In the given homogeneous system in parametric vector form, where the solution set is x = x₁ x₂ x₃, x = x₃. where the value of x₁ = 5k, x₂ = 5k and x₃ = 10k.
The system is
x₁ + x₂ - 15x₃ = 0 .......(1)
2x₁ + x₂ - 15x₃ = 0 .......(2)
-x₁ + x₂ = 0
We write the system in the matrix form or AX = 0
Where A = ( 1 2 -15) and X = ( x₁)
( 2 1 -15) (x₂)
( -1 1 0) (x₃)
Now, | A | = [15] -2 [-15] -15 [2+1]
= 15 + 30 - 45
= 0
∴ Since, | A | = 0, ∴ Given system has non-trivial solution.
Let x₃ = k
∴ From equation (1), x₁ + x₂ = 15k
from equation (2), 2x₁ + x₂ = 15k × 2
4x₁ + 2x₃ = 30k - x₁ + 2x₂ = 15k
= 3x₁ = 15k
∴ x₁ = 5k
and x₂ = 15k - 2x₁ = 15k - 10k
∴ x₂ = 5k.
To find x₃ = x₁ + x₂
x₃ = 5k + 5k
∴ x₃ = 10k.
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Bobby walked 15 miles in 6 hours fill out the blanks in the table of equivalent ratios.
Miles. Hours
5. ____
15. _____
____. 12
Answer:
5. 2
15. 6
30. 12
Step-by-step explanation:
For every 5 miles walked, he walks for 2 hours each.
For example:
Miles. Hours
5. 2
10. 4
15. 6
20. 8
25. 10
30. 12
Complete the table of values for the function f(x)=1/2
The table of values for the function f(x)=1/2 are shown as follow:
What is function?
A function is an expression or relationship that involves one or more variables. It has several inputs and outputs. Each input has a single output. The function describes how the inputs relate to the output.
The function f(x) = 1/2 is a horizontal line that crosses the y-axis at the point (0, 1/2). Since the value of f(x) is constant for all values of x, the table of values for this function will have the same output value of 1/2 for every input value of x.
The table of values for the function f(x)=1/2 are shown as follow:
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what are the real solutions to y=x^2-25
Answer:
[tex]x=-5,5[/tex]
Step-by-step explanation:
[tex]y=x^2-25\\0=x^2-25\\0=(x+5)(x-5)\\x=-5,5[/tex]
Kennedi is feeding her 5 puppies. She has half a bag of dog food left. How much of the bag
will each puppy receive?
Answer:
1/10
Step-by-step explanation:
Puppy food portions.
Unish Baskota
Kennedi is feeding her 5 puppies. She has half a bag of dog food left. How much of the bag
will each puppy receive?
If Kennedi has half a bag of dog food left and she has 5 puppies to feed, then each puppy will receive 1/5th of the remaining dog food.
To calculate this, you can divide the remaining dog food (which is half a bag) by the number of puppies (which is 5):
1/2 bag of dog food ÷ 5 puppies = 1/10 bag of dog food per puppy
So each puppy will receive 1/10th of the bag of dog food.
Each puppy would receive 1/10 of food
25 points. Please show working out!
Answer:
42 m
Step-by-step explanation:
Equilateral triangle means that all sides are the same thus if the perimeter is 12 m
12m/3 = 4m for each side
The length of the boom will then how many triangles make the length which looks to be about 10.5
So 10.5 x 4 = 42 m
what is the percent of change from 75 to 24?
Please help been stuck on this
The measures of the unknown angles are
x = 46
and
y = 88
Calculating the measures of the unknown angles in the triangleFrom the question, we are to determine the measures of the unknown angles in the triangle.
From the diagram, we can observe that the triangle is an isosceles triangle as the legs of the triangle are congruent to each other.
Since the triangle is an isosceles triangle, the base angles of the triangle must be equal
The base angles are 46° and x°
Thus,
x = 46
Also,
We can write that
y° + 46° + x° = 180° (Sum of angles in a triangle)
y° + 46° + 46° = 180°
y° + 92° = 180°
y° = 180° - 92°
y° = 88°
Thus,
y = 88
Hence, x = 46 and y = 88
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The sum of the measures of the interior angles of a decagon (10 sided polygon) is 1,440.
