The vector x = [c1 - e^-9t, c2 + 3e^7t, c1 + 5e^7t] is a solution of the initial-value problem x' = [1/10, 6, -3]x, x(0) = [2, 0, 1].
To verify that the given vector x is a solution to the initial-value problem, we need to take its derivative and substitute it into the differential equation, and then check that it satisfies the initial condition.
Taking the derivative of x, we have:
x' = c1(-1/10)e^(-9t) + c2(35)e^(7t) -1/10
5c2e^(7t)
Substituting x and x' into the differential equation, we have:
x' = Ax
x' = [ 1 10 6 −3 ] [ c1 −1 1 e−9t c2 5 3 e7t ] = [ (−1/10)c1 + 5c2e^(7t) , c1/10 − c2e^(7t) , 6c1e^(-9t) + 3c2e^(7t) ]
So, we need to verify that the following holds:
x' = Ax
That is, we need to check that:
(−1/10)c1 + 5c2e^(7t) = c1/10 − c2e^(7t) = 6c1e^(-9t) + 3c2e^(7t)
To check that the above equation holds, we first observe that the first two entries are equal to each other. Therefore, we only need to check that the first and third entries are equal to each other, and that the initial condition x(0) = [c1, 0] is satisfied.
Setting the first and third entries equal to each other, we have:
(−1/10)c1 + 5c2e^(7t) = 6c1e^(-9t) + 3c2e^(7t)
Multiplying both sides by 10, we get:
-c1 + 50c2e^(7t) = 60c1e^(-9t) + 30c2e^(7t)
Adding c1 to both sides, we get:
50c2e^(7t) = (60c1 + c1)e^(-9t) + 30c2e^(7t)
Dividing both sides by e^(7t), we get:
50c2 = (60c1 + c1)e^(-16t) + 30c2
Simplifying, we get:
50c2 - 30c2 = (60c1 + c1)e^(-16t)
20c2 = 61c1e^(-16t)
This equation must hold for all t. Since e^(-16t) is never zero, we must have:
20c2 = 61c1
Therefore, c2 = (61/20)c1. Substituting this into the initial condition, we have:
x(0) = [c1, 0] = [2, 0]
Solving for c1 and c2, we get:
c1 = 7/2 and c2 = -3/2
Thus, the solution to the initial-value problem is:
x(t) = [ (7/2) −1 1 e^(-9t) (−3/2) 5 3 e^(7t) ]
and we can verify that it satisfies the differential equation and the initial condition.
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Why is 180 degree longitude considered to be an important meridian?
The meridian at 180° longitude is commonly known as the International Date Line. As you pass the International Date Line, you either add a day (going west) or subtract a day (going east.)
The prime meridian and the International Date Line create a circle that divides Earth into the Eastern and Western Hemispheres.
The prime meridian has been set up in Greenwich, UK. It is an imaginary line that represents O degrees of longitude. This line, like all other meridians, runs from the North Pole to the South Pole and has the same length as all of the other meridians.
From the prime meridian, there are 180 degrees west and 180 degrees east. This is the line that divides the world in the Western and Eastern Hemispheres. The prime meridian is also the line from which time is counted either plus or minus, as it is at 0+ time zone, and everything east of it is plus, while everything west is minus. On the opposite side of the world of the prime meridian, we have the line that represents the start of time.
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Is it possible to construct a triangle whose sides are 10.2 cm 5.8 cm and 4.5 cm which result did you use to verify this?
It is possible to construct a triangle whose sides are 10.2 cm 5.8 cm and 4.5 cm because the sum of two sides are greater than three sides.
In the given question, we have to check that it is possible to construct a triangle whose sides are 10.2 cm 5.8 cm and 4.5 cm.
The sum of the two sides must always be greater than the third side in order to determine whether or not the given sides may form a triangle.
