The makeup artist will need approximately 7.296π square inches of gift wrap to wrap 3 lipsticks. Rounded to two decimal places, the answer is 19.59π square inches.
How to calculate how many total square inches of gift wrap will the makeup artist need to wrap 3 lipsticksThe formula for the surface area of a cylinder is:
SA = 2πr² + 2πrh
where r is the radius and h is the height of the cylinder.
For one lipstick, the surface area is:
SA = 2π(0.4)² + 2π(0.4)(2.2)
SA = 1.024π + 1.408π
SA = 2.432π square inches
To wrap 3 lipsticks, the total surface area would be:
SA = 3(2.432π)
SA = 7.296π square inches
Therefore, the makeup artist will need approximately 7.296π square inches of gift wrap to wrap 3 lipsticks. Rounded to two decimal places, the answer is 19.59π square inches.
So the correct option is: 19.59π square inches.
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A builder wishes to fence in 80000 m2 of land in a rectangular shape. for security reasons, the fence along the front part of the land will cost $60 per meter, while the fence for the other three sides will cost $20 per meter.
how much of each type of fence should the builder buy to minimize the cost of the fence?
determine the length of the fence along the front part of the land that will be cost $60 per meter.
(give your answer as a whole or exact number.)
To minimize the cost of the fence, the builder should use the expensive fence along the shorter side of the rectangular shape, as this will require less length of the expensive fence. Let's say the length of the rectangle is x meters and the width is y meters. Then the area of the rectangle is given by:
A = xy = 80000
And the perimeter of the rectangle is:
P = 2x + 2y
We are given that the cost of the fence along the front part of the land will cost $60 per meter, while the fence for the other three sides will cost $20 per meter. So the total cost of the fence is:
C = 60x + 20(2x + 2y)
Simplifying this expression, we get:
C = 100x + 40y
We can now use the area equation to eliminate one of the variables. Solving for y, we get:
y = 80000/x
Substituting this expression for y into the cost equation, we get:
C = 100x + 40(80000/x)
Simplifying this expression, we get:
C = 100x + 3200000/x
To minimize this function, we need to take its derivative and set it equal to zero:
dC/dx = 100 - 3200000/x^2 = 0
Solving for x, we get:
x = sqrt(32000) = 178.89
So the length of the rectangle should be approximately 178.89 meters, and the width should be:
y = 80000/178.89 = 446.68
Therefore, the amount of expensive fence needed is 178.89 meters, and the amount of cheap fence needed is:
2(178.89) + 2(446.68) - 178.89 = 893.36 meters
Finally, the length of the fence along the front part of the land that will be cost $60 per meter is simply the width of the rectangle, which is:
y = 446.68 meters (rounded to two decimal places)
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point (4, -13) lies on the graph of the equation y = kx + 7
what is value of k?
Answer:
-5
Step-by-step explanation:
(4, -13) = (x, y)
y = kx + 7
-13 = k(4) + 7
4k = -13-7
4k = -20
k = -5
#CMIIW3x − 15y = 11 in slope intercept form
Answer:
To convert the equation 3x - 15y = 11 into slope-intercept form, we need to solve for y.
First, we'll subtract 3x from both sides:
-15y = -3x + 11
Next, we'll divide both sides by -15:
y = (3/15)x - (11/15)
Simplifying the fraction:
y = (1/5)x - (11/15)
This is the slope-intercept form, where the slope is 1/5 and the y-intercept is -11/15.
Use the image below to find x: Show your steps and identify the TRIG RATIO that you used to find x.
The measure of the angle x in the circle is 65 degrees
Solving for x in the circleFrom the question, we have the following parameters that can be used in our computation:
The circle
On the circle, we have the angle at the vertex of the triangle to be
Angle = 100/2
Angle = 50
The sum of angles in a triangle is 180
So, we have
x + x + 50 = 180
Evaluate the like terms,
2x = 130
So, we have
x = 65
Hence, the angle is 65 degrees
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Is The number of insects feeding on a tree leaf discrete or continious
The number of insects feeding on a tree leaf is a discrete variable.
The number of insects feeding on a tree leaf is a countable variable that can only take on integer values (0, 1, 2, 3, etc.). It cannot take on fractional or continuous values. This is because each insect can either feed on the leaf or not, and there cannot be a fractional or continuous number of insects feeding on the leaf.
