Answer:
x is 3 and y is -3
Step-by-step explanation:
[tex]x + y = 0 \\ x + y - y = 0 - y \\ x = - y \\ \\ 8x + 9y = 3 \\ (8 \times - y) + 9y = 3 \\ - 8y + 9y = 3 \\ y = 3[/tex]
X = -3
g if the determinant of a matrix is , and the matrix is obtained from by adding times the second row to the first, then
Therefore the solution to the given matrix problem is
a)det(B) = det(A) = 3 ,det(B) = 3*det(A) = 9 and c)det(B) = -det(A) = -3
What is matrix ?A matrix is a rectangular array or table that contains numbers, symbols, or expressions that are arranged in rows and columns to represent a mathematical object or a characteristic of such an entity.
Here,
The following will be demonstrated using determinant properties:
A)Det(B) = 3
B) Det(B) = 9
C) det(B) = -3
Assuming A is a 4x4 matrix, we can see that:
det(A) = 3.
a) In this case, we are adding one row to another row by performing a transformation to the data in A. These operations don't alter the matrix's determinant, therefore in this instance:
det(B) = 3 = det(A)
b) If K is used to multiply a row of A to produce B, then:
B = det(K*det) (A).
In this situation, we have:
det(A) = 3*det(B) = 9
c) If we switch two rows around once (this is an odd permutation), the determinant's sign changes:
det(B) = -det(A) = -3
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The complete question is "If the determinant of a 4 by 4 matrix A is det(A) = 6, and the matrix D is obtained from A by adding 9 times the third row to the first, then det(D) is equal to?"
Ms.Jones bought a piece of cloth measuring 3 3/4 meters by 2 2/7. She bought the cloth at $14 per square meter. How much did Ms.Jones pay for the cloth?
Answer:
$120
Step-by-step explanation:
1) rewrite the mixed numbers into improper fractions. We do this by multiplying the denominator by the whole number, and then adding the numerator. The numerator stays the same:
[tex]3 \frac{3}{4} = \frac{(4 \times 3) + 3}{4} = \frac{15}{4} [/tex]
[tex]2 \ \frac{2}{7} = \frac{(7 \times 2) + 2}{7} = \frac{16}{7} [/tex]
So the length of the cloth = 15/4 meters
width of the cloth = 16/7 meters
2) Calculate the area of the cloth:
area = length × width
[tex]area \: = \: \frac{15}{4} \times \frac{16}{7} = \frac{240}{28} = \frac{60}{7} (simplest form)[/tex]
area = 60/7 m²
3) The cloth costs $14 per square meter, so the total cost of the cloth is:
the area of the cloth × cost of cloth per square meter
[tex]cost \: = \: \frac{60}{7} \times 14 = \frac{840}{7} = 120[/tex]
the cost is $120
Find a parametric representation for the part of the sphere (a surface, not a solid) of radius 4 centered at the origin that lies (a) inside the cone z= 3(x2 +y 2) (b) inside the cone x= 31(y 2 +z 2 )
c) inside the cone y= x +z 2
Hint: you may want to use spherical-like coordinates, where the roles of x,y and z are permuted.
Answer:
Read carefully below
Step-by-step explanation:
We can use spherical coordinates to represent the points on the sphere. These coordinates are typically denoted as (r, θ, φ), where r is the radius of the sphere, θ is the polar angle, and φ is the azimuthal angle.
In order to find a parametric representation of the part of the sphere that lies inside the given cones, we can use the following approach:
First, we need to find the intersection of the sphere and the cone. This is the set of points that satisfy the equations of both the sphere and the cone.
Then, we can use spherical coordinates to represent these points on the sphere.
Finally, we can use these spherical coordinates as parameters in a parametric equation of the form x = x(r, θ, φ), y = y(r, θ, φ), z = z(r, θ, φ) to represent the points on the part of the sphere that lies inside the cone.