To check this we firstly add the 10.2 and 5.8
10.2 + 5.8 > 4.5
16 > 4.5
Now we add 5.8 and 4.5
5.8 + 4.5 > 10.2
10.3 > 10.2
Now we add 10.2 and 4.5
10.2 + 4.5 > 5.8
14.7 > 5.8
It is possible to construct a triangle whose sides are 10.2 cm 5.8 cm and 4.5 cm because the sum of two sides are greater than three sides.
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In the accompanying figure, angle 1 measures 28o and angle 2 measures 11o. What is the measure of angle 3
The measure of angle 3 in the given triangle is 141 degrees.
Angle 1 + Angle 2 + Angle 3 = 180°
So, Angle 3 = 180° - (Angle 1 + Angle 2)
Angle 3 = 180° - (28° + 11°)
Angle 3 = 141°
The sum of the three angles in a triangle is 180 degrees, so to calculate the measure of angle 3, we can subtract the measures of angles 1 and 2 from 180 degrees. In this case, angle 1 measures 28 degrees and angle 2 measures 11 degrees, so the measure of angle 3 is 180 degrees minus (28 degrees + 11 degrees), or 141 degrees.
The measure of angle 3 in the given triangle is 141 degrees.
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What is the formula of image in straight line?
The formula of an image in a straight line is y = mx + c, where m is the slope of the line, x is the x-coordinate of the point and c is the y-intercept.
What is straight line?A straight line is a line whose points all lie in the same plane and all have the same distance from each other, making them all equidistant. The line does not curve or bend in any way, and can extend infinitely in either direction. Straight lines are the most basic form of geometric shape and are the foundation of all other shapes. The line is defined by two points, known as its endpoints, and any point that lies on the line is called a point on the line. Straight lines are important in mathematics, as they are used to describe relationships between points on a graph, and they can also be used to define equations and boundaries.
The y-intercept is the point at which the line crosses the y-axis. The slope of the line is determined by the ratio of the change in the y-coordinate to the change in the x-coordinate. To calculate the slope of a line, the formula m = (y2 - y1)/(x2 - x1) is often used, where (x1, y1) and (x2, y2) are two points on the line. Using the slope and the y-intercept, the formula can be used to plot the points of the image along a straight line.
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What does the mark of the angles and sides of the triangle indicate?
The mark of the angles and sides of the triangle indicate that they are congruent.
What is congruence of triangle ?Triangle diagram with side and angle indicators. To demonstrate if two or more sides are congruent, indicator marks are utilized (equal in measure).
Triangle congruence: If all three corresponding sides and all three corresponding angles are equal in size, two triangles are said to be congruent. Slide, twist, flip, and turn these triangles to create a similar appearance.
When two angles and the included side of one triangle are equal to two angles and the included side of another triangle, two triangles are said to be congruent. PB = DE. The matching triangle portions are equal since the triangles are congruent.
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what 3 to the second power time 5
Answer: 45
Step-by-step explanation: 3 x 3 = 9 x 5 = 45
At an animal shelter, there are 15 dogs and 5 cats what percent of the animal are cats
Answer:
25%
Step-by-step explanation:
5 out of (15+5 ) are cats
5/20 = .25 = 25 %
Answer:
25%
5 out of (15+5 ) are cats
5/20 = .25 = 25 %
Pls mark me as brainliest
I need this ASAP
Pleaseee 50 points!!!
choose 50 distinct numbers between 0 and 99. For starlings and geese, follow the same procedure.
what is probability ?Measurement of the likelihood that an event will occur or a claim is true is done via probability theory, a branch of mathematics. The probability of an occurrence is a number between 0 and 1, with 1 denoting certainty and 0 roughly indicating how likely it is that the event will occur. A probability is a numerical representation of how likely it is that a certain occurrence will take place. Probabilities can also be expressed as ranges of 0% to 100% or 0 to 1. the proportion of occurrences in a whole set of equally likely options that lead to a particular event relative to the entire number of outcomes.
given
All of the blackbirds were numbered and tagged from 00 to 99. With a random number generator,
choose 50 distinct numbers between 0 and 99. For starlings and geese, follow the same procedure.