Therefore, the number of insects feeding on a tree leaf is a discrete variable. This is in contrast to a continuous variable, which can take on any value within a certain range. For example, the weight of the insects on the leaf would be a continuous variable since it can take on fractional values.
In mathematical terms, the number of insects feeding on a tree leaf can be represented as a discrete random variable X, where X can take on any non-negative integer value.
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Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting Carmen needs 2 cups of her frosting mixture how many cups of strawberry frosting and how many cups of lemon frosting will Carmen need
Carmen needs (6/7) cups of strawberry frosting and (1 1/7) cups of lemon frosting to make 2 cups of the frosting mixture.
To determine the amount of strawberry frosting and lemon frosting that Carmen needs to make 2 cups of the frosting mixture, we need to use a proportion.
Let x be the amount of strawberry frosting needed in cups, and y be the amount of lemon frosting needed in cups.
From the given information, we know that Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting. Thus, the ratio of the amounts of strawberry frosting to lemon frosting is:
x/y = (1/4)/(1/3)
Simplifying this ratio, we get:
x/y = 3/4
We also know that the total amount of frosting needed is 2 cups, so:
x + y = 2
Using substitution, we can solve for x:
x + (4/3)x = 2
(7/3)x = 2
x = (6/7) cups
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4. what is the difference in the measures of center?
5. what is the variability of grades each week?
6. what conclusions can you draw about the test?
Measures of center are statistical tools used to determine the central tendency of a dataset, including mean, median, and mode.
The difference between these tools is how they capture the central tendency.
Variability of grades refers to how much grades fluctuate from week to week, which can be measured using statistical tools such as range, variance, and standard deviation. Without specific information about the test, it is not possible to draw conclusions.
However, analyzing the measures of center and variability can provide insights into student performance and grading consistency.
Further analysis, such as comparing grades to class averages or identifying patterns over time, may reveal additional information.
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Find the area of the shaded region. Provide an answer accurate to the
nearest tenth.
18 ft
10 ft
Thus, the area of the shaded part is found to be 50 sq. ft.
Define about area of the shaded region:The shaded region's area is most frequently found in common geometry problems. Such problems always have a minimum of two forms, and you must determine the area for each shape as well as the darkened zone by deducting the smaller shape's area from the larger.
Rectangle's area :
Area has two dimensions: length and width. Square units like square inches, square feet, or square metres are used to measure area.
Multiply its length by the width to determine the area of a rectangle. A is equal to L * W, where * denotes multiplication, L is the length, W is the breadth, and A is the area.Length of shaded part = 5 ft
width of shaded part = 10 ft
Area = 5*10
Area = 50 sq. ft
Thus, the area of the shaded part is found to be 50 sq. ft.
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Correct question:
For the given figure find the area of the shaded region.
Length BC = 18 ft
Length CD = 10 ft
Simplify m^8m^−6.
one over m to the forty eighth power
m^2
one over m squared
−m^14
It should be noted that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².
How to calculate the valueUsing the product of powers rule for exponents, we can simplify m⁸m as follows:
m⁸m⁻⁶ = m⁸⁻⁶) = m²
Therefore, m⁸⁻⁶ is equal to m².
Now, we can further simplify by expressing m² as (m²)¹. Multiplying the exponents, we get:
(m²)¹ = m²
We can say that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².
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A figure with parallel Lines m and n is shown.
The measure of angles A, B, and C for the given parallel lines will be 53°,90°, and 143° corresponding.
What is an example of a parallel line?
In terms of geometry, parallel lines are two separate lines that never cross each other and are located in the same plane. Both vertical and horizontal can be used. A zebra crossing, rows of notebooks and nearby railway tracks are just a few instances of parallel lines that we encounter every day.
As per the parallel lines m and n.
The adjacent angle of B = 37° (corresponding angle same)
m∠B = 180° - (53° + 37°) = 90°
m∠A = 53° (corresponding angle same)
m∠C = 180° - 37° = 143°
Hence "The measure of angles A, B, and C for the given parallel lines will be 53°,90°, and 143° corresponding".