Let's use this approach to find a parametric representation of the part of the sphere that lies inside the cone z = 3(x^2 + y^2) in the first case.
To find the intersection of the sphere and the cone, we need to solve the following system of equations:
x^2 + y^2 + z^2 = 16 (equation of the sphere)
z = 3(x^2 + y^2) (equation of the cone)
Substituting the second equation into the first equation, we get:
x^2 + y^2 + 9(x^4 + y^4) = 16
This equation represents an ellipse in the xy-plane. We can use the parametric equations of an ellipse to represent the points on this ellipse:
x = a * cos(t)
y = b * sin(t)
where a and b are the semi-major and semi-minor axes of the ellipse, respectively, and t is a parameter that varies from 0 to 2π.
We can now use spherical coordinates to represent the points on the part of the sphere that lies inside the cone. We can let r be the radius of the sphere (r = 4), θ be the polar angle (0 ≤ θ ≤ π), and φ be the azimuthal angle (0 ≤ φ ≤ 2π).
The parametric equations of the sphere in spherical coordinates are:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
Substituting the equations for x and y from the ellipse into the equations for x and y from the sphere, we get:
x = 4 * sin(θ) * cos(φ) = a * cos(t)
y = 4 * sin(θ) * sin(φ) = b * sin(t)
z = 4 * cos(θ) = 3(x^2 + y^2) = 3(a^2 * cos^2(t) + b^2 * sin^2(t))
We can solve for a and b in terms of r and θ:
a = 4 * sin(θ)
b = 4 * sin(θ)
Therefore, the parametric equations for the part of the sphere that lies inside the cone z
Solve the problem by finding x. I don’t know how to? Do you.
The solution of the logarithmic equation:
ln(x - 7) = 3
Is x = 27.09
How to solve the logarithmic function?Here we have the following logarithmic function:
ln(x - 7) = 3
And we want to solve this for x.
Remember that the logarithmic function is the inverse of the exponential function, so now we can use the exponential function in both sides so we get:
exp( ln(x - 7) ) = exp(3)
x - 7 = exp(3)
Now we can add 7 in both sides so we isolate x, we will get:
x = exp(3) + 7
x = 27.09
That is the solution of the logarithmic equation.
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A 6 percent, semiannual coupon bond has a yield to maturity of 7.4 percent and a Macaulay duration of 5.7. The bond has a modified duration of _____ and will have a _____ percentage increase in price in response to a 25 basis point decrease in the yield to maturity5.4966; 1.37
25 bps decrease in interest rate, bond price increases by 1.425% when A bond with a 6 percent semiannual coupon has a 5.7.
Given that,
A bond with a 6 percent semiannual coupon has a 5.7 Macaulay duration and a 7.4 percent yield to maturity. A 25 basis point drop in yield to maturity will result in a ____ percentage price rise for the bond, which has an adjusted duration of .
We have to fill the blank.
We know that,
% increase in bond price = Duration of bond × % change in interest rate
= 5.70×0.25%
= 1.4250%
Therefore, 25 bps decrease in interest rate, bond price increases by 1.425% when A bond with a 6 percent semiannual coupon has a 5.7.
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Under an anticockwise rotation about the point O, AOAB is mapped onto AOA'B'. Determine the angle of rotation.
Check the picture below.
let's notice that the triangle is an equilateral triangle, meaning all its sides are congruent and likewise, all its angles are also congruent.
An electric car battery, when fully charged, can travel 260 miles. The car uses 232 miles of charge on a drive.
Enter the percentage of miles the car has left in battery charge?
The percentage of miles the car has left in battery charge is 10.77%
Total distance traveled when fully charged = 260 miles
The total charged used by the car = 232 miles of charges
The remaining charges left in the car battery = Total distance traveled when fully charged - The total charged used by the car
Here we have to use the subtraction
Substitute the values in the equation
= 260 - 232
= 28 miles of charges
The percentage of miles the car has left in battery charge = (28/260) × 100
= 10.77%
Therefore, 10.77% miles of charge left in the battery
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Read the following prompt and type your response in the space provided.