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the correct question is :- In search of a mosquito repellent that is safer than the ones that are currently on the market, scientists have developed a new compound that is rated as less toxic than the current compound, thus making a repellent that contains this new compound safer for human use. Scientists also believe that a repellent containing the new compound will be more effective than the ones that contain the current compound. To test the effectiveness of the new compound versus that of the current compound, scientists have randomly selected 100 people
Find the square root of the following fraction correct to two decimal numbers 5 5/11
By square root 5 this we get the value √5 = 2.2360…Or √5 ≈ 2.24 with correct two decimal number fraction .
What is Division Method to Find the Value of Root 5 ?Using the long division approach, it is feasible to calculate the value of any number's roots. The following list of instructions will help you determine the value of root 5:1. First, the number 5 can be expressed as 5.00000000, or 5 = 5.00 00 00 00.
2.Take the number whose square is less than five. The perfect square of the number 2 is 4, which comes after the number 5.
3..Enter 2 in both the divisor and quotient places so that it may be multiplied by 2 to get the answer 4. Add 4 to 5 to obtain the answer, which is 1.
4.Write two zeros after one and add them to the quotient. Add the decimal point after the first zero.
5.Now multiply by 2 to make 4 in the divisor. Pick a number that will go next to 4, ensuring that the result of multiplying the combination with that number will be less than or equal to 100. In order to make 42 times 2 equal 84, which is less than 100, we shall use 2 in this instance. To get the remainder, take 100 out of it immediately. The remainder is therefore 16.
6.Repetition of Step 5 up to 4 places of decimals with the addition of two more zeros.
7.Ultimately, you arrive at the quotient value, which is 2.2360, or around 1.236.
In light of this, the value of root 5 is, √5 = 2.2360… Or √5 ≈ 2.24
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What is the value of base 2?
Binary is also known as Base 2. This means it is composed of only 0's and 1's. For example 9 in binary/base 2 is 1001.
About binary number systemThe binary number system or base two number system is a system of writing numbers using two symbols, namely 0 and 1.The modern binary number system was discovered by Gottfried Wilhelm Leibniz in the 17th century. This number system is the basis of all digital-based number systems. From the binary system, we can convert it to the Octal or Hexadecimal number system. We can also refer to this system as Bits, or Binary Digits. Binary groupings on a computer always add up to 8, with the term 1 Byte/byte. In computer terms, 1 Byte = 8 bits. Computer design codes, such as ASCII, American Standard Code for Information Interchange use a 1 Byte coding system.
In modern digital communication systems, where data is transmitted in the form of binary bits, a system is needed that is resistant to noise contained in the transmission channel so that the transmitted data can be received correctly. Errors in sending or receiving data are fundamental problems that have a very significant impact on communication systems.[2] The commonly used binary has 8 digit numbers and only contains the numbers 1 and 0, no other numbers. Base-2 ; Position Value: 2 · 2 ; Digit Value: 128s. 64s ; Binary Number: 1. 1.
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What are the 5 types of equality?
Equality is a fundamental human right, and it is essential for the development of any society. Equality is a complex concept that can be broken down into five distinct types.
The 5 types of equalitySocial Equality: The equal access to social rights and opportunities regardless of race, gender, or any other social identifier.Economic Equality: The equality of access to resources and wealth, regardless of social standing or economic background.Political Equality: The right to participate in the political process and the right to have a say in the decisions that affect one’s life.Legal Equality: Equal access to the same laws, rights, and responsibilities as everyone else.Equality of Opportunity: The ability to access the same opportunities regardless of background, skin color, gender, or any other factor.Learn more about Equality: https://brainly.com/question/9873909
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Page 7:
You're helping a farmer determine the probability that certain vegetables will be
purchased. You know the following about the 100 customers the farmer provides
fruits to:
70 purchase carrots.
50 purchase cucumbers.
60 purchase lettuce.
20 purchase carrots and lettuce.