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What is 2 9 as a percentage? give your answer rounded to one decimal place.
2/9 as a percentage is approximately 22.2%.
To convert the fraction 2/9 to a percentage, you simply need to divide the numerator (2) by the denominator (9) and then multiply the result by 100.
1. Divide the numerator by the denominator: 2 ÷ 9 ≈ 0.2222
2. Multiply the result by 100: 0.2222 × 100 = 22.22%
Now, to round the answer to one decimal place, we consider the second digit after the decimal point. In this case, it's 2. Since it's less than 5, we can round down.
So, 2/9 as a percentage rounded to one decimal place is approximately 22.2%.
In summary, converting a fraction to a percentage involves dividing the numerator by the denominator and then multiplying the result by 100. Rounding to a specific decimal place helps in presenting the result in a more easily understandable form, especially when dealing with non-integer values.
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Gianna keeps track of the number of people inside a music hall to attend a concert by looking at the number of scanned tickets. She plotted the data on the graph below, where x = 0 x=0 represents the time at 6 p.m., then drew a line of best fit. What does the point ( 1 , 92 ) (1,92) represent?
Note that the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.
How is this so?This is based on the given graph.
Note tha the horizontal axis = x = 0
which corresponds to 6pm
the vertical is the number of scanned tickets.
Also, the pont (1, 92) is the line of best fit so that means that at 7pm which is a n hour after 6pm, there were about 92 persons still in the hall.
Hence we are correct to state that the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.
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Full Question:
See attached image/graph
Find the values of a and b, if the function defined by f(x) = x^2 + 3x + a , x <= 1
bx + 2, x >= 1 is differentiable at x = 1
To find the values of a and b, we need to ensure that the function is differentiable at x = 1. Thus, the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
First, we need to check that the function is continuous at x = 1. Since the function has different definitions for x <= 1 and x >= 1, we need to check that the limit of the function as x approaches 1 from both sides is the same.
Limit as x approaches 1 from the left (x <= 1):
f(x) = x^2 + 3x + a
lim x->1- f(x) = lim x->1- (x^2 + 3x + a) = 1^2 + 3(1) + a = 4 + a
Limit as x approaches 1 from the right (x >= 1):
f(x) = bx + 2
lim x->1+ f(x) = lim x->1+ (bx + 2) = b + 2
For the function to be continuous at x = 1, these two limits must be equal.
4 + a = b + 2
a = b - 2
Now we need to check that the derivative of the function at x = 1 exists and is equal from both sides.
Derivative of the function for x <= 1:
f(x) = x^2 + 3x + a
f'(x) = 2x + 3
f'(1) = 2(1) + 3 = 5
Derivative of the function for x >= 1:
f(x) = bx + 2
f'(x) = b
f'(1) = b
For the function to be differentiable at x = 1, these two derivatives must be equal.
5 = b
Substituting b = 5 into the equation we found earlier for a, we get:
a = 5 - 2 = 3
Therefore, the values of a and b that make the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
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9. define a relation r on the integers, ∀m, n ∈ z, mean if m n is even. is r a partial order relation? prove or give counterexample.
No, the relation r is not a partial order relation.
To prove this, we need to show that r is not reflexive, not antisymmetric, or not transitive.
r is reflexive if ∀a∈Z, a a holds, which means that any integer is related to itself. This is true for r since a a = 2 × a = even.r is antisymmetric if whenever a b and b a, then a = b. This is not true for r since, for example, 2 6 and 6 2, but 2 ≠ 6.r is transitive if whenever a b and b c, then a c. This is not true for r since, for example, 2 6 and 6 4, but 2 is not related to 4.Since r fails to satisfy the antisymmetric property, it is not a partial order relation.
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A soccer field (football pitch) has a length of 102. 9 m and a width of 66. 3 m. Find the total area of the field in square meters (m2) and convert this measurement to square yards (yd2). Use the fact that 1 yard = 0. 9144 m. Round your answer to the nearest whole number
The total area of the soccer field is approximately 8150 square yards.
We'll find the total area of the soccer field in square meters first, and then convert it to square yards using the conversion factor provided.