Look at the problem and work shown. In which line was an error made and what should have been done differently?
Six less than 4 times a number is 50. What is the number?
6 – 4n = 50 (line 1)
6 – 6 – 4n = 50 – 6 (line 2)
–4n = 44 (line 3)
–4n÷ –4 = 44÷ –4 (line 4)
n = –11 (line 5)
.
Answer: Line 1
Step-by-step explanation:
Without looking at the context given, everything seems to be normal, everything is correct. Inputting n = -11 into line 1 makes it true. But when we take a look at the context given, we can see that the equation that it said "6 less than 4 times a number" meaning we were supposed to take 6 and subtract it from 4 times the number, meaning the actual equation is
4n - 6 = 50
Therefore, line 1 made an error
Considered the father of sociology, a French philosopher who in 1822 used the term sociology or social physics, highlighting its scientific nature by comparing it to the queen of natural sciences.
The French character who is considered the father of sociology has been Auguste Comte.
¿Who was Auguste Comte?This character was a Frenchman, he himself was a philosopher and is recognized mainly by:
Start what is positivism and its associated elements.Start scientific sociology, that is, sociology with scientific elements and components.Due to the above, Auguste Comte is considered the initiator of the social sciences, which are currently very important sciences for the study of various events.
¡Hope this helped!
20. Beatrice bought a new car for $28,000. She put down $5,000 and paid $550 for 60 months. What's the total finance charge to Beatrice?
A. $12,000
B. $10,000
C. $16,000
D. $23,000
Answer: B. $10,000
Step-by-step explanation:
$550x60+5,000=38,000
Subtract $38,000 by purchase price.: $28,000.
total is $10,000.
Question 12
The plans for a zipline are shown. Use two points to determine the slope of the zipline. Then verify that the slope is the same by choosing a
different set of points.
y
B
A
C
The slope of the zipline is Enter your answer as a fraction or decimal.
The slope of the zipline in the given diagram is; -1/3
How to find the slope of the line?The slope of a line is defined as the change in y values divided by change in corresponding x-values.
The formula to find slope here will be;
Slope = (y₂ - y₁)/(x₂ - x₁)
Let us pick two points along line BD namely points C and D.
Coordinate of point C is (7, 2)
Coordinate of point D is; (10, 1)
Thus;
Slope = (1 - 2)/(10 - 7)
slope = -1/3
Let us pick point B and point D now.
Coordinate of point B is; (1, 4)
Thus slope of BD = (4 - 1)/(1 - 10)
= -3/9 = -1/3
The slopes are equal and as such that is the slope of the zip line
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There are 15 dogs in an obedience class. Five of the dogs are Dalmatians. Aside from that, the rest are all different breeds. Assume that the instructor can only distinguish between the dogs by their breeds. (In other words, he can’t tell the Dalmatians apart!) At the beginning of class all of the dogs are lined up in a row.
a. As far as the instructor can tell, how many ways can the dogs be lined up in a row? (Remember,the Dalmations are indistinguishable.)
b. Assume that the dogs are given sweaters that completely disguise what breed they are. 3 wear yellow sweaters, 4 wear red sweaters, 3 wear blue sweaters, and 5 wear green sweaters. At this point, the instructor can only distinguish between the dogs by their sweater colors. (In other words, all the dogs with the same color sweater look alike to him!) As far as the instructor can tell, how many ways can the dogs be lined up in a row? (You may assume that the dogs will wear the sweaters without objection.)
c. Assume the sweater scenario in part (b). What is the probability that the instructor sees all of the dogs with the same sweater color sitting next to each other (for example: RRRRYYYGGGGGBBB)?
a) The number of ways in which the dogs can be lined up in a row is = 15! / 5!
b) The number of ways in which the dogs can be distinguished by their sweater colours is 15! / (3! × 4! × 3! ×5!)
c) The probability that the instructor sees all of the dogs with the same sweater colour sitting next to each other is
(a) The total number of dogs is 15, of which 5 dogs are Dalmatians and others are different.