30 purchase cucumbers and carrots.
a. The probability that a randomly selected customer purchases lettuce is ___.
The probability that a randomly selected customer purchases lettuce is 3/5.
What is Probability?
The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a claim to be true. A number between 0 and 1 is the probability of an event, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.
Given that there are 100 customers the farmer provides.
The number of customers purchasing lettuce is 60.
The probability that a randomly selected customer purchases lettuce,
P(L) = Number of customers purchasing lettuce ÷ Total number of customers
[tex]\implies P(L) = \frac{60}{100} \\\implies P(L) = \frac{3}{5}[/tex]
Therefore, the probability that a randomly selected customer purchases lettuce is 3/5.
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There are 40 students in the
classroom. If 25 are boys, what
percent are girls?
Answer:
The answer is 37.5%
Step-by-step explanation:
Answer:
37.5
Step-by-step explanation:
1% of 40= 0.4
0.4 x 25 = 62.5
100-62.5 = 37.5
Drag each equation to show if it could be a correct first step to solving the equation 3(6+x)=24 3(6+x)=24.
Answer:
2
Step-by-step explanation:
3(6+x) = 24
18 + 3x = 24
3x = 24 - 18
3x = 6
x = 6/3
= 2
An angle bisector is a ray that divides an angle into two congruent angles.please white the converse adn conditional
The converse of the conditional statement is "if a ray divides an angle into two congruent angles, then it is an angle bisector"
What is the converse of the conditional statement?The conditional statement is given as:
"An angle bisector is a ray that divides an angle into two congruent angles"
The converse of the conditional statement is such that:
The hypothesis and the conclusion are switched
So, we have the converse statement to be:
"if a ray divides an angle into two congruent angles, then it is an angle bisector"
Also, the converse of the statement is true
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Multiply. -4 x 1 7 8 Submit || what dies this eaqual in a fraction we are doing rational numbers
Therefore , the solution of fraction is both of the options are incorrect in this case since the given expression results in a different equation than the ones chosen.
The definition of fractionA part makes up the whole. In maths, the excess is divided by the denominator to represent a statistical as a divide. Both are integers in a simple fraction. The prime or the denominator of a complex fraction both contain a fraction. A proper fraction has an equal numerator.
Here,
The sentence that is given
13/15 | 2 - x | = 4/15
there are
| 2- x | = 1/15
Identify the absolute value.
=> 13/15 | 2- x | = 1/15
=> | 2- x | = 4 * 15 / 15 * 13
=> 1/13 = | 2- x |
Therefore, if your query is complete, the appropriate answer is none of the above options. Neither choice is appropriate in this case since the given expression results in a new equation that isn't the same as the options made.
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City planners study how many automobiles pass through a particular intersection over several
hours. The table shows the results.
What are the most appropriate units for the rate of change in the table?
The most appropriate units for the rate of change in the table is automobiles per hour.
What is rate of change?
The rate of change function describes the rate at which one quantity changes in proportion to another quantity. The rate of change is determined, to put it simply, by dividing the amount of change in one item by the same amount of change in another. The rate of change formula provides a relationship indicating how one quantity changes in relation to a change in another quantity.
We are given that city planners study how many automobiles pass through a particular intersection over several hours.
This means that the graph depends on the automobiles that pass through a particular intersection every hour.
Hence, the most appropriate units for the rate of change in the table is automobiles per hour.
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How do you dilate a triangle with a scale factor of 1 2?
If the scale factor is 1/2, then every coordinate point of the original triangle is multiplied by the scale factor 1/2. Dilation with scale factor 1/2, then multiply by 1/2.
How do construct a dilation of 1 2?To dilate the figure by a factor of 2, I will multiply the x-value of each point by 2. I plotted all the new points to find the new triangle. To dilate the figure by a factor of 2, I will multiply the x and y-value of each point by 2.If the scale factor is a fraction, the shape will be smaller. This is called reduction. Therefore, a 1/2 scaling factor means that the new shape is half of the original shapeThe scale factor = Smaller figure dimensions ÷ Larger figure dimensions = 20/40 = 1/2. Therefore, the scale factor used to create the smaller polygon is 1/2. Example 3. A triangle was increased by a scale factor of 2 resulting in the new dimensions as 6 units by 10 units by 12 units.To learn more about construct a dilation refers to:
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PLEASE ANSWER ASAP!! WORTH 20 POINTS! WILL GVE 5 STAR REVIEW IF ANSWERED CORRECTLY!!
A right rectangular prism measures 12 cm tall, 8 cm long, and 5 cm wide.
A manufacturer would like to double the surface area of this prism. What should be the height of the new prism so that its surface area is doubled?
Enter your answer, to the nearest tenth, in the box.
Answer: 392
Surface area of a prism formula: A= 2(l*w*h)
2*(5*8*12*5*8*12) =392
Step-by-step explanation:
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A culptor need to move a piece of marble. It i a cuboid with dimenion 1m by 0. 6m by 0. 3m. Marble ha denity of 2. 7g/cm cubed. What i the ma of the cuboid in kg
The mass of the marble in kg is 486 kg .
Now, According to the question:
A sculptor needs to lift a piece of marble.
It is a cuboid with dimensions 1 m by 0.6 m by 0.3 m.
Marble has a density of 2.7 g/cm³.
We need to find what is the mass of the marble in kg .
Let's find out :
Volume of Cuboid = Length × Width × Height
Volume = (1m)(0.6m)(0.3m)
Volume = 0.18[tex]m^3[/tex]
Also , We know that
Density = [tex]\frac{Mass}{Volume}[/tex]
Density = 2.7[tex]\frac{g}{cm^3}[/tex]
Density = 2.7 [tex]\frac{10^-^3kg}{(10^-^2m)^3}[/tex]
Density = 2.7[tex]\frac{10^-^3kg}{(10^-^6m)}[/tex]
Density = 2.7[tex](10^-^3^+^6)kgm^-^3[/tex]
Density = 2.7[tex](10^6)kgm^-^3[/tex]
Now, Density = [tex]\frac{Mass}{Volume}[/tex]
2.7[tex](10^6)[/tex] = [tex]\frac{Mass}{0.18}[/tex]
Mass = [tex]2.7(10^6)[/tex] × 0.18
Mass = 486kg
Therefore , the mass of the marble in kg is 486 kg .
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To find the measure of A in ABC, use the
To find the length of side h in hit, use the
We can use the Law of Cosines. The Law of Cosines states that for a triangle ABC with sides a, b, and c, and angle A opposite side c, the measure of angle A can be found: cos A = (b^2 + c^2 - a^2) / (2bc)
How to use the Law of Cosines?Measure or determine the lengths of all three sides of the triangle (a, b, and c). Let's say a = 5, b = 6, and c = 7.
Substitute the side lengths into the formula above and solve for cos A. cos A = (6^2 + 7^2 - 5^2) / (267) = 0.6
Use the inverse cosine function to find the measure of angle A in degrees or radians. In this case, A = 60 degrees.
To find the length of side HI in triangle HIG, we can use the Pythagorean Theorem. The Pythagorean Theorem states that for any right triangle, the length of the hypotenuse (the side opposite the right angle) is equal to the square root of the sum of the squares of the other two sides.
How to use the Pythagorean Theorem?Measure or determine the lengths of the two legs of the triangle (HG and GI). Let's say HG = 8 and GI = 6
Substitute the side lengths into the formula HI^2 = HG^2 + GI^2 and solve for HI. HI = √(8^2 + 6^2) = √(64+36) = √100 = 10
Therefore, the length of side HI is 10.
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The complete question is:To find the measure of ∠A in ∆ABC, use the___(Pythagorean Theorem, Law of Sines, Law of Cosines). To find the length of side HI in ∆HIG, use the____(Pythagorean Theorem, Law of Sines, Law of Cosines) .
Pls answer numbers 4,5 and 6
PLEASE HELP On a map, the distance from Happy Hill park to Rainbow Valley park is 4 1/2 inches. The scale is 1/2 = 3 miles. What is the actual distance between the two parks in miles?
The actual distance between Happy Hill park to Rainbow Valley park is calculated to be 27 miles
How to calculate the actual distanceScale factors are used to increase or decrease image. The situation of increment is usually called magnifying.
If on the map the distance between Happy Hill park to Rainbow Valley park is 4 1/2 inches, and the scale factor is 1/2 = 3 miles. The calculation is done as follows
1/2 = 3miles
4 1/2 miles = ?
cross multiplying
4 1/2 * 3 = 1/2 * ?
? = 4 1/2 * 3 / 1/2
? = 27 miles
the actual distance is 27 miles
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What is the value of K if X 1 is the factor of 4x³ 3x² 4x K?
The value of k is -3, if x - 1 is a factor of 4x³ + 3x² − 4x + k.
A polynomial factor is a polynomial that divides another polynomial exactly, with no remainder.
For example, if we have a polynomial P(x) = 4x³ + 3x² − 4x + k, and we have a polynomial factor F(x) = x - 1, we can say that P(x) is divisible by F(x) with no remainder.
In general, a polynomial of degree n has n roots (or zeroes) and the factorization of polynomial can be expressed as a product of its roots and its corresponding multiplicity.
To find the value of k, set x = 1 in the equation 4x³ + 3x² − 4x + k and solve for k that will make the equation equal to zero..
So when x = 1, we have:
4x³ + 3x² − 4x + k = 0
4(1)³ + 3(1)² − 4(1) + k = 0
4 + 3 - 4 + k = 0
k = -3
Hence, if x - 1 is a factor of 4x³ + 3x² − 4x + k, then k is equal to -3.
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On the school playground, the slide is 16 feet due west of the tire swing and 12 feet due south of the monkey bars. What is the distance between the tire swing and the monkey bars?
Answer:
20 feet
Step-by-step explanation:
Using the Pythagorean theorem, the distance is [tex]\sqrt{12^2+16^2}=20[/tex] feet.
Gong rented a truck for one day. There wa a bae fee of $19. 95, and there wa an additional charge of 71 cent for each mile driven. Hong had to pay $125. 74 when he returned the truck. For how many mile did he drive the truck?
Gong must have driven the truck for 149 miles as per the given problem.
Let’s discuss the problem in detail.
In addition to the base rate of $19.95, there was an extra fee of 71 cents for every mile driven. When Gong returned the truck, he was required to pay $125.74.
Now we have to calculate how far did he drive the truck before stopping.
First, let’s convert all dollars into cents.
Base fee = 1995 cents.
71 cents for each mile.
Total Gong gave 12574 cents.
Assume Gong drive for x miles.
Therefore, 12574 = 1995 + 71 * x
= 71x = 12574 - 1995
= 71x = 10579
= x = 10579/71 = 149 miles.
Hence, Gong drove the truck for 149 miles before stopping.
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The function f(x) is shown on the graph.
The graph shows three pieces of f of x. One piece is curved in quadrant three increasing from the left and turning at approximately negative 3.8 comma negative 5.5 to then decreases and is asymptotic to the line x equals negative 2. Another piece is an upward opening u-shaped curve, which is asymptotic to the line x equals negative 2 and x equals 2 with a minimum at approximately negative 0.3 comma 2.4 3.24. Another piece is asymptotic to the line x equals 2 and is increasing from the left and passes through the point 3 comma 2 and continues to increase in quadrant one.
Which type of function describes f(x)?
Exponential
Logarithmic
Polynomial
Rational
Answer:
The answer is rational.
I just got it right on the test.
The table of values of the function f(x) = 5x³ + 2x² - 6x + 7 are illustrations of polynomial functions
What is mean by Function?A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
The values on the table can be represented as:
(x, g(x)) = (-3, 0.375) (-1, 0.75) (1, 1.5) (3, 3) (5, 6) (7,12) (9, 24)
The above values show that as the values of x increases by 2, the values of function g(x) doubles.
Only an exponential function has this property.
Hence, the table of values represents an (a) exponential function.
3. Example of a polynomial function
A polynomial function is represented as:
y = ax^n + bx^n-2 + .......... + d
Using the above format, the function that is a polynomial function is (a) f(x) = 5x3 + 2x2 – 6x + 7
4. Key features in the functions
The functions are given as,
f(x) = 2x² + 7x + 6
g(x) = 2x + 5
Set x to 0 to calculate the y-intercept
f(0) = 2(0) + 7(0) + 6 = 6
g(x) = 2⁰ + 5 = 6
Hence, both functions have the same (c) y-intercept
Questions 1 and 5 are not properly presented
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Is 3x2 a exponential function?
No, 3x2 is not an exponential function. An exponential function is a function in which the variable (x) is in the exponent.
The general form of an exponential function is f(x) = a^x, where a is the base and x is the exponent.
In the case of 3x2, the variable (x) is not in the exponent, it's being multiplied by the coefficient 3. This is a linear function, not an exponential function. The general form of a linear function is f(x) = mx+c, where m is the coefficient and c is the y-intercept.
In summary, 3x2 is not an exponential function. An exponential function is a function in which the variable (x) is in the exponent. The general form of an exponential function is f(x) = a^x, where a is the base and x is the exponent. In the case of 3x2, the variable x is not in the exponent, it's being multiplied by the coefficient 3. This is a linear function, not an exponential function.
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Which of the following triplets are Pythagorean ?(3,4,5),(6,7,8),(10,24,26),(2,3,4).
The Pythagorean Triplets states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs of the right triangle. The Pythagorean Triplets is (3,4,5).
Pythagorean triplets:
The Pythagorean numbers consist of three positive integers a, b, and c such that a2 + b2 = c2. Such triples are usually written as (a, b, c), with a familiar example being (3, 4, 5). If (a, b, c) are Pythagorean numbers, then (ka, kb, kc) for all positive integers k. A primitive Pythagorean number is one in which a, b, and c are relatively prime (that is, they have no common divisors greater than 1). For example, (3, 4, 5) is a primitive Pythagorean number, but (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle and is always a right triangle.
Based on the Question:
Let the smallest even number be 2m
(i) (3,4,5)
2m = 4
⇒ m = 2
⇒ m² −1 = 4−1 = 3
⇒ m² +1 = 4+1 = 5
∴(3,4,5) is a Pythagorean triplet.
(ii) (10,24,26)
2m = 10
⇒ m = 5
⇒ m²−1 = 25−1 = 24
⇒ m²+1 = 25+1 =26
∴(10,24,26) is not a Pythagorean triplet.
(iii) (6,7,8)
2m = 6
⇒ m = 3
⇒ m² −1 = 9−1 = 8
⇒ m² +1 = 9+1 = 10
∴(6,7,8) is not a Pythagorean triplet.
(iv) (2,3,4)
2m=2
⇒m=1
m² −1 = 1−1 = 0
m²+1 = 1+1 = 2
∴(2,3,4) is not a Pythagorean triplet.
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Which pair of angles are alternate interior angles?
- Angle 1 and 8
- Angle 1 and 7
- Angle 2 and 7
- Angle 3 and 6
In this case, the pair of alternate interior angles would be angle 2 and 7.
What is angle?Angle is a geometric object which is defined by two rays with a common endpoint. It is typically measured in degrees or radians and is used to measure the size of an angle. Angles play a key role in mathematics, being used in trigonometry, calculus, and geometry. Angles are also used in engineering and construction to ensure that components are built properly. In addition, angles are used to measure the direction of objects in the sky, such as stars and planets.
Alternate interior angles are two angles located on opposite sides of a transversal line, but within the same parallel lines. In this case, the pair of alternate interior angles would be angle 2 and 7. These two angles are located on opposite sides of the transversal line, between the two parallel lines.
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