The measure of each interior angle of a decagon is 144 degrees.
1. A decagon is a 10 sided polygon.
2. The sum of the interior angles of any polygon with n sides is (n - 2) * 180 degrees.
3. For a decagon, n = 10, therefore the sum of the interior angles is (10 - 2) * 180 degrees = 8 * 180 degrees = 1440 degrees.
4. Therefore, the measure of each interior angle of a decagon is 1440 degrees / 10 = 144 degrees.
The measure of each interior angle of a decagon is 144 degrees.
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if 'a' is the Geometric mean of 2 and 1/4 then find 'a'
The geometric mean of two numbers is the square root of their product. Hence, the geometric mean 'a' of 2 and 1/4 is approximately equal to 0.70710678118.
Finding the Geometric Mean of Two NumbersRecall that the geometric mean of two numbers is the square root of their product. This means that we need to multiply the two numbers (2 and 1/4) and then take the square root of the result.
Multiply 2 and 1/4:
2 × 1/4 = 1/2
Take the square root of the result:
a = √(1/2)
Simplify the square root:
a = √(1/2) = √(1)/√(2) = 1/√(2)
Rationalize the denominator by multiplying the top and bottom by √(2):
a = (1/√(2)) × (√(2)/√(2)) = √(2)/2
Approximate the result to a desired level of accuracy, if necessary. In this case, rounding to 11 decimal places gives:
a ≈ 0.70710678118
Therefore, the geometric mean 'a' of 2 and 1/4 is approximately equal to 0.70710678118
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In New York State, the minimum wage has grown exponentially. In 1966, the minimum wage was $1.25 an hour and in 2015, it was $8.75. Algebraically determine the rate of growth to the nearest percent.
The rate of growth of the minimum wage in New York State from 1966 to 2015 is approximately 6.41%.
To determine the rate of growth of the minimum wage in New York State from 1966 to 2015, we can use the following formula:
rate of growth = ((final value / initial value)^(1/number of years) - 1) * 100%
where the initial value is the minimum wage in 1966, the final value is the minimum wage in 2015, and the number of years is the difference between the two years.
Plugging in the values, we get:
rate of growth = [tex](($8.75 / $1.25)^{1/(2015-1966)) - 1} * 100%[/tex]
= [tex]((7)^{1/49) - 1} * 100%[/tex]
=[tex](1.0641 - 1) * 100%[/tex]
= 6.41%
Therefore, the rate of growth of the minimum wage in New York State from 1966 to 2015 is approximately 6.41%.
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can the tangent constraint be applied between a line and an arc?
Compare the rate of change of the two graphs shown.
The function for graph (1 or 2) has a higher rate of change than the function of graph (1 or 2).
.
Answer: The function of graph 2 has a higher rate of change than the function of graph 2
Step-by-step explanation:
If y* = y1/2(y2 + 1), then what is -3(16* - 9*)?
If y*=y1/2(y2+1) then -3(16* - 9*) = -10.5y1(y2 + 1), called polynomial.
What is a polynomial?A polynomial is an expression that consists of variables, numbers, and operations (like addition, subtraction, multiplication, and division). It is similar to a mathematical equation, but instead of an equals sign, it uses coefficients and variables in a power form. The power of a polynomial is determined by the highest power of the variables in the expression.
For example, if the polynomial is x2 + 5x + 2, then the power is 2. Polynomials are used in many areas of mathematics, such as calculus, algebra, and number theory.
If y*= y1/2(y2+1) then -3(16* - 9*)
= -3(16(y1/2(y2 + 1)) - 9(y1/2(y2 + 1)))
= -3(8y1(y2 + 1) - 4.5y1(y2 + 1))
= -3(3.5y1(y2 + 1))
= -10.5y1(y2 + 1)
In this problem, 16* is equal to 16(y1/2(y2 + 1)) and 9* is equal to 9(y1/2(y2 + 1)). When -3 is multiplied to 16* - 9*, the result is -3 times the difference of 16* and 9*. The expression then simplifies to -10.5y1(y2 + 1).
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what is 187cm in feet?
