We can prove this statement by using the Pigeonhole Principle.
Let's say we have a collection of n integers and we want to demonstrate that at least one of them, or the sum of several of them, is a multiple of n.
Consider the 0, 1, 2,..., n-1 possible remainders when these integers are divided by n.
There must be at least one remainder that appears more than once since there are n numbers and only n potential remainders.
Case 1: Choose two integers from the set, say a and b, such that a b (mod n), that is, a and b have the same remainder when divided by n, if a remainder occurs more than once.
Since their remainders cancel out, their difference a - b is a multiple of n.
As a result, we have identified an integer that is a multiple of n in the set.
Case 2: The sum of any two integers in the set may be taken into consideration if there are no remainders that appear more than once.
Two integers a and b can be added together using the formula a + b = qn + r, where q is the quotient and r is the residual after a + b is divided by n.
The Pigeonhole Principle states that two pairs of integers must have the same remainder when combined modulo n since there are n potential remainders but only n-1 possible values for the remainder r (from 0 to n-1).
These pairings should be (a1, b1) and (a2, b2), where (a1 b1 (mod n) and (a2 b2) (mod n).
Then, there is:
(a1 + b1) + (a2 + b2) = (a1 + a2) + (b1 + b2) ≡ 0 (mod n)
thus the remainders of a1 + a2 and b1 + b2 are equal modulo n.
As a result, n is a multiple of the sum of these four numbers.
In either scenario, we have demonstrated that any collection of n integers contains either an individual integer that is a multiple of n or an aggregate of many numbers that is a multiple of n.
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What is 20% of 50 with a model to show
Answer:
10
Step-by-step explanation:
20% = 0.2
What is 20% of 50?
We take
50 x 0.2 = 10
So, 20% of 50 is 10
PLEASE HELP DUE ASAP !
The four points on the graph of this function f(x) = √(x + 1) + 4 has been plotted on the graph shown in the image attached below.
What is a graph?In Mathematics, a graph can be defined as a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis respectively.
Next, we would use an online graphing calculator to plot the given absolute value function f(x) = √(x + 1) + 4 as shown in the graph attached below.
By observing critically the graph (see attachment), the four ordered pairs (points) that fit on its axes include the following:
Ordered pair = (-1, 4), which is the leftmost.
Ordered pair = (0, 5).
Ordered pair = (3, 6).
Ordered pair = (8, 7).
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Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie. (a) Describe and compare how the runners run the race. (b) At what time is the distance between the runners the greatest? (c) At what time do they have the same velocity?
(a) - A runs the race at a constant speed, never speeding up or slowing down. B accelerates throughout the race, starting out slower than A and, by the end, running faster than A.
(b) - Based on the graph, it appears that they are furthest apart after 8 seconds, when they are approximately 30 meters apart.
(c)- The two graphs appear to have the same slope (i.e., velocity) 9 or 10 seconds into the race.
graph attached below,
constant speed
When an object travels the same distance in the same period of time, it is said to be traveling at a constant speed. At constant speed, an object travels a uniform distance in an equal interval of time. The equation of the speed can be given as: S = d t.
Slope
Slope is a measure of the steepness of a line.
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Solve the equation 2x+4 1/5 =9 Explain the steps and properties used
Answer:
x = 2.4
Step-by-step explanation:
[tex]2x + 4 \times \frac{1}{5} = 9 \\ = 2x + \frac{21}{5} = 9 \\ = 5 \times 2x + 5 \times \frac{21}{5} = 9 \times 5 \\ = 10x + 21 = 45 \\ = 10x = 45 - 21 \\ = 10x = 24 \\ \frac{10x}{10} = \frac{24}{10} \\ \times = 2.4[/tex]
The graph of g(x), shown below in pink, has the same shape as the graph of
f(x) = x², shown in gray. Which of the following is the equation for g(x)?