So, by using permutation the number of dogs can be arranged in 15! / 5! ways.
(b) There are 3 dogs wearing yellow sweater, 4 dogs wearing red sweater, 3 dogs wearing blue sweater, and 5 dogs wearing green sweater.
Thus, the number of dogs can be arranged in 15! / (3! × 4! × 3! ×5!) ways.
(c) The 15 dogs can be arranged in 15! ways.
There are sweaters of 4 different colours, so the 4 colours can be arranged in 4! ways.
There are 3 dogs wearing yellow sweater, so they can be arranged in 3! ways.
There are 4 dogs wearing red sweater, so they can be arranged in 4! ways.
There are 3 dogs wearing blue sweater, so they can be arranged in 3! ways.
There are 5 dogs wearing red sweater, so they can be arranged in 5! ways.
Therefore, the probability that the instructor sees all of the dogs with the same sweater colour sitting next to each other is
4! (3! × 4! × 3! ×5!) / 15!
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(XA) 26. In a certain community, 65% people speak Newar language and 45% people speak Gurung language. Find percentage of people who can speak (i) both the languages. (ii) Newari only (iii) Gurung language only.
Answer: (i) 10% (ii) 60% (iii) 40%
Step-by-step explanation:
(i) Newar language = 65%
Gurung language = 45%
Total covers = 110 but to sum it up to base that is 100% then 110-100 is 10.
so, 10% of people speak both the language.
(ii) people who speak only Newari only would be 60% as 5% among them speak Gurung language.
(iii) people who speak only Gurung only would be 40% as 5% among them speak Newari language.
use the central limit theorem to approximate the probability that the total number of magazines that charlie reads in one full year (12 months of 30 days each) is between and . (note that you are asked about the number of magazines, not the number of minutes of total reading time.) (give an answer accurate to at least 3 decimal places.) (please answer the question as is, even if you have doubts.)
Applying central limit theorem number of books reads in one full year is equal to 720.
As given in the question,
Let X be the number of magazine read by Charlie.
Number of days in 12 months consider 30 days in each month 'n'
= 12 × 30
= 360 days
Number of magazine lies between 3500 and 3600
Consider μ = 2 and σ = √2
Central limit theorem , we have
z =( [tex]\bar{X}[/tex] - μ )/ σ/√n
[tex]\bar{X}[/tex] = X / n
Probability of total number of magazine reads between ( 3500 to 3600)
= P ( 3500 ≤ X ≤ 3600 )
= P( 3500/360 ≤ X/ n ≤3600/360)
= P( 9.72< X/n≤ 10 )
= P [ (9.72 - 2)/ √2/√360 ≤ ( X/n - μ)/σ/√n ≤(10 - 2)/ √2/√360]
= P(130.2 ≤ Z ≤107.4)
= P( 0≤Z≤107.4) - P (0≤Z≤130.2)
= φ( 130.2) + φ(107.4)
= 1 + 1 { φ(a) = 1 , a>1}
= 2
Number of books Charlie reads in one year =
∑2ₙ where n = 1 to 360
= 2+ 2+ 2+ ....+ 2 ( 360 times)
= 720 books
Therefore, using central limit theorem number of books reads in one full year is equal to 720.
The above question is incomplete , the complete question is :
Use the central limit theorem to approximate the probability that the total number of magazines that Charlie reads in one full year (12 months of 30 days each) is between 3500 and 3600. (Note that you are asked about the number of magazines, NOT the number of minutes of total reading time.) (Please answer the problem as written.) (Give an answer accurate to at least 3 decimal places.)
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the graphical solution method of linear programming employs the corner point method to find the optimal solution.