Find the area in square meters (m²):
Area = Length × Width
Area = 102.9 m × 66.3 m
Area ≈ 6816.47 m²
Convert the area to square yards (yd²):
Use the conversion factor: 1 yard = 0.9144 meters
1 m² = (1/0.9144)² yd²
1 m² ≈ 1.19599 yd²
Now, multiply the area in m² by the conversion factor to get the area in yd²:
Area ≈ 6816.47 m² × 1.19599 yd²/m²
Area ≈ 8150 yd² (rounded to the nearest whole number).
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x Which two choices are equivalent to this expression?
2√75 +3√50
x
x
A
25√6
B 10√3+15/2
C 25√3+25√2
D 2√25-3+3√25-2
E
3√25+2√25
Can someone help me please
Answer:
A, 48 degrees
Step-by-step explanation:
180 - 42 - 90 = 48 (all interior angles add up to 48)
Answer:
it’s A
Step-by-step explanation:
it’s a because the right triangle is always 90°.As you can see there is 42° so you subtract to find the missing number.
PLS HELP____________
Answer:
the answer is the 1st one
2+2=4
3+1=4
4+0 = 4
The line on a coordinate plane makes an angle of depression 32 degrees. What is the slope of the line
The slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.
To find the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees,:
Step 1: Determine the angle of elevation. Since the angle of depression is 32 degrees, the angle of elevation is also 32 degrees, because they are alternate angles.
Step 2: Use the tangent function to find the slope. The tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle (rise) to the side adjacent to the angle (run). In this case, the tangent of the angle of elevation (32 degrees) is equal to the slope of the line.
Step 3: Calculate the tangent of 32 degrees. Using a calculator or a trigonometric table, you can find that tan(32°) ≈ 0.625.
So, the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.
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Ghost riders co. has an eps of $1.65 that is expected to grow at 8.5 percent per year. if the pe ratio is 19.15 times, what is the projected stock price in 4 years?
The projected stock price of Ghost Rider Co. in 4 years is $45.24.
First, we need to calculate the future EPS of Ghost Rider Co. in 4 years. We can do this using the formula for the future value of an annuity:
[tex]FV = PV x (1 + r)^n[/tex]
where FV is the future value, PV is the present value, r is the growth rate, and n is the number of years.
Using this formula, we get:
[tex]FV = $1.65 x (1 + 0.085)^4 = $2.36[/tex]
Next, we can use the following formula to determine the anticipated stock price:
Estimated stock price = EPS x PE ratio
When we enter the values we have, we obtain:
Projected stock price = $2.36 x 19.15 = $45.24
Therefore, the projected stock price of Ghost Rider Co. in 4 years is $45.24.
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Expand up to the 4th term
√1+3x
Answer:
1+3x
Step-by-step explanation:
[tex] \sqrt{1 } = 1 \\ 1 + 3x = 1 + 3x[/tex]
Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles.
What is the area of the plot?
Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles. The area of the plot is 384 sq meters.
The Pythagorean theorem is a fundamental geometric idea that deals with the connections between the sides of right triangles. The square of the length of the hypotenuse (c) of a right triangle is equal to the sum of the squares of the lengths of the other two sides, according to the theorem (a and b). This may be stated mathematically as follows:
c² = a² + b²
Pythagoras, the ancient Greek mathematician who is credited with inventing the theorem, is named for him. It is employed in domains like physics, astronomy, and surveying and has extensive applications in mathematics, science, and engineering.
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Answer:
Step-by-step explanation:
The plot is in the shape of a trapezium with two sides measuring 32 m and 24 m, and two other sides measuring 25 m each.
To find the area of the plot, we need to first find the height of the trapezium. We can use the Pythagorean theorem to do this.
The side opposite to the right angle is the hypotenuse of the right-angled triangle formed by the two sides measuring 25 m each. So,
h² = 25² - 24²
h² = 625 - 576
h² = 49
h = 7
Therefore, the height of the trapezium is 7 m.
The area of a trapezium is given by the formula:
Area = (sum of parallel sides) x (height) / 2
In this case, the sum of the parallel sides is:
32 + 24 = 56
So, the area of the plot is:
Area = 56 x 7 / 2
Area = 196 m²
Therefore, the area of the plot is 196 square meters.
salvador recorded in this list the heights in millimeters of each of his bean plants.
52, 46, 51, 32,50
which 2 inequalities best describe, h, the plant heights in millimeters?
h < 32, h > 52
h> 32, h < 52
h < 46, h > 52
h < 46, h > 52
The two inequalities that best describe the plant heights in millimeters are: h > 32 and h < 52. This is because all the recorded heights fall within this range. The other options do not include all the recorded heights or include heights that are not recorded.
To find the best inequalities that describe the plant heights (h) in millimeters, we need to determine the minimum and maximum heights from the given list.
List of plant heights: 52, 46, 51, 32, 50
Minimum height: 32 mm
Maximum height: 52 mm
Now we can write the inequalities that best describe the plant heights:
h > 32 (heights are greater than 32 mm)
h < 52 (heights are less than 52 mm)
Your answer: h > 32, h < 52
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Use your mouse or finger to split the
trapezoid into two triangles and a rectangle.
I ready
Trapezoids can be split into two triangles and a rectangle.
Trapezoid is also known as a trapezium which is a closed shape having 4 sides with one pair of parallel sides. Trapezium is quadrilateral with 4 sides The parallel sides of a trapezium are known as the bases, and its non-parallel sides are called legs. A trapezium can also have parallel legs. The parallel sides can be horizontal, vertical, or slanting. Few real-life objects example of trapezium is a lamp, popcorn box etc.
A trapezoid consists of two triangles and one rectangle figure shows how can we cut the trapezium to split the trapezium.
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A circular mirror has a radius of 3. 4 feet rosalinda is decorating the edge of the mirror with Washington tape if she has exactly enough washi tape which measurement is closest to the length of the piece of washi tape in feet
The measurement closest to the length of the piece of washi tape needed is approximately 21.36 feet.
The circumference of the circular mirror can be calculated using the formula C = 2πr, where r is the radius. Plugging in the given radius of 3.4 feet, we get C = 2π(3.4) = 21.36 feet (rounded to two decimal places). Since Rosalinda is decorating the edge of the mirror with washi tape, she needs a piece of tape that is equal in length to the circumference of the mirror. Therefore, the length of the piece of washi tape needed is closest to 21.36 feet.
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88 POINTS!!!!!
Joe and Tommy are playing a game of chicken. Tommy has a mass of 91 kg, Joe has a mass of 97 kg. Tommy is running towards Joe with a velocity of 5 m/s, Joe is running towards Tommy with a velocity of 6 m/s. Neither "chickens" out. After collision, Joe is standing still. How fast does Tommy bounce off Joe? Round to four decimal places
Tommy bounces off Joe at a speed of 5.9451 m/s after the collision.
We need to use the conservation of momentum, which states that the total momentum of a closed system remains constant. In this case, the closed system is the two players, Joe and Tommy.
We can start by calculating the initial momentum of the system, which is given by:
[tex]p_{initial} = m_{Tommy} * v_{Tommy} + m_{Joe} * v_{Joe}[/tex]
where m_Tommy and m_Joe are the masses of Tommy and Joe, respectively, and v_Tommy and v_Joe are their initial velocities.
Plugging in the given values, we get:
[tex]p_{initial} = 91 kg * 5 m/s + 97 kg * 6 m/s[/tex]
p_initial = 1123 kgm/s
After the collision, Joe is standing still, which means his velocity is zero. Let's call Tommy's velocity after the collision v_Tommy', which we need to find.
The final momentum of the system is given by:
[tex]p_{final} = m_{Tommy} * v_{Tommy'} + m_{Joe} * 0[/tex]
where we set Joe's velocity to zero since he's standing still.
Since the momentum is conserved, we can equate p_initial and p_final:
p_initial = p_final
[tex]m_Tommy * v_Tommy + m_Joe * v_Joe = m_Tommy * v_Tommy'[/tex][tex]m_{Tommy} * v_{Tommy} + m_{Joe} * v_{Joe} = m_{Tommy} * v_{Tommy'}[/tex]
91 kg * 5 m/s + 97 kg * 6 m/s = 91 kg * v_Tommy'
Solving for v_Tommy', we get:
[tex]v_{Tommy'} = (91 kg * 5 m/s + 97 kg * 6 m/s) / 91 kg[/tex]
v_Tommy' = 5.9451 m/s (rounded to four decimal places)
In summary, we used the conservation of momentum to calculate the velocity of Tommy after bouncing off Joe. We found that he bounces off Joe at a speed of 5.9451 m/s.