in
OA. g(x) = (x-3)²-1
OB. g(x) = (x + 1)² - 3
OC. g(x)=(x-1)²-3
OD. g(x) = (x+3)2-1
f(x)
(0,0)
g(x)
(3,-1)
on solving the provided question we can say that the function at (h, k) = (1, - 3), will be [tex]g(x) = (x - 1)^2 - 3- > B[/tex]
what is function?Mathematics deals with numbers and their variants, equations and associated structures, forms and their positions, and locations where they might be found. The term "function" describes the connection between a group of inputs, each of which has a corresponding output. A function is an association between inputs and outputs where each input results in a single, unique outcome. A domain and a codomain, or scope, are assigned to each function. Functions are often denoted by the letter f. (x). The input is an x. On functions, one-to-one functions, many-to-one functions, within functions, and on functions are the four primary categories of functions that are available.
The graph of g(x) has its vertex at (1, - 3)
The equation of a parabola in vertex firm is
[tex]y = a(x - h)^2 + k[/tex]
(h, k) =coordinates of the vertex
and
a is a multiplier
(h, k) = (1, - 3),
[tex]g(x) = (x - 1)^2 - 3- > B[/tex]
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Can you find the area of these shapes
Answer:
1. 40
2. 351
3. 49
Step-by-step explanation:
just multiply the 2 numbers!
5 times 8 is 40
13 times 27 is 351
7 times 7 is 49
(square is the same on every side so every side is 7)
If r(t) is the position vector for a smooth curve C, and Î (t), Ñ(t), and B(t) are unit tangent vector, principal unit normal vector, and binormal unit vector, respectively, then 1. B(t) · Î (t) = 2. Þ(t) · B(t) = 3. ÎN(t) · (B(t) – 5ÊN(t)) = 4. Þ(t) x Î (t) = (enter an upper case T for Î(t), N for ÎN(t), and B for B(t))
If r(t) is the position vector for a smooth curve C, and Î (t), Ñ(t), and B(t) are unit tangent vector, principal unit normal vector, and binormal unit vector, respectively, then (1) B(t) · Î(t) = 0, (2) Þ(t) · B(t) = 0, (3) ÎN(t) · (B(t) – 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0 and (4) Þ(t) x Î (t) = B(t).
(1) Since B(t) is the cross product of Î(t) and Ñ(t), it is perpendicular to both Î(t) and Ñ(t). Therefore, B(t) · Î(t) = 0.
(2) Þ(t) is the derivative of r(t), and B(t) is defined as the cross product of Î(t) and Ñ(t). Therefore, Þ(t) and B(t) are both orthogonal to Î(t). Hence, Þ(t) · B(t) = 0.
(3) ÎN(t) is the cross product of Î(t) and Ñ(t), and B(t) is also the cross product of Î(t) and Ñ(t). Therefore, B(t) - 5ÊN(t) is parallel to ÎN(t). Hence, ÎN(t) · (B(t) - 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0.
(4) The cross product of two vectors is orthogonal to both of the vectors. Therefore, Þ(t) x Î(t) is orthogonal to both Þ(t) and Î(t), and hence it is parallel to Ñ(t). Therefore, Þ(t) x Î(t) is equal to B(t). So the answer is B.
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Calculate the covariance and correlation between the random variables X and Y. 1.0 1 1/4 1.5 2 1/8 1.5 3 1/4 2.5 6 1/4 3.0 5 1/8 Round your answers to two decimal places (e.g. 9.87). OXY=
The correlation between X and Y is -0.648, rounded to two decimal places.
First, we need to calculate the means of X and Y:
mean(X) = (1 + 1.5 + 1.5 + 2.5 + 3) / 5 = 2
mean(Y) = (1/4 + 1/8 + 1/4 + 1/4 + 1/8) / 5 = 0.15
Next, we can calculate the covariance using the formula:
cov(X, Y) = E[(X - mean(X)) * (Y - mean(Y))] = Σ[(X - mean(X)) * (Y - mean(Y))] / (n - 1)
where n is the number of data points. Substituting the values, we get:
cov(X, Y) = [(1 - 2) * (1/4 - 0.15) + (1.5 - 2) * (1/8 - 0.15) + (1.5 - 2) * (1/4 - 0.15) + (2.5 - 2) * (1/4 - 0.15) + (3 - 2) * (1/8 - 0.15)] / 4
= -0.035
Therefore, the covariance between X and Y is -0.035.