True, the graphical solution method of linear programming employs the corner point method to find the optimal solution.
Linear Programming
The most straightforward method of problem optimization is linear programming. We can turn a real-world issue into a mathematical model using this approach. Linear programming can be used to handle a wide range of issues in many different fields, although it is typically applied to issues where the goal is to maximize profit, cut costs, or use resources as sparingly as possible.
Explanation:
The corner point method of graphical linear programming involves plotting the linear equations representing the constraints of the problem onto a graph and then finding the points of intersection of the equations. These points of intersection are called the corner points, and the optimal solution will be located at the corner point that produces the greatest value.
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What is √x⁵y⁶ expressed in simplified form?
x²y³√x
xy√xy
x²y²√xy
x²y√x
The simplified form of the expression [tex]\sqrt{x^5y^6}[/tex] is [tex]x^2y^3\sqrt{x}[/tex]
The given expression is [tex]\sqrt{x^5y^6}[/tex]
The expression the mathematical statement that consist of different types of variables, numbers and the mathematical operators. The mathematical operators are addition, subtraction, division and multiplication. The equal sign and the inequality sign will not be the part of the expression
Here the given expression is
[tex]\sqrt{x^5y^6}[/tex]
We know that
[tex]\sqrt{x^2}[/tex] = x
Therefore, for each square values we can take that term outside of the square root
Here you can take x^4 take outside as x^2 and one x will remain in the square root, similarly you can take y^6 outside of the square root as y ^3
The final result = [tex]x^2y^3\sqrt{x}[/tex]
Therefore, the simplified form is [tex]x^2y^3\sqrt{x}[/tex]
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Company A has three times as many customers as company B while company C has 10 customers more than company B. Let x represent the number of customers held by company B, what is the expression so that it represents the total number of customers held by all three companies?
The expression that represents the total number of customers held by all three companies is 5x + 10.
What is a mathematical expression?A mathematical expression is a combination of numbers, variables, and mathematical operands (multiplication, division, subtraction, and addition).
What differentiates a mathematical expression from an equation is the equal symbol because a mathematical expression does not have it.
In simple terms, a mathematical expression is like a phrase in the English language without the complete form of a sentence.
The number of customers held by Company B = x
The number of customers held by Company A = 3x
The number of customers held by Company C = x + 10
The total number of customer for all three companies = x + 3x + x + 10
= 5x + 10
Thus, mathematically, we can conclude that the three companies have 5x + 10 customers.
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The expression that represents the total number of customers held by all three companies is; 5X + 10
What is an Expression?An expression consists of one or more numbers or variables along with one more operation.
X + 1 is an example of an expression.
"X" is the variable, "+" is the operation and 1 is a number. Think of "X" as any number that is unknown.
Representing the variables;
X, Y and Z represents number of customers
Let Company A = Y
Let Company B = X
Let Company C = Z
But,
Company A = 3X
Company C = 10 + X
Therefore;
Total number of customers = Y + X +Z
Substituting;
Total number of customers = 3X + X + 10 + X
Total number of customers = 5X + 10
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If P = (-1,-1), find the image
of P under the following rotation.
90° counterclockwise about the origin
([?], [])
Enter the number that belongs in
the green box.
The image of P after the rotation of 90° counterclockwise is (-1, 1)
How to find the image after the rotation?
Here we have the point P = (-1, -1), notice that both values are negative, thus, the point is on the third quadrant.
For any point (x, y) on the third quadrant, if we apply a rotation of 90° counterclockwise the new coordinates of the point (coordinates of the image) after the reflection are (y, -x)
In this case the original coordinates are (-1, -1), so the coordinates after the reflection are (-1, -(-1)) = (-1, 1)
The image after the rotation is (-1, 1).
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One man can paint a garage in 28 hours. After working for 12 hours, he was given a new job, and a second man finished the painting in 18 hours. How long would it take the second man to paint the whole garage alone? Show the setup/equation used to solve.
