 2x), and normalizing it.
To normalize, we divide by the magnitude, which is sqrt(1^2 + (2x)^2) = sqrt(1 + 4x^2).
Let's use x = 6:
y = 2 * 6 * (x - 6) + 36
y = 72
So, the direction vector is (1, 12) and its magnitude is sqrt(1 + 144) = sqrt(145) = 7.07.
The unit vector that is parallel to the tangent line is (1/7.07, 12/7.07) = (0.141, 1.68).
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What is the y-intercept of the line?
Interpret the y-intercept in the context of the problem.
The point on a line where it crosses the y-axis is known as the y-intercept. To put it another way, it is the value of y when x is equal to 0. This can sometimes have real significance for the model that the line offers, but it can also have no significance. In this section, we will see examples of both categories.
How should the line's y-intercept be interpreted?The y-intercept can frequently be used as a beginning point when the line in question represents a group of data or observations.
How can the slope of a least squares regression line be recognised and interpreted?The formula m = r(SDy/SDx) can be used to get the slope of a least squares regression.
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Li gives 3 clues for a mystery polygon. It has 4 sides it has no pairs of parallel sides. Its angles measure 90 degrees 63 degrees 120 degrees and 87 degrees. Which best describes lis mystery polygon?
According to the given information, the mystery polygon is most likely a right trapezoid.
Li's mystery polygon has 4 sides and no pairs of parallel sides, which means it is a quadrilateral. Based on the angles given, 90 degrees, 63 degrees, 120 degrees, and 87 degrees, the mystery polygon is a trapezoid. A trapezoid is a quadrilateral with exactly one pair of parallel sides and no two angles that add up to 180 degrees.
A trapezoid with two right angles (90 degrees) is called a right trapezoid. Since Li's mystery polygon has one angle that measures 90 degrees, it is likely a right trapezoid. The other angles, 63 degrees, 120 degrees, and 87 degrees, do not match the angles of any other specific type of quadrilateral. Therefore, based on the information given, the mystery polygon is most likely a right trapezoid.
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if tyler paid 16 dollars per 4 tickets hw much would 1000 tickets cos 250 or 4000 dollars im cunfused
Answer:
16 per 4 tickets so that 4 a ticket so multiply 4 bye 1000 to get 4000
Step-by-step explanation:
Answer:
$4000 because the amount it costs is larger than the quantity, so the price will be more.
Step-by-step explanation:
Hope it helps! =D
an edge of a square can be expressed as $4x-15$ meters or as $20-3x$ meters. what is its area in square meters?
The area in a square meter is 5$. We can equate the two expressions for the length of the edge of the square and solve for x:
$4x - 15 = 20 - 3x$
$7x = 35$
$x = 5$
The length of one edge of the square is 5 meters, so the area of the square is $5^2 = 25$ square meters. Therefore 5$ is the solution for the $4x-15$ and 20 - 3x$ equation. 15 and 3 will exchange their places to form 7$ and 35$ and 5$ is the end result of the problem that is given.
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if f(3) = 15, f ' is continuous, and 7 f '(x) dx 3 = 16, what is the value of f(7)? f(7) =
The value of f(7) is 31
Now, According to the question:
First fundamental theorem of integral calculus states that “Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function. Then A′(x) = f (x), for all x ∈ [a, b]”.
"Information available from the question"
If f(3) = 15, f ' is continuous, and 7 f '(x) dx 3 = 16,
Since ∀x, f'(x) = f(x) , f is a primitive function of f'.
Therefore,
[tex]=\int\limits^7_3f' {x} \, dx = f(x)|^7_3[/tex]
= f(7) - f(3)
= f(7) - 15 = 16
by the fundamental theorem of calculus (part 2) (because f' is continuous and thus integrable).
= f(7) = 16 + 15
=> f(7) = 31
Hence, The value of f(7) is 31
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how do the values of the 4s in 64.723 and 9.048 compare? the value of the 4 in 64.723 is times the value of the 4 in 9.048
The 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
The values of the 4 in 64.723 and 9.048 are not directly comparable because they represent different places in the number and have different values as a result.
In 64.723, the 4 is the hundredths place, which means it represents 0.04. In this case, the value of the 4 is 0.04.
In 9.048, the 4 is the thousandths place, which means it represents 0.004. In this case, the value of the 4 is 0.004.
To compare the two values of 4, we can divide the value in 64.723 by the value in 9.048:
0.04 / 0.004 = 10
So the value of the 4 in 64.723 is 10 times the value of the 4 in 9.048. This means that the 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
Therefore, the 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
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The 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
The values of the 4 in 64.723 and 9.048 are not directly comparable because they represent different places in the number and have different values as a result.
In 64.723, the 4 is the hundredth place, which means it represents 0.04. In this case, the value of the 4 is 0.04.
In 9.048, the 4 is the thousandths place, which means it represents 0.004. In this case, the value of the 4 is 0.004.
To compare the two values of 4, we can divide the value in 64.723 by the value in 9.048:
0.04 / 0.004 = 10
So the value of the 4 in 64.723 is 10 times the value of the 4 in 9.048. This means that the 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
Therefore, the 4 in 64.723 has a value that is 10 times greater than the value of the 4 in 9.048.
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Use the Rational Root Theorem to determine all POSSIBLE rational zeros. f(x)=2x^3-6x^2+9x-27
Answer:
[tex]\displaystyle{\text{possible rational roots} = \pm 1, \pm \dfrac{1}{2}, \pm 3, \pm \dfrac{3}{2}, \pm 9, \pm \dfrac{9}{2}, \pm 27, \pm \dfrac{27}{2}}[/tex]
Step-by-step explanation:
Rational Root theorem is when we divide the factors of last term by the factors of first term:
[tex]\displaystyle{\text{possible rational roots} = \dfrac{\text{factors of the coefficient of last term (constant)}}{\text{factors of lead coefficient (coefficient of first term)}}}[/tex]
Our constant (last term) is -27. The factors of -27 can be ±1, ±3, ±9, ±27
Our lead coefficient is 2. The factors of 2 can be ±1 and ±2.
Hence:
[tex]\displaystyle{\text{possible rational roots} = \dfrac{\pm 1, \pm 3, \pm 9, \pm 27}{\pm 1, \pm 2}}\\\\\displaystyle{\text{possible rational roots} = \pm 1, \pm \dfrac{1}{2}, \pm 3, \pm \dfrac{3}{2}, \pm 9, \pm \dfrac{9}{2}, \pm 27, \pm \dfrac{27}{2}}[/tex]
All of these are possible roots, meaning that not all of them are real roots of the equation, just a possibility.
A commission is a percentage of a sale. People who conduct sales earn a portion of their income based on how much they sell. Zachary earned 31% commission on a sale of $635. What is the amount of the commission?
The amount of the commission earned by Zachary on the $635 sale is $196.85.
What is the amount of the commission earned by Zachary?A commission is a percentage of a sale earned by a sales person.
Given that;
Commission = 31%Sale price = $635Commission amount = ?Since commission is a percentage of a sale earned
Commission amount = sale price × commission
Plug in the given values and simplify
Commission amount = $635 × 31%
Note that; 31% is the same as 31/100
Commission amount = $635 × 31/100
Commission amount = $635 × 0.31
Multiply $635 and 0.31
Commission amount = $196.85
Therefore, the amount of commission is $196.85.
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how many yards are in 18 miles
there are 31680 yards in 18 miles
Answer:
31680
Step-by-step explanation:
plss help me i need helppp
The missing value of the first coordinate x is 9.
What is the missing x-coordinate?Slope is simply expressed as change in y over the change in x.
Slope m = ( y₂ - y₁ )/( x₂ - x₁ )
Given the data in the question;
Point A (x,8)
x₁ = xy₁ = 8Point B (4,-7)
x₂ = 4y₂ = -7Slope m = 3
Plug the given coordinates into the slope formula and and solve for x.
Slope m = ( y₂ - y₁ )/( x₂ - x₁ )
3 = ( -7 - 8 )/( 4 - x )
Cross multiply
3( 4 - x ) = ( -7 - 8 )
12 - 3x = -7 - 8
12 - 3x = -15
-3x = -15 - 12
-3x = -27
x = -27 / -3
x = 9
Therefore, the value of x is 9.
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Which of the following statements about a boxplot is always false? (A) The confidence interval for the mean lies outside the box. (B) The median lies outside the span of the box. (C) The plot shows only a box, with no outliers or "whiskers". (D) The plot shows no outliers. (E) None of the above.
The median lies outside the span of the box is always false.
Boxplots are particularly helpful for immediately determining a collection of data's location, dispersion, symmetry, or skewness as well as for comparing these characteristics between two or more data sets. Building a boxplot is straightforward.
A box and whisker plot, often known as a box plot, shows a data set's five-number summary. The minimum, first quartile, median, third quartile, and maximum make up the five-number summary.
In a box plot, we draw a box from the first quartile to the third quartile. A vertical line goes through the box at the median. The whiskers go from each quartile to the minimum or maximum.
Thus, The median lies outside the span of the box is always false.
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in a continuous probability distribution, the random variable: group of answer choices can take on an infinite number of values. may have a probability greater than 1.00. has a limited number of values. is discrete.
The correct answer is option 1. In a continuous probability distribution, the random variable can take on an infinite number of values within a given range.
This range is typically represented by the interval between two points on the number line, such as 0 and 100. The probability of a particular value occurring is defined as the area under the curve of the probability density function for that distribution, rather than a single value.
Option 2, "may have a probability greater than 1.00" is incorrect as probabilities are always between 0 and 1, inclusive.
Option 3, "has a limited number of values" is incorrect as, by definition, continuous random variables can take on an infinite number of values.
Option 4, "is discrete" is incorrect as a continuous random variable is not limited to a finite or countable number of values.
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Factor out the greatet common factor. If the greatet common factor i 1, jut retype the polynomial. 30b^96b^818b^7–6
The 5x factored out will be 5x(4x+3)
What is the Greatest Common Factor?
The greatest common factor, which is the product of two or more numbers, is the largest number (GCF). A natural number is created by dividing them by the biggest number (factor). There are a few factors that both share once all the number's factors have been identified. The greatest common factor is the one with the largest number among the common factors. The highest common factor is another name for the GCF (HCF)
The factors of 20x2 (all the values that will divide into 20x2:
1, 2, 4, 5, 10, 20, x, 2x, 4x, 5x, 10x, 20x, x2, 2x2, 4x2, 10x2, 20x2
The factors of 15x:
1, 3, 5, 15, x, 3x, 5x, 15x
The greatest ("largest") one that is common ("in both lists"):
5x
So, with the 5x factored out: 5x(4x+3)
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The full question
Factor out the greatest common factor.
20x2 + 15x
What is moment of inertia and how to calculate It for a rod?
I = (1/12) ML² is a common way to write the moment of inertia of a rod whose axis passes through its center and has mass (M) and length (L).
What is moment of inertia?A body's moment of inertia, which is determined by multiplying each particle's mass by the square of its distance from the rotational axis, is what causes it to resist angular acceleration.A measurement of an object's resistance to changes in its rotating motion is the moment of inertia. A body's resistance to variations in rotation is measured by its cross-moment section's of inertia. Depending on how far away from the center of the body each component is. The moment of inertia is calculated as the sum of the mass of the section and the square of the distance from the centroid of the section to the reference axis.Therefore,
When the rod's axis passes through the end of the rod, the moment of inertia can also be stated using a different formula. In this instance, we apply I = 13 ML².
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draw the graph of y=2x-3 for value of x from -1 to +3 use a scale of 2cm to the unit on both axis
The standard equation of a line is expressed as y = mx + b
How to graph the equation?m is the slope
b is the y-intercept
Given the equation y = 2x - 2
If x = -2
y = 2(-2) - 2
y = -4-2
y = -6
If x = 4
y = 2(4) - 2
y = 8-2
y = 6
This shows that the point (-2, -6) and (4, 6 )must be on the line graph. FInd the graph attached below.
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-2x^{3}\left(-x^{3}\right)
The required simplified solution of the given expression is 2x⁶.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given expression,
= -2x³(-x³)
Since we know that, -a × - a = a²,
= 2x³⁺³
= 2x⁶
Thus, the required simplified solution of the given expression is 2x⁶.
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Find an equation of the tangent plane to the given surface at the specified point.
z =( ex − y 5), (3, 3, 6)
Please explain steps
This is an equation is e⁻¹³⁸ˣ - 5e⁻¹³⁹ × 81y + z = 6
What do you mean by tangent?In mathematics, a tangent is a line that touches a curve at a single point and has the same slope as the curve at that point. The tangent can be thought of as an approximation of the curve at the point where it touches it.
The concept of the tangent is used in calculus and other branches of mathematics to study the properties of curves and to analyze their behavior. The derivative of a function at a point is defined as the slope of the tangent to the curve of the function at that point. The derivative is a fundamental concept in calculus and is used to study the rate of change of a function, the optimization of functions, and to solve problems in many different fields.
To find the equation of the tangent plane to the surface z = e^(x - y^5) at the point (3, 3, 6), we need to find the normal vector of the surface at that point, and use that to find the equation of the plane.
The normal vector can be found using the gradient of the surface, which is given by the partial derivatives of the function:
∇f = ∇(e^(x - y^5)) = (e^(x - y^5), -5e^(x - y^5)y^4, 1)
Evaluating this at the point (3, 3, 6), we get:
∇f(3, 3, 6) = (e^(3 - 3^5), -5e^(3 - 3^5)3^4, 1) = (e^-138, -5e^-138 * 81, 1)
So the normal vector of the surface at the point (3, 3, 6) is (e^-138, -5e^-138 * 81, 1).
To find the equation of the tangent plane, we need to find a point on the plane, and use that along with the normal vector. We already know that the point (3, 3, 6) is on the surface, so we can use that as a point on the plane.
Using the point-normal form of a plane, the equation of the tangent plane is given by:
ax + by + cz = d
Where (a, b, c) is the normal vector and (x⁰, y⁰, z⁰) is a point on the plane.
Substituting the values, we get:
e⁻¹³⁸ˣ - 5e⁻¹³⁹ × 81y + z = 6
This is the equation of the tangent plane to the surface z = e^(x - y^5) at the point (3, 3, 6).
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