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66. Which value of m makes the inequality true?
A. 4
B. 5
3m-4 < 11
C. 6
D. 7
Answer:
The answer to the question provided is choice A, 4.
The value of m which makes the inequality true is, 4
What is Inequality?A relation by which we can compare two or more mathematical expression is called an inequality.
Given that;
The inequality is,
⇒ 3m - 4 < 11
Now,. We can simplify as;
⇒ 3m - 4 < 11
⇒ 3m < 11 + 4
⇒ 3m < 15
⇒ m < 5
Thus, The value of m which makes the inequality true is, 4
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A spring with an m-kg mass and a damping constant 5 (kg/s) can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons. If the spring is stretched 1 meters beyond its natural length and then released with zero velocity, find the mass that would produce critical damping. m = kg
The mass can be any value greater than zero.
To find the mass that would produce critical damping, we first need to find the damping coefficient, which is given by:
c = damping constant * 2 * √m
where m is the mass in kg.
In this case, c = 5 * 2 * √m = 10√m.
Next, we can use the equation for the displacement of a damped harmonic oscillator to find the value of m that produces critical damping:
x = e^(-ct/2m) * (A + Bt)
where x is the displacement from equilibrium, t is time, A and B are constants determined by the initial conditions, and c and m are the damping coefficient and mass, respectively.
For critical damping, we want the system to return to equilibrium as quickly as possible without oscillating, so we set the damping coefficient equal to the critical damping coefficient:
c = 2 * √km
where k is the spring constant.
Since the spring can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons, we know that the spring constant is:
k = F/x = 2/0.5 = 4 N/m
Substituting this value into the equation for critical damping, we get:
10√m = 2 * √(4m)
Squaring both sides and simplifying, we get:
100m = 16m
84m = 0
Since this is a contradiction, there is no value of m that produces critical damping. Therefore, the mass can be any value greater than zero.
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find the general solution of the following linear system. y′ = [2 2 −4 2 −1 −2 4 2 −6] y with λ = −1,−2,−2
To find the general solution of the linear system y' = [2 2 -4; 2 -1 -2; 4 2 -6] y, we need to first find the eigenvectors and eigenvalues of the coefficient matrix A = [2 2 -4; 2 -1 -2; 4 2 -6].
Using the characteristic equation, we can find the eigenvalues:
det(A - λI) = 0
=> det([2-λ 2 -4; 2 -1-λ -2; 4 2 -6-λ]) = 0
=> (2-λ)[(-1-λ)(-6-λ) - 4] - 2[(-2)(-6-λ) - 8] + 4[2(-1-λ) - 4] = 0
=> λ^3 - + 8λ - 4 = 0
=> (λ-1)(λ-2[tex])^2[/tex] = 0
Thus, λ = 1, 2 (with multiplicity 2). For each eigenvalue, we need to find a corresponding eigenvector.
For λ = 1, we need to find the null space of the matrix (A - λI):
A - λI = [1 2 -4; 2 -2 -2; 4 2 -7]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3]
So, we have a basic eigenvector of the form [4,-2,1]^T. To obtain a linearly independent eigenvector, we use the method of generalized eigenvectors. We need to find a vector v such that (A - λI) v = u, where u is the basic eigenvector.
(A - λI) v = u
=> [1 2 -4; 2 -2 -2; 4 2 -7] v = [4; -2; 1]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9] v = [4; -2; 1]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3] v = [4; -2; 1]
=> -6v2 + 6v3 = -2
=> 3v3 = 1
=> 2v2 - 4v3 = -2
=> v2 = 0
So, we have v = [0; 1/3; 2/3[tex]]^T[/tex] as the second eigenvector corresponding to λ = 1.
For λ = 2, we need to find the null space of the matrix (A - λI):
A - λI = [0 2 -4; 2 -3 -2; 4 2 -8]
=> R1 <-> R2
[2 -3 -2; 0 2 -4; 4 2 -8]
=> R3 <- R3 - 2R1
[2 -3 -2; 0 2 -4; 0 8 -12]
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