The number of hours taken by man to paint the whole garage alone is 31.5 hours.
Given that, one man can paint a garage in 28 hours.
What is time and work?The basic concept of Time and Work is similar to that across all Arithmetic topics, i.e. the concept of Proportionality. Efficiency is inversely proportional to the Time taken when the amount of work done is constant.
Let x be number of hours taken by second man paint the whole garage.
Work done by first man be 1/28
So, work completed by first man =1/28 ×12
= 12/28
= 3/7
Remaining work is 1-3/7 =4/7
1/x ×18 =4/7
⇒ 18/x=4/7
⇒ 4x=126
⇒ x=126/4
⇒ x=31.5 hours
Therefore, the number of hours taken by man to paint the whole garage alone is 31.5 hours.
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I tried but i doubt that's the answer (NO USELESS ANSWERS I WILL REPORT YOU )
Answer:
Step-by-step explanation:
So, they want the equation of the 4 colored lines, not a coordinate.
a --> x=-4
b --> x=4
c --> y=4
d --> y=-2
bc they are horizontal and vertical lines so only one value is staying the same, depending on whether it's vertical or horizontal.
To show 4 dimensional matching is NP complete, we want to reducte from 3 dimensional matching. The trick is to just pad each triple with an extra coordinate. We need to do this in a way that we don't really change the problem. One way to do this is to just repeat the last coordinate - so that the triple [a, b, c] is mapped to the quadruple [a, b, c, c].
NP finished. Only two jobs share resources with the reduction provided in section A.
what is coordinate?to put in the same order or rank. : to bring into a common action, movement, or condition : harmonize. coordinate schedules. She'll be coordinating the relief effort.
The overall issue is NP-complete. From Independent Set, this is a decrease. We substitute jobs for the graph's vertex points. If there is an edge connecting vertices v and w, we establish a resource called r v,w that v and w can share.
If k = 2, we are simply checking to see if there are any jobs that are resource-exclusive. By checking every pair of jobs, we can accomplish this using sheer force.
NP finished. Only two jobs share resources with the reduction provided in section A.
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Which of the following is the correct representation of permutation?
A.rPn
B .P (n,r)
C. C(n, r)
D .nCr
(Will mark brainlist)
Answer:
P (n,r)
Step-by-step explanation:
The correct representation for a permutation is P (n,r).
Again, please just please help
The perimeter of ΔMNP is calculated as; 130
How to find the perimeter of the triangles?
From the given image, it is clear that some sides are congruent to others as indicated.
Thus, we see the following;
QM is parallel and congruent to RS.
Similarly, we see that QR is parallel and congruent to MS.
PR is parallel and congruent to QS.
NR = PR because of line segment division.
Thus, we can plug in the relevant values to get;
PR = QS = 22 = NR
Similarly, MS = SP = QR = 25
RS = x + 4 and so;
x + 4 + x + 4 = 5x - 34
42 = 3x
x = 14
MN = 5(14) - 34
MN = 36
Thus, the perimeter of ΔMNP = 36 + 25 + 25 + 22 + 22
= 130
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Help quick!! Very confused, create equivalent expression.
Answer:
Either A or B
Step-by-step explanation:
Brainliest, Please!Answer:
B
Step-by-step explanation:
(1.3^4 / 1.2^3)^7
1.2^ (3 · 7) = 1.2^(21)
1.3 ^ (4 · 7) = 1.3^(28)
1.3^(28)/1.2^(21)
In the diagram below, AB || CD, AD || BC, m/ABC = 110°,
m/CDE = 37° and m/ECB = 43°. Find m/DEC.
A
B
110°
E
D
37°
43°
C
Use photo to understand better PLEASE HELP
Which of the following designs involves repeated measurement of a variable before and after some event?