187 centimeter is approximately equal to 6.1351 feet
To convert 185cm to feet, we can use the following formula:
1 cm = 0.0328 feet
The conversion is the process of changing the unit of one quantity to another units
The conversion factor is defined as the number that is used to change one unit to another units by multiplying or dividing
Therefore,
The distance in feet = conversion factor × The distance in centimeter
Substitute the values in the equation
187 cm = 187 x 0.0328
Multiply the numbers
= 6.1351 feet ( rounded to four decimal places )
Therefore, 187 cm is 6.13517 feet.
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What is the conversion of 77 f to c?
The conversion of 77 f to c is equal to 25°C.
To convert temperatures in degrees Fahrenheit to Celsius, deduct 32 and multiply by .5556 ( or 5/9).
Temperatures, across the world, are measured either in Fahrenheit( the US temperature scale) or Celsius( the metric scale), two of the most popular and extensively used scales.
Temperature Conversion- Degrees Fahrenheit into Degrees Celsius.
To convert a temperature from Fahrenheit (°F) to Celsius (°C), we can use the formula:
°C = (°F - 32) x 5/9
By applying this formula, we can convert 77 °F to Celsius as follows
°C = (77 - 32) x 5/9
°C = 45 x 5/9
°C = 25
Therefore, 77°F is equal to 25°C.
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Elle sews fleece baby blankets, and each blanket requires 3 ft. of material. She has 52 ft. of fleece in her workshop and would like to make at least 16 blankets. Write an inequality that represents how many blankets Elle can make in her workshop
The final inequality that represents how many blankets Elle can make in her workshop would be:
3x ≤ 52 and x ≥ 16
What is Inequality ?In mathematics, an inequality is a statement that compares two values or expressions using an inequality symbol such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).
An inequality indicates that one value or expression is smaller, greater, or not equal to the other value or expression.
The inequality "y <= 10" states that the value of "y" is less than or equal to 10.
Let "x" be the number of blankets Elle can make in her workshop.
Since each blanket requires 3 ft. of material, the total amount of fleece required for "x" blankets would be 3x ft.
We know that Elle has 52 ft. of fleece in her workshop, so we can set up the inequality:
3x ≤ 52
This inequality states that the amount of fleece required for "x" blankets must be less than or equal to the amount of fleece Elle has available.
To satisfy the condition of making at least 16 blankets, we can add the requirement that "x" must be greater than or equal to 16.
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The graph of y=f(x) is shown in the xy-plane
which equation defines function f?
the equation of the line will be 5y = 7x -7. Option c is correct.
What is the Linear equation?A linear equation is an algebraic equation of the form y = mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. Sometimes, the aforementioned is referred to as a "linear equation of two variables," with y and x serving as the variables.
Given, a linear equation that passes through the points (5,0) and (0, -7)
The slope-intercept form of the equation of a line
y=mx+b,
where m is the slope
b is the y-intercept
since, slope = (y - y')/(x -x')
In our case,
slope = (-7 -0)/(0-5)
slope = 7/5
b = -7
thus,
the equation of the line will be:
y = 7/5x -7
multiply by 5 on both sides
5y = 7x -7
Therefore, the equation of the line will be 5y = 7x -7.
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Consider the expansions for expressions of the form (cy-3)^4, where c is equal to or less than 1.
For the case where c=1, determine the expansion of (y-3)^4.
For the case where c>1, how many times greater will the third term in the expansion of (cy-3)^4 be than in the expansion of (y-3)^4? Explain your answer.
The third term in the expansion of [tex](cy - 3)^{4}[/tex] is c² times greater than the third term in the expansion of [tex](y - 3)^{4}[/tex].
What is Expression ?A mathematical expression is made up of a statement, at least two integers or variables, and one or more arithmetic operations. This mathematical operation enables the multiplication, division, addition, or subtraction of numbers.
Now in the given question ,
When c = 1, we have:
[tex](y - 3)^{4}[/tex] =[tex]y^4 - 12y^3 + 54y^2 - 108y + 81[/tex]
When c > 1, we have:
[tex](cy - 3)^4 = c^4y^4 - 12c^3y^3 + 54c^2y^2 - 108cy + 81[/tex]
Expanding [tex](y - 3)^4[/tex] and [tex](cy - 3)^4[/tex], we can see that the third term in each expansion is:
[tex]54y^2[/tex] for [tex](y - 3)^4[/tex]
[tex]54c^2y^2[/tex] for [tex](cy - 3)^4[/tex]
To compare the third terms, we divide the third term in [tex](cy - 3)^4[/tex] by the third term in [tex](y - 3)^4[/tex]:
[tex](54c^2y^2) / (54y^2) = c^2[/tex]
So the third term in the expansion of [tex](cy - 3)^4[/tex] is c² times greater than the third term in the expansion of[tex](y - 3)^4[/tex].