Finally, we can calculate the correlation coefficient using the formula:
corr(X, Y) = cov(X, Y) / (stddev(X) * stddev(Y))
where stddev is the standard deviation. We can calculate the standard deviations as follows:
stddev(X) = sqrt([Σ(X - mean(X))^2] / (n - 1)) = 0.8292
stddev(Y) = sqrt([Σ(Y - mean(Y))^2] / (n - 1)) = 0.0632
Substituting the values, we get:
Corr(X, Y) = -0.035 / (0.8292 * 0.0632) = -0.648
Therefore, the correlation between X and Y is -0.648, rounded to two decimal places.
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n people line up to board a plane. each has a boarding pass with assigned seat. however, the first person has lost the boarding pass and takes a random seat uniformly. after that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats uniformly at random otherwise. denote by pnthe probability that the last person to board sits in the assigned seat. show that pn
The probability that the last person to board sits in the assigned seat is always 1/2, regardless of the number of people n in the line.
The issue can be tackled utilizing numerical enlistment.
Base Case:
At the point when n = 2, there are two travelers and two seats. The principal traveler takes an irregular seat, and the subsequent traveler will sit in his doled out seat with likelihood 1/2 or take the other seat with likelihood 1/2. Consequently, p2 = 1/2.
Inductive Speculation:
Accept that for some certain number k, pk = 1/2.
Inductive Step:
Think about n = k+1 travelers. The principal traveler takes an irregular seat. There are two cases to consider:
Case 1: The primary traveler sits down relegated to the k+1-th traveler. For this situation, the last traveler will be ensured to sit in his doled out seat, and the leftover k travelers can be considered as a subproblem with similar circumstances. By the inductive speculation, the likelihood that the last traveler in the subproblem sits in his alloted seat is 1/2. Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is 1/(k+1) + (k/(k+1)) * 1/2 = (k+2)/(2(k+1)).
Case 2: The primary traveler sits down other than the one doled out to the k+1-th traveler. For this situation, the issue lessens to the case with k travelers, and the likelihood that the last traveler in the decreased issue sits in his doled out seat is pk = 1/2 (by the inductive speculation). Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is (k/(k+1)) * 1/2 = k/(2(k+1)).
Joining the two cases, we have:
pk+1 = (k+2)/(2(k+1)) + k/(2(k+1)) = (k+1)/(2(k+1)) = 1/2.
Consequently, by numerical acceptance, the likelihood that the last individual to board sits in the appointed seat is dependably 1/2, no matter what the quantity of individuals n in the line.
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let v be the (real) vector space of all functions f from r into r. which of the following sets of functions are subspaces of v?
The following sets of functions are subspaces of v, if v be the (real) vector space of all functions f from r into r is: all f such that f(x²) = f(x²), all f which are continuous.
V is a Vector- Space of all real- valued functions over field of real numbers R and W consists of all real- valued even functions which are bounded also as a subset of V.
Let f , g belong to W then f , g both are even & bounded. Hence ;
(1) (f + g ) is even & bounded because ,
(f + g )(-x ) = f(- x )+ g(-x) = f( x) + g( x )
=( f+g)( x) and for all x€ R and
= | f (x) + g(x) | </= |f (x) | + | g (x) |
</= (c +d) = C(constant)
and similarly for any scalar k€ R , (kf ) will be an even function and it will be bounded also.
A set whose elements, frequently termed vectors, can be added to and multiplied ("scaled") by figures known as scalars is referred to as a vector space (also known as a linear space). Real numbers make up scalars most of the time, but they can also be complex numbers or, more broadly, components of any field. Certain conditions, referred to as vector axioms, must be met by the operations of vector addition and scalar multiplication.
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the standard deivation expressed as a percent of the mean is the group of answer choices mean standard error standard error of the mean z-score coefficient of variation
The standard deviation expressed as a percent of the mean is the D: coefficient of variation.
The coefficient of variation (CV) is a statistical measure that expresses the standard deviation of a data set as a percentage of the mean. It is particularly useful for comparing the variability of different data sets with different units or scales, and for identifying the degree of variation relative to the mean. The formula for the coefficient of variation is:
CV = (standard deviation / mean) x 100%
So, the coefficient of variation is a dimensionless quantity that measures the relative dispersion of the data.
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According to a recent study, 21% of peanut M&M's are brown, 13% are yellow, 3% are red, 24% are blue, 16%
are orange, and 24% are green. Assume these proportions are correct and suppose you randomly select four
peanut M&M's from an extra-large bag of the candies. Calculate the following probablities. Also calculate
the mean and standard deviation of the distribution. Round all solutions to four decimal places, if
necessary.
P (X=4) = 0.0028
P(X=3) or P(X=4) = 0.0326
P (X<=4) = 0.9999
P(X>=4) 0.0029
The standard deviation is 0.8198
What is Standard Deviation?Standard deviation is a statistical measure of how to spread out a set of data is from its mean or average value. It measures the degree of variation or dispersion of a dataset, which helps in understanding how much the individual data points deviate from the average.
To solve for:
1. P(X=4) = [tex](\frac{5}{4}) (0.16)^4 (1-0.16)^5^-^4[/tex]
= 0.0028.
2. P(X=3) or P(X=4) = [tex](\frac{5}{3}) (0.16)^3 ((1-0.16)^2\\[/tex]
= 0.0326
3. P(X<=4) = [tex](\frac{5}{x})(0.16)^x(1-0.16)5^-^x[/tex]
= 0.9999
4. P(X>=4) = P(X=4) + P(X=5)
= [tex](\frac{5}{4}) (0.16)^4(0.84)^1 + (\frac{5}{5})(0.16)^5[/tex]
= 0.0029.
5. u = np = 0.8
[tex]\sqrt{np(1-p)}[/tex]
= 0.8198
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A normal distribution (X) has a mean of 100 and a standard deviation of 10.
What is the probability that X is between 90 and 110?
The probability that X is between 90 and 110 = 0.2 if a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40.
What is a normal distribution?The majority of the observations are centered around the middle peak of the normal distribution, which is a continuous probability distribution that is symmetrical around its mean.
The probabilities for values that are farther from the mean taper off equally in both directions. Extreme values in the distribution's two tails are likewise rare. Not all symmetrical distributions are normal, even though the normal distribution is symmetrical.
A normally distributed random variable x has a mean of 100
P(x < 90) = 0.4
The probability that X is between 90 and 110
Mean is 100
Hence P (< 100) = 0.5 as Mean is centered
Hence
P (90 < X < 100) = 0.5 - 0.4 = 0.1
The difference between 100 and 90 is 10
The difference between 100 and 110 is 10
and 100 is Mean
Hence P (100 < X < 110) = P (90 < X < 100) = 0.1
Adding Both
P ( 90 < X < 110) = 0.1 + 0.1 = 0.2
The probability that X is between 90 and 110 = 0.2
Using z score table.
Z = ( Value - Mean) /SD
Z score for 0.4 = -0.254
-0.253 = (90 - 100)/SD
SD ≈ 39.53
Z = ( 110 - 100)/39.53
Z = 0.253
For z score < 0.253, the probability is 0.6
Hence Probability in Between is 0.6 - 0.4 = 0.2
The probability that X is between 90 and 110 = 0.2
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Can you help me solve problem #3???
The two-column method table to prove that the triangles ΔLMN and ΔLJK are similar, ΔLMN ~ ΔLJK, where LM/LJ = LN/LK can be completed as follows
Statements [tex]{}[/tex] Reasons
LM/LJ = LN/LK [tex]{}[/tex] Given
∠L ≅ ∠L [tex]{}[/tex] Reflexive property of congruency
ΔLMN ~ ΔLJK [tex]{}[/tex] SAS similarity theorem
What are similar triangles?Similar triangles are triangles that have the same shape, and interior angles, but which may have different side lengths such that the proportion of the corresponding sides are equivalent.
The details of the reasons used to prove the similarity of triangles ΔLMN and ΔLJK can be presented as follows;
Reflexive property of congruency
The reflexive property of congruency states that a figure, such as an angle or a length is congruent to itself
SAS similarity theorem
The SAS, Side-Angle-Side similarity theorem states that if two sides in one triangle are proportional to two sides in another triangle, and the included angle between the two sides in both triangles are congruent, then the two triangles are similar.
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Write a function that models the data.
The function that models the data is [tex]y = 42.(\frac{1}{2})^x[/tex].
What is the formula for exponential growth and exponential decaying function?The formula for exponential growth is [tex]y = y_0e^{(kt)}.[/tex]
The formula for exponential decay is [tex]y = y_0e^{(-kt)}.[/tex]
Let, The given exponential be [tex]y = a(b)^x[/tex] .
Now, At (0, 42).
42 = ab⁰.
a = 42.
At (1, 21).
21 = 42.b¹.
b = 1/2.
Therefore, [tex]y = 42.(\frac{1}{2})^x[/tex]is the required function.
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The double dot plot below shows the number of hours Kayla and Carmen studied during a two week period in college. Determine the most appropriate measure of variation for each data set. What is the difference between the centers?
The measure of variation include range, variance, iqr etc
What are the measure of variationIn statistics, a measure of variation is a numerical value that describes how spread out or dispersed a set of data is. The most common measures of variation are:
Range: The range is the difference between the maximum and minimum values in a data set. It gives an idea of the spread of the data, but is sensitive to outliers.
Interquartile range (IQR): The IQR is the range of the middle 50% of the data. It is less sensitive to outliers than the range.
Variance: The variance is the average of the squared differences of each data point from the mean. It measures how much the data is spread out from the mean.
Standard deviation: The standard deviation is the square root of the variance. It is a common measure of variation and indicates the typical amount that each data point deviates from the mean.
Coefficient of variation: The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. It is used to compare the variation of data sets with different means.
An overview was given due to incomplete information.
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How many solutions does each system of {y+4x=7 −2y−4=8x
The system of equations y+4x=7 and −2y−4=8x has no solutions.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given system of equations are
y+4x=7 ...(1)
−2y−4=8x...(2)
2y+8x=-4
From equation 1
y=7-4x
Simplifying and solving for x, we get:
-14 + 8x - 4 = 8x
-18 = 0
This is a contradiction, since -18 is not equal to 0. Therefore, there is no solution that satisfies both equations.
Hence, the system of equations has no solutions.
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Answer to a question
The type and degree of association is As the time a basketball player practices increases, the number of points scored in a game increases with a strong nonlinear association, the correct option is B
What does correlation coefficient convey?The correlation coefficient is the degree of association between two quantities in term of linear relation.
The range of correlation coefficient is -1 to 1
when the correlation is -1, then that means as the one quantity increases, the other quantity decreases (linearly)
when the correlation is 0, then there is no linear relationship between two variables.
when the correlation is 1, then that means as the one quantity increases, the other quantity increases(linearly) and vice versa for decrement.
Given the graph
Now, a function is an expression, that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) .
Linear function, the graph is a straight line
Therefore, by correlation coefficient answer will be B
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Please help !! I can’t find the answer for this
Answer:
400.059
How do you write this problem as a decimal?
So the first part of this question is four hundred:
400.
But then.. It says fifty nine hundredths. So lets write it out.
The closer we get to the decimal it gets " smaller " example the hundredths is all the way in back and the tenths is all the way to the front! So with that knowledge lets figure it out:
( to make it nice and simple for you ):
0.059
The " 9 " is in the hundredths zone and the " 5 " is in the tenths zone. So just like on top it looks like that.
0.059+400=
400.059
Thus, your answer is 400.059
(3x³-4x² + x + 7) + (x-1)
Answer:3x^3−4x^2+2x+6
Step-by-step explanation:
1).subtract the numbers
2).combine like terms
recipe for 36 cupcakes calls for ¾ of a cup of butter. If Rob wants to make 24 cu cups of butter does he need?
3/12 =1/4 hopefully this answer works for you
Mike is making macaroni salad each bowl
The numbers of cups that macaroni will he use will he use if he wishes makes 27 bowls is 9 cups.
How do we find the numbers of cups that macaroni will he use?If Mike makes 27 bowls of macaroni salad, we can find out how many cups of macaroni he will need by multiplying the amount of macaroni needed for one bowl by the number of bowls:
= 1/3 cup of macaroni per bowl x 27 bowls
= 9 cups of macaroni
Therefore, Mike will need 9 cups of macaroni to make 27 bowls of macaroni salad.
Full question "Mike is making macaroni salad. For each bowl of macaroni salad, he needs 1/3 cup of macaroni. How many cups of macaroni will he use will he use if he makes 27 bowls of?"
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Graph the function f(x) = 3√x -2.
the graph of the function is attached below.
What is the function?A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.
Given a function, f(x) = 3√x -2
To graph the function f(x) = 3√x -2, we can follow these steps:
Choose some x values to evaluate the function. It's a good idea to pick values that will give us an idea of the general shape of the graph. Let's choose x = 0, 1, 8, and 27.Plug in these values of x into the function to find the corresponding y values:f(0) = 3√0 - 2 = -2
f(1) = 3√1 - 2 = 1
f(8) = 3√8 - 2 ≈ 4.9
f(27) = 3√27 - 2 ≈ 7.2
Plot the points (0, -2), (1, 1), (8, 4.9), and (27, 7.2) on a coordinate plane.Draw a smooth curve through the points to show the shape of the graph.Here, the graph of f(x) = 3√x -2 is attached below.
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Find a plane through the points P1(12,3),P2(3,2,1) and perpendicular to the plane 4−y+2z=7.
The equation of the plane passing through P1(12,3), P2(3,2,1), and perpendicular to the plane 4−y+2z=7 is -3x + 51y - 8z = -39.
To find a plane that passes through the points P1(12,3) and P2(3,2,1) and is perpendicular to the plane 4−y+2z=7, we can follow these steps:
Find the normal vector of the given plane by taking the coefficients of x, y, and z in the equation 4−y+2z=7. The normal vector is (1, -1, 2), because the coefficients of x, y, and z are 1, -1, and 2 respectively.
Find the direction vector of the line passing through P1 and P2 by subtracting the coordinates of P1 from P2. The direction vector is (-9, -1, -2), because P2 - P1 = (3-12, 2-3, 1-0) = (-9, -1, -2).
Find the cross product of the normal vector of the given plane and the direction vector of the line passing through P1 and P2 to get the normal vector of the plane we are looking for. The cross product is:
(1, -1, 2) x (-9, -1, -2) = (-3, -17, -8)
Plug one of the given points, say P1(12,3), and the normal vector we found in step 3 into the point-normal form of the equation of a plane:
-3(x - 12) - 17(y - 3) - 8(z - 0) = 0
Simplifying, we get:
-3x + 51y - 8z = -39
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the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, find its perimeter.
If the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, the perimeter of the rectangle is 80 cm.
Let's first use the given information to write two equations relating the length and width of the rectangle:
The length L is four times the width W: L = 4W
The area A is 256 cm²: A = LW = 256
Substituting equation 1 into equation 2, we get
4W^2 = 256
Solving for W, we get:
W^2 = 64
W = 8 (since we're looking for a positive value)
Substituting this value back into equation 1, we get:
L = 4W = 32
Therefore, the length and width of the rectangle are 32 cm and 8 cm, respectively.
The perimeter of the rectangle is twice the sum of its length and width:
P = 2(L + W) = 2(32 + 8) = 80 cm
So the perimeter of the rectangle is 80 cm.
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which pair of lines are parallel
The pairs of linear equations that are parallel are:
2 and 4.
Which pair of lines are parallel?Two linear equations:
y = a*x + b
y = c*x + d
Are parallel if the two slopes are equal and the y-intercepts are different, then:
a = c
b ≠ d
Let's write all the lines in the slope-intercept form:
1) 4x + 3y = 15
3y = 15 - 4x
y = (-4/3)*x + 15/3
y = (-4/3)*x + 5
2) 3x - 4y = -8
-4y = -8 - 3x
y = 2 + (3/4)x
So we can see that lines 2 and 4 are parallel.
If we rewrite line 3 we will get:
y + 1 = (4/3)*(x - 6)
y = (4/3)*x - 8 -1
y = (4/3)*x - 9
Then we conclude that only lines 2 and 4 are parallel.
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Jeremiah wanted to create step-function that increases more gradually than the greatest integer function, f (r) IlrIl: He decided that the function g(=) [Iellwould work Help Jeremiah by filling in the missing data below; then determine if he was right or wrong that glx) increases more gradually than flx): flx) g(x) 0.51 1.5/1 2.5 3.513 Jeremiah was right to say that g(r) increases more gradually than f()
Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).
It seems like some of the numbers in the table are cut off. Nevertheless, we can determine if Jeremiah was correct in saying that g(x) increases more gradually than f(x).
From the given information, we know that f(x) = IlxIl (the greatest integer function). This function increases by 1 whenever x crosses an integer value.
Jeremiah's function is g(x) = [x] (the floor function), which is the greatest integer less than or equal to x. This function increases by 1 only when x is an integer.
Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).
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let t(n) be number of all the positive divisors of n. prove that t(n) is odd only if n is a perfect square
We must establish both of the following statements in order to demonstrate that t(n) is unusual if and only if n is a perfect square.
First instruction: If t(n) is odd, n is a perfect cube.
Assume that t(n) is strange. All the positive divisors of n should be d 1, d 2, ldots, and d k. So, we understand that k=t(n) is unusual. The divisors can be combined into frack2 pairs, with a sum of n for each pair:
(d 1, d k), (d 2, d k-1)
If k is odd, only one divisor remains, which, if n is a perfect cube, is the square root of n. The conclusion is that n must be a perfect square if t(n) is unusual.
Second instruction: If t(n) is odd, then n is a perfect square and n is.
Let's assume that n is a perfect square, such as n=m2. Then, the positive divisors of n appear in pairs, denoted by (d, fracnd), where d spans all the divisors of m. We only need to tally the divisor d when d=fracnd because the product d cdot fracnd = n is not a perfect square if d is not equal to fracnd. Since m has an odd number of divisors, t(n) is only odd if and only if m.
We can look at m's prime factorization to understand why it has an odd amount of divisors. Write m=p 1,p 2,a 1,a 2,ldots,p k where p 1,p 2,a 1,a 2,ldots,p k are distinct prime numbers and a 1,a 2,ldots,a k are positive integers.
As a result, we have demonstrated that t(n) is odd only when n is a perfect square.
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What is the height h for the base that is 5/4 units long?
The height of the triangle is 3/5.
What is a right-angled triangle?A triangle is said to be right-angled if one of its inner angles is 90 degrees, or if any one of its angles is a right angle.
Given:
A right-angled triangle.
Let the values of the hypotenuse in parts be m and n.
So,
(1)² = (5/4)m
m = 4/5
And n = 9/16 x 4/5
n = 9/20
So, the height of the triangle is,
h² = 9/20 x 4/5
h² = 9/25
h = 3/5
Therefore, h = 3/5.
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A scientist places a cell in a Petri dish. At the end of 1 hour, the cell divides so
that there are 2 cells in the Petri dish. At the end of 2 hours, those cells divide so
there are 4 cells in the Petri dish. The cells continue to divide this way every hour.
Use the expression 2 to the 10th power to find the number of cells in the Petri dish after 10 hours.
Show your work.
The number of cells in the Petri dish after 10 hours is 2¹⁰ = 1024 cells in the dish.
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,
The number of cells in the Petri dish after each hour can be represented by the expression 2ⁿ, where n is the number of hours that have passed.
So, after 1 hour, there are 2¹ = 2 cells in the dish.
After 2 hours, there are 2² = 4 cells in the dish.
After 3 hours, there are 2³ = 8 cells in the dish.
And so on.
Therefore, after 10 hours, there are 2¹⁰ = 1024 cells in the dish.
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