Group of answer choices
nonequivalent control group design
interrupted time-series design
matched group factorial design
multiple regression design
The design that involves repeated measurement of a variable before and after some event will be matched group factorial design. Hence, option C is correct.
What is Matched group factorial design?In this kind of experimental design, the research's participants are divided into groups, and key factors are matched to each group. The variables that are used to match the respondents must have an impact on the original study conclusion (the dependent variable).
The benefit of using matched group factorial design is,
Fewer people are needed for this kind of investigation, which could also produce more accurate results and results that are based on more information.
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9
Tom and Adil are the two runners in a 200-metre race.
Tom completes the race in 24 seconds.
Adil completes the race at an average speed of 28.8 kilometres per hour.
Who wins the race?
You must show your working.
[3
Answer:
Since Adil takes 25 seconds to complete the race, and Tom takes 24 seconds, Tom wins the race.
Step-by-step explanation:
To determine who wins the race, we need to first convert the given information into comparable units. The speed of Adil is given in kilometers per hour, while the time taken by Tom to complete the race is given in seconds. We can convert Adil's speed to meters per second by dividing by 3.6:
28.8 km/hr = 28.8 km/hr * (1 hour / 3600 seconds) * (1000 meters / 1 km)
= 8 meters/second
We can then use this value to find the time it takes Adil to complete the race:
time = distance / speed
= 200 meters / 8 meters/second
= 25 seconds
Since Adil takes 25 seconds to complete the race, and Tom takes 24 seconds, Tom wins the race.
Can someone help please? Picture is already attached. If its wrong please correct it. This isn’t mine by the way, I’m posting it for someone else to double check if it’s right
The value of the other points in the number line for each case are;
a) M = -2¹/₂
N = -1
R = 4
b) M = -1
P = 1
R = 1.6
c) M = -125
P = 125
N = -50
d) N = -6
P = 15
R = 24
How to interpret Number Lines?a) We are given that P = 2¹/₂
From the point 0 of the number line to point P is 5 units and as such;
Each unit = 2¹/₂/5 = ¹/₂
Thus;
M = -2¹/₂
N = -1
R = 4
b) We are given that N = -0.4
From the point 0 of the number line to point N is 2 units and as such;
Each unit = 0.4/2 = 0.2
Thus;
M = -1
P = 1
R = 1.6
C) We are given that R = 200
From the point 0 of the number line to point R is 8 units and as such;
Each unit = 200/8 = 25
Thus;
M = -125
P = 125
N = -50
D) We are given that M = -15
From the point 0 of the number line to point N is 5 units and as such;
Each unit = 15/5 = 3
Thus;
N = -6
P = 15
R = 24
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Find all the roots for this polynomial using the steps below.
P(X)=x^3+3x^2+3x+2
List the 4 possible rational roots.
Plug each possible rational root into P(x) to determine which is a root of the polynomial.
Divide P(x) by the root you found in part 2 using synthetic division.
Write the fully factored form of P(x).
Use the quadratic formula to find the solutions of the quadratic factor.
List the 3 solutions of the P(x).
Answer:
[tex]p(x)=(x+2)(x^2+x+1)[/tex]
[tex]x=-2[/tex]
[tex]x=\dfrac{-1 + \sqrt{3}\:i}{2}[/tex]
[tex]x=\dfrac{-1- \sqrt{3}\:i}{2}[/tex]
Step-by-step explanation:
Given polynomial:
[tex]p(x)=x^3+3x^2+3x+2[/tex]
Rational Root Theorem
If P(x) is a polynomial with integer coefficients and if p/q is a root of P(x), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).