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An electronics store sells used game consoles for $79.00 and new ones for $149. After two weeks the store made a profit of $1,456 by selling 14 to total game consoles. How many of each type of console were sold?
The number of used games consoles is 9 and the number of new game consoles is 5
Finding the numbers of each game:Here we use the Linear equation method to solve the problem. Since we don't know the number games, represent them with variables x and y.
Now for two linear equations according to the conditions as given in the problem. Solve the equations for the values of both x and y by using the substitution method.
Here we have
An electronics store sells used game consoles for $79.00 and new ones for $149.00
Let 'x' and 'y' be the number of used and new game consoles respectively
Cost of 'x' used games consoles = 79x
Cost of 'y' new game consoles = 149y
Given total profit =$ 1456
=> 79x + 149y = 1456 ----- (1)
The total number of game consoles = 14
=> x + y = 14
=> x = 14 - y ---- (2)
Substitute (2) in (1)
=> 79(14 - y) + 149y = 1456
=> 1104 - 79y + 149y = 1456
=> 70y = 350
=> y = 5
From (2) => x = 14 - 5 = 9
Therefore,
The number of used games consoles is 9 and the number of new game consoles is 5
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Sergio wants to buy the newest Jordans. They cost $210. Sergio plans to save $12 each month. His mom is going to give him $50. Write an equation to find x, the number of months it will take him to save the needed money.
Answer:
It will take him 13 Months to save up $12 a month to buy the shoes.
Hope this helps
Step-by-step explanation:
A cheetah can run 105 miles in hours. At that rate, how far can the cheetah run in 6 hours?
A cheetah could run 630 miles in 6 hours
Answer:
It can run 630 miles in 6 hours.
Step-by-step explanation:
If the cheetah can run 105 miles per hour
then,
in 6 hours, it can run 105*6 miles.
105*6= 630
Hope this helps.
find the surface area of the above solid a.81cm b.78cm c.84m d.72cm
Viola drives 200 meters up a hill that makes an angle of 3 degrees with the horizontal. To the nearest tenth of a meter , what horizontal distance has she covered ?
Viola has covered a horizontal distance of 199.7 meters when driving 200 meters up a hill that makes an angle of 3 degrees with the horizontal.
What horizontal distance has she covered?Viola drives 200 meters up a hill that makes an angle of 3 degrees with the horizontal. To find the horizontal distance she covered, we can use the cosine formula:
Cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is the horizontal distance we are trying to find, the hypotenuse is the distance she drove (200 meters), and the angle is 3 degrees. Plugging in these values, we get:
Cos(3º) = adjacent/200
Rearranging the equation to solve for the adjacent side, we get:
adjacent = 200*cos(3)
Using a calculator, we find that cos(3) is approximately 0.9986. Multiplying this by 200 gives us:
adjacent = 199.72
Therefore, the horizontal distance that Viola covered is approximately 199.72 meters.
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Using the equation f=9/5 c+32 which best describes if the relation c,f is a function?
Answer:
F = (9/5)C + 32 is a function because every Celsius temperature is associated with only one Fahrenheit temperature.
hope this helps!!
1. A toy rocket is launched from a 4. 9 m high platform in such a way that its height, h (in meters), after t seconds is given by the equation h=−4. 9t2+33. 6t+4. 9. How long will it take for the rocket to hit the ground?
he toy rocket will hit the ground after approximately enter your response here seconds.
2. The length of a rectangle is 7 centimeters less than five times its width. Its area is 24 square centimeters. Find the dimensions of the rectangle.
The width is enter your response here cm.
Part 2
The length is enter your response here cm.
3. Jeff and Kirk can build a 75-ft retaining wall together in 10 hours. Because Jeff has more experience, he could build the wall by himself 1 hour quicker than Kirk. How long would it take Kirk (to the nearest minute) to build the wall by himself?
It will take Kirk approximately enter your response here hours enter your response here min to build the wall himself.
(Round to the nearest minute as needed. )
4. On the last three physics exams a student scored 87, 85, and 91. What score must the student earn on the next exam to have an average of at least 90?
The student must score at least enter your response here%
In the given question, the rocket will hit the ground in 7 seconds. The width of the rectangle is 3 cm. Kirk makes the wall in 5.5 hours. The student needs 97 marks.
The relation between the height and the time of the toy rocket has been given in the question as,
h = -4.9t²+33.6t + 4.9
where,
h= height of the rocket after getting launched
t= time after which the rocket has been launched
So, if we have to calculate how much time the rocket takes to reach the ground,
we have to take h=0 (at ground).
So, the equation becomes
0 = -4.9t²+33.6t + 4.9
Now, the roots of this equation will give the time at which the rocket will hit the ground.
So, after calculating we get
t = -0.1428 sec
or
t = 7 sec
since negative time is not possible,
the time at which the rocket will hit the ground is 7 seconds.
The length of the rectangle is = 5b-7 cm
Hence, the area is = b(5b-7) = 24
Hence, the width comes out to be = 3 cm
Kirk would be able to make the wall in 5.5 hours. This is because Jeff can make it one hour earlier than him i.e., K-1 hours. So, we get -
= (K-1) + K = 10
= 2K = 11
= K = 5.5 hours
The student already has 87 + 85 + 91 = 263 marks. He needs at least -
= (263 + x) / 4 = 90
= 97 marks
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Solve. It’s all real numbers are solutions, click on all reels if there’s no solution click on no solution.
Answer:
no solution
Step-by-step explanation:
7 | y + 6 | + 3 < - 18 ( subtract 3 from both sides )
7 | y + 6 | < - 21 ( divide both sides by 7 )
| y + 6 | > - 3
the absolute value function cannot be negative
thus the equation has no solution
A student is taking a multiple-choice math test. He has two questions left to answer. The first question has the four answer choices of A, B, C, and D. The second question has the five answer choices A, B, C, D, and E. The student randomly selects an answer for each question. Construct a sample space showing all the possible outcomes the student could answer the two questions
There are 4 possible answers for the first question and 5 possible answers for the second question, so there are a total of 4 x 5 = 20 possible outcomes.
What is Sample Space ?A set of potential outcomes from a random experiment is known as a sample space. The sample space is identified by the letter "S." Events are the subset of possible experiment results.The results in a sample area could vary depending on the experiment. Discrete or finite sample spaces are those that have a finite number of outcomes.
The sample space showing all possible outcomes of the two questions is the Cartesian product of the two sets of answer choices, which is:
{(A, A), "(A, B), "(A, C), "(A, D), "(A, E), "(C, A), "(C, B), "(C, C), "(C, D), "(C, E")," "(D, A), "(D, B), "(D, C), "(D, D), "(D, E)}.
There are 4 possible answers for the first question and 5 possible answers for the second question, so there are a total of 4 x 5 = 20 possible outcomes. Each outcome is a pair of answers, where the first element of the pair is the answer to the first question and the second element of the pair is the answer to the second question.
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People enter line for an escalator at rate modeled by the function given by r(t) {~Gios)' ( 300 ,_ for 0 < t < 300 for [ > 300, where r(t) is measured people per second and IS measured In seconds_ As people get on the escalator; they exit the line at constant rate of 0.7 person per second. There are 20 people in line at time (a) How many people enter the line for the escalator during" the time interval 0 < / < 300 (b) During the time interval 0 < 0 < 300. there are always people in line for the escalator: How many people are Mn Line at time 300 (c) For t > 300. what is the first time that there are no people in line for the escalator? (d) For 0 < t < 300_ at what time is the number of people in line minimum? To the nearest whole number; find the number of people in line at this time. Justify vour answer:
270 people enter the line for the escalator during" the time interval 0 < / < 300 and 80 people are at Mn Line at time 300. 414.285714 sec. is the first time that there are no people in line for the escalator and 33. 0133 time is the number of people in line minimum.
X(t) = 44 ( t/100)³ * (1-t/300)⁷
for 0≤t≤300
for t≥ 300
a) Total number of people = ∫³⁰⁰₀ r(t)*dt
∫³⁰⁰₀ 44(t/100)³ * (1-t/300)⁷ * dt
∫³⁰⁰₀ (11 (1-t/300)⁷ / 250000 * t³) * dt
∫³⁰⁰₀ -11 / 250000 * (t-300)⁷ * t³ * (1/300)⁷ dt
= -11 (1/300)⁷ / 250000 ∫³⁰⁰₀ (t-300)⁷ * t³ * dt
Now Integrate with respect to 't'
using substitution
Let u = t-300
du = dt
∫ u⁷ * (u + 300)³ * du
∫ (u¹⁰ + 900 u⁹ +270000 u⁸ + 27000000 u⁷) * du
= u¹¹/11 + 90 u¹⁰ + 30000 u⁹ + 3375000 u⁸
Undo the substitution
u= (t-300)
= [ (t-300)¹¹ / 11 + 90 (t-300)¹⁰ + 30000(t-300)⁹ + 337500(t-300)⁸]³⁰⁰₀
Substitutes the limits
= [0- ((-300)¹¹/ 11 + 90 * (-300)¹⁰ + 30000 * (-300)⁹ + 3375000 * (-300)⁸]
= 270 people
b) Constant rate of 0.7 people per second
This is the rate of exit
at t = 0.20 people in line
People in a line at t = 300 is = 20+ ∫³⁰⁰₀ ((r(t) - 0.7 ) * dt
= 20 + ∫³⁰⁰₀ 44(t/100)³ * (1-t/300)⁷- 0.7 dt
= 20 + 270 - 0.7 * 300
= 80 people
c) t = 300 sec
people = 80
when t>300 , r(t) = 0
rate of exit remains the same
= 80 + ∫ ( 0-0.7) * dx
= 80 + [-0.7x] = 0
= 80 - [0.7x] = 0
= 80 - 0.7 t + 0.7 * 300 = 0
= 80 - 0.7t + 210 = 0
= - 0.7 t = -290
t = 414.285714 sec.
d) f(t) = 20 + ∫⁶₀ (r(x) - 0.7) * dx
f(t) = r(t) - 0.7
f(t) = 0 (for critical point)
r(x) = 44 (t/100)³ * (1-t/300)⁷
r(t) = 0
we get t= 33.0133 sec.
t = 166.575 sec.
Now check for the value
when t=0
f(t) 20
when t= 33.0133
f(t) = 3.803 ≈ 4
when t = 166.575
f(t) = 158.07 ≈ 158
when t= 300
f(t) = 80
so the minimum number of people is 4 at the time t= 33. 0133
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ABCD is a parallelogram. Find angle y.
Therefore , the solution of the given problem of parallelograms comes out to be ∠y = 62 degree.
What is the purpose of parallelograms?A simple square with two sets of equal intervals is referred to as a parallelogram in Euclidean geometry. The opposite ends of a parallelogram, a special type of quadrilateral, were both straight and equal. There are four distinct parallelogram types, and two of them are exclusive of one another. The four different shapes are rhombuses, equilateral triangles, triangles, and rectangles. A quadrilateral is a parallelogram if it has two pairs of parallel sides.
Here,
Given :
A parallelogram ABCD and a triangle AED
So,
To find angle y:
∠D + 118 = 180
=> ∠D = 180 -118
=>∠D= 62 Degree
So,
According to alternate interior angle :
∠D= ∠y
=> 62 = ∠y
Thus ,∠y = 62 degree.
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What is the conversion of 22 kg to lbs ?
The conversion of 22 kg to lbs is 48.50164 lbs.
Kilogram (kg) and pound (lbs) are both units of mass or weight.
Kilogram is the base unit of mass in the International System of Units (SI). It is defined as the mass of the International Prototype of the Kilogram (IPK), a platinum-iridium cylinder kept at the International Bureau of Weights and Measures.
To convert 22 kilograms (kg) to pounds (lbs), we can use the conversion factor of 1 kg = 2.20462 lbs.
So, we can multiply 22 kg by 2.20462 lbs/kg to get the equivalent weight in pounds:
22 kg × 2.20462 lbs/kg = 48.50164 lbs (rounded to five decimal places)
Therefore, 22 kg is approximately equal to 48.50164 lbs.
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