Possible p-values
Factors of the constant term: ±1, ±2
Possible q-values
Factors of the leading coefficient: ±1
Therefore, all the possible values of p/q:
[tex]\sf \dfrac{p}{q}=\dfrac{\pm 1}{\pm 1}, \dfrac{\pm 2}{\pm 1}=\pm1,\pm2[/tex]
Substitute each possible rational root into the function:
[tex]x=-1 \implies p(-1) =(-1)^3+3(-1)^2+3(-1)+2=1[/tex]
[tex]x=1 \implies p(1) =(1)^3+3(1)^2+3(1)+2=9[/tex]
[tex]x=-2 \implies p(-2) =(-2)^3+3(-2)^2+3(-2)+2=0[/tex]
[tex]x=2 \implies p(2) =(2)^3+3(2)^2+3(2)+2=28[/tex]
Therefore, x = -2 is a root of the polynomial since f(-2) = 0.
Divide the polynomial by the root using synthetic division:
[tex]\begin{array}{c|crrr}-2 & 1 & 3 & 3 & 2\\\cline{1-1} & \downarrow &-2&-2&-2\\ \cline{2-5} & 1&1&1&0\end{array}[/tex]
The bottom row (except the last number) gives the coefficients of the quotient. Therefore, the quotient is:
[tex]x^2+x+1[/tex]
So the fully factored form of p(x) is:
[tex]p(x)= (x+2)(x^2+x+1)[/tex]
[tex]\boxed{\begin{minipage}{3.6 cm}\underline{Quadratic Formula}\\\\$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$\\\\when $ax^2+bx+c=0$ \\\end{minipage}}[/tex]
Therefore, the solutions to the quadratic factor are:
[tex]\implies x=\dfrac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{-3}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3\cdot -1}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3} \sqrt{-1}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3}\:i}{2}[/tex]
The 3 solutions of p(x) are:
[tex]x=-2[/tex][tex]x=\dfrac{-1 + \sqrt{3}\:i}{2}[/tex][tex]x=\dfrac{-1- \sqrt{3}\:i}{2}[/tex]You are doing a Diffie-Hellman-Merkle key exchange with Aisha using generator 7 and prime 437. Your secret number is 227. Aisha sends you the value 308. Determine the shared secret key.
Answer:
382^2 mod
Step-by-step explanation:
To determine the shared secret key, we need to use the formula (g^a mod p)^b mod p = (g^b mod p)^a mod p, where g is the generator, p is the prime, a is our secret number, and b is the value sent by Aisha. Plugging in the values, we get (7^227 mod 437)^308 mod 437 = (7^308 mod 437)^227 mod 437.
To solve for the shared secret key, we first need to calculate (7^308 mod 437). This can be done by raising 7 to the 308th power and then taking the remainder when divided by 437. We can do this by repeatedly squaring 7 and taking the remainder each time. This results in the following sequence:
7^2 mod 437 = 49
7^4 mod 437 = 161
7^8 mod 437 = 267
7^16 mod 437 = 9
7^32 mod 437 = 49
7^64 mod 437 = 161
7^128 mod 437 = 267
7^256 mod 437 = 9
7^512 mod 437 = 49
Since 512 is greater than 308, we can stop here and use the value 49 as the result of 7^308 mod 437. We can now plug this value into the formula to calculate the shared secret key: (49^227 mod 437)^308 mod 437 = (49^308 mod 437)^227 mod 437.
To solve for the shared secret key, we need to find the value of 49^308 mod 437. This can be done using the same method as before, by repeatedly squaring 49 and taking the remainder each time. This results in the following sequence:
49^2 mod 437 = 67
49^4 mod 437 = 382
49^8 mod 437 = 221
49^16 mod 437 = 67
49^32 mod 437 = 382
49^64 mod 437 = 221
49^128 mod 437 = 67
49^256 mod 437 = 382
Since 256 is greater than 308, we can stop here and use the value 382 as the result of 49^308 mod 437. We can now plug this value into the formula to calculate the shared secret key: 382^227 mod 437 = 382^227 mod 437.
To find the shared secret key, we need to calculate 382^227 mod 437. This can be done using the same method as before, by repeatedly squaring 382 and taking the remainder each time. This results in the following sequence: