(a) The non-identity isometry given by a reflection followed by a glide reflection can be classified as a translation. In Euclidean geometry, any reflection followed by a glide reflection is equivalent to a translation in a specific direction.
(b) The non-identity isometry that fixes two points can be classified as a rotation. In Euclidean geometry, any isometry that fixes two distinct points is a rotation about the midpoint between those two points.
(a) Let's denote the reflection as R and the glide reflection as G. If G is applied after R, we have GR. Since the isometry does not have any fixed points, GR cannot be a reflection or a translation. Therefore, the only possibility left is that GR is a glide reflection. However, in Euclidean geometry, a glide reflection is equivalent to a translation. Hence, the non-identity isometry in this case is a translation.
(b) Since the isometry fixes two points P and Q, let's denote the isometry as F. Fixing two distinct points implies that the isometry must be a rotation about the midpoint between those two points. Therefore, the non-identity isometry in this case is a rotation.
In summary, the non-identity isometry in (a) is a translation, and the non-identity isometry in (b) is a rotation.
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Let λ be an eigenvalue of an invertible matrix a. show that λ^−1 is an eigenvalue of A^−1. [hint: suppose a nonzero x satisfies Ax=λx.]
Let λ be an eigen value of an invertible matrix. Then, [tex]\lambda^{-1}[/tex] is surely an eigenvalue of [tex]A^{-1}[/tex].
What is an invertible matrix?
For a matrix to be invertible, it must have a unique matrix that, when multiplied with the original matrix, gives the identity matrix.
[tex]A * B = B * A = I[/tex]
Suppose A is an invertible matrix and λ is an eigenvalue of A with a corresponding nonzero eigenvector x, i.e., Ax = λx.
To show that [tex]\lambda^{-1}[/tex] is an eigenvalue of [tex]A^{-1}[/tex], we need to find a nonzero vector y such that [tex]A^{-1}y[/tex] = [tex]\lambda^{-1}y[/tex].
Let's start by multiplying both sides of the equation Ax = λx by [tex]A^{-1}[/tex]:
[tex]A^{-1}(Ax) = A^{-1}(\lambda x)[/tex]
(x is nonzero, so we can divide by x)
[tex]A^{-1}(Ax/x) = A^{-1}(\lambda x/x)\\A^{-1}(A(x/x)) = A^{-1}(\lambda)[/tex]
Since [tex]A^{(-1)}A = I[/tex] (identity matrix), and x/x = 1, we have:
[tex]A^{(-1}(I) = A^{(-1)}[/tex] λ
[tex]A^{(-1)}[/tex] = λ[tex]A^{(-1)}[/tex]
Now, let y = A^(-1)x. We can rewrite the equation above as:
[tex]A^{(-1)}x[/tex] = λ[tex]A^{(-1)}y[/tex]
([tex]A^{(-1)}x[/tex]/λ) = [tex]A^{(-1)}y[/tex]/λ
(x is nonzero, so we can divide by x)
([tex]A^{(-1)}x/x[/tex])/λ = [tex](A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}(x/x)[/tex])/λ = ([tex]A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}(1)[/tex])/λ = ([tex]A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}[/tex])/λ = ([tex]A^{(-1)}y[/tex])/λ
Since [tex]A^{(-1)}[/tex] is a matrix and λ is a scalar, we can rearrange the equation as follows:
([tex]A^{(-1)}[/tex])/λ = [tex]A^{(-1)}[/tex]y/λ
(1/λ)[tex]A^{(-1)}[/tex] =[tex]A^{(-1)}[/tex]y/λ
This shows that 1/λ is an eigenvalue of [tex]A^{(-1)}[/tex] with the corresponding eigenvector y. Therefore, we have shown that if λ is an eigenvalue of A, then [tex]\lambda^{(-1)}[/tex] is an eigenvalue of [tex]A^{(-1)}[/tex]
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Which value of x satisfies the equation below? 1/2 (3x + 17) = 1/6 (8x-10)
Choice answers:
A. -61
B -55
C. -41
D-35
For the differential equation s" + bs' +9s = 0, find the values of b that make the general solution overdamped, underdamped, or critically damped. (For each, give an interval or intervals for b for which the equation is as indicated. Thus if the the equation is overdamped for all b in the range -1
The general solution to the differential equation s" + b s' + 9s = 0 can be written as:
[tex]s(t) = c1*e^(-bt/2)*cos(({4b-36)/2)t} 4b-36)/2)t) + c2e^(-bt/2)*sin\sqrt{(4b-36)/2)*t)} (4b-36)/2)*t)[/tex]
where c1 and c2 are constants determined by the initial conditions.
The behavior of the solutions to this equation depends on the value of the parameter b. Specifically, there are three cases to consider:
Overdamped: If b > 6, then the roots of the characteristic equation[tex]s^2 + bs + 9 = 0[/tex] are real and distinct, i.e., [tex]b^2 - 4ac[/tex] > 0. In this case, the general solution is a linear combination of two decaying exponentials, and the system is said to be overdamped. To find the interval for b for which the equation is overdamped, we solve the inequality b > 6, which gives the interval (6, infinity).
Critically damped: If b = 6, then the roots of the characteristic equation are real and equal, i.e., [tex]b^2 - 4ac[/tex]= 0. In this case, the general solution is a linear combination of two decaying exponentials, where one of the exponentials has an additional factor of t. The system is said to be critically damped. To find the interval for b for which the equation is critically damped, we solve the equation b = 6, which gives the singleton set {6}.
Underdamped: If b < 6, then the roots of the characteristic equation are complex conjugates, i.e., [tex]b^2 - 4ac[/tex] < 0. In this case, the general solution is a linear combination of two decaying exponentials, where the exponentials have a sinusoidal factor. The system is said to be underdamped. To find the interval for b for which the equation is underdamped, we solve the inequality b < 6, which gives the interval (-infinity, 6).
Therefore, the interval for b that makes the general solution overdamped is (6, infinity), the singleton set {6} makes the general solution critically damped, and the interval for b that makes the general solution underdamped is (-infinity, 6).
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add the following fraction give me the answer in lowest terms and mixed numbers if necessary. 10/12 +1/2 =
Answer:
[tex]1\frac{1}{3}[/tex]
Step-by-step explanation:
Honestly help I’m slow
Answer:
46 degrees
Step-by-step explanation:
x equals 23 so just multiply by 2
Answer:
[tex]46^{o}[/tex]
Step-by-step explanation:
Bottom-right angle is 65 as 180 - 90 - 25 = 65
Top-left angle is also 65 as angles opposite a point are equal (when divided by straight lines)
Angles around a point = 360 so 360 - 65 - 65 - 90 - 25 = 115
5x = 115
x = 23
23 x 2 = 46
x(x-4)=12 solve for x
Answer:
x=6 and x=-2
Step-by-step explanation:
so
x(x-4)=12
first distribute
then move the terms
and the u get
x=6 and x=-2
hope this helped
Answer:
x=6, x=-2
Step-by-step explanation:
x(x-4)=12
distributive property, x^2-4x=12
x^2-4x-12=0
(x-6)(x+2)
therefore, x=6, x=-2
Is the President doing a good job? We will examine this by taking a random sample of n = 4220 adults and asking whether they feel the president is doing a good job? Of these sampled adults, x = 2222 said the President was doing a good job. (Assume nobody lies.) Let p be the (unknown) true proportion of adults who feel the President is doing a good job. We want to estimate p. X is the random variable representing the number of sampled adults who say the president is doing a good job. have?
a) What type of probability distribution does X have?
O binomial
O gamma exponential
O Weibull
O Poisson
b) What was the sample proportion, P^, of sampled adults who say the President is doing a good job? _____
b) What is the R formula for the expected value of X in terms of n and p?
O sqrt(n*p*(1-p))
O n*p*(1 - p)
On^2
O n*p
O 1/p
d) What is the z critical value that we would use to construct a classical 90% confidence interval for p? _______
e) Construct a 90% classical confidence interval for p? (_____,_____)
f) How long is the 90% classical confidence interval for p? ______
g) If we are creating a 90% classical confidence interval for p based upon the sample size of 4220, then what is the longest possible length of this interval? _____
(a) Binomial probability distribution does X have. The option 1 is correct answer.
(b) 0.5265 is the sample proportion [tex]\hat{P}[/tex] of sampled adults who say the President is doing a good job.
(c) n * p is the R formula for the expected value of X in terms of n and p. The option 4 is correct answer.
(d) 1.645 is the z critical value that we would use to construct a classical 90% confidence interval for p.
(e) A 90% classical confidence interval for p is 0.5104, 0.5428.
(f) 0.0324 is the 90% classical confidence interval for p.
(g) 0.0357 is the longest possible length of this interval.
a) The random variable X, representing the number of sampled adults who say the President is doing a good job, follows a binomial probability distribution. Therefore, the correct answer is option 1.
b) The sample proportion, [tex]\hat{P}[/tex], of sampled adults who say the President is doing a good job can be calculated by dividing the number of adults who said the President was doing a good job (x = 2222) by the total sample size (n = 4220):
[tex]\hat{P}[/tex] = x / n
= 2222 / 4220
= 0.5265
c) The expected value of X is given by
n*p,
where n is the sample size and
p is the true proportion of adults who feel the President is doing a good job.
Therefore, the correct answer is option 4.
d) To construct a classical 90% confidence interval for p, we need to find the z critical value. This value can be found using a z-table or calculator and is approximately 1.645.
e) Using the sample proportion, [tex]\hat{P}[/tex], the z critical value, and the sample size, a 90% classical confidence interval for p can be calculated. This is done using the formula:
[tex]\hat{P} \pm z \times \sqrt{\frac{\hat{P} \times (1 - \hat{P})}{n}}[/tex]
The interval is (0.5104, 0.5428).
f) The length of the 90% classical confidence interval for p can be found by subtracting the lower limit from the upper limit: 0.5428 - 0.5104 = 0.0324.
g) The longest possible length of the 90% classical confidence interval for p can be found by using the formula:
[tex]2z \sqrt{\frac{\hat{P} ( 1 - \hat{P})}{n}[/tex]
Plugging in the values from the sample, we get
21.645 √(0.5266(1-0.5266)/4220)
= 0.0357.
This means that the interval can be at most 0.0357 in length.
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HOW MANY TABLESPOONS ARE IN 400 MILLIMETERS? 1 TSP = 5mL
Answer:
80 tsp.
Step-by-step explanation:
400 divided by 5 is 80, so 80 tsp's.
Answer:
80 tsp.
Step-by-step explanation:
400 mL = 80 tsp
25x+20y=200 in slope intercept form.
Answer:
5x+4y-40=0
Step-by-step explanation:
How much Interest(in dollars) is earned by Investing $2200 at a simple interest rate of 8% for 12 years? Write the correct answer.
A = $4,312.00
I = A - P = $2,112.00
hey!
Equation:
A = P(1 + rt)
Calculation:
First, converting R percent to r a decimal
r = R/100 = 8%/100 = 0.08 per year.
Solving our equation:
A = 2200(1 + (0.08 × 12)) = 4312
A = $4,312.00
The total amount accrued, principal plus interest, from simple interest on a principal of $2,200.00 at a rate of 8% per year for 12 years is $4,312.00.
-------------------------------------------------
have a good day! hope i helped in some way
A certain county is 25% African American. Suppose a researcher is looking at jury pools, each with 200 members, in this county. The null hypothesis is that the probability of an African American being selected into the jury pool is 25%. a. How many African Americans would the researcher expect on a jury pool of 200 people if the null hypothesis is true? b. Suppose pool A contains 17 African American people out of 200, and pool B contains 39 African American people out of 200. Which will have a smaller p-value and why?
Answer: a. 50 African Americans
b. Pool B will have a smaller p-value because that pool's number of AA people is further from the hypothesized number of AA people.
Step-by-step explanation:
There were 32 volunteers to donate blood. Unfortunately, n of the volunteers did not meet the health
requirements, so they couldn't donate. The rest of the volunteers donated 470 milliliters each.
How many milliliters of blood did the volunteers donate?
Write your answer as an expression.
12. from the slope of your best-fit line, what is the velocity of the pacific plate, as expressed in cm/yr? (2 significant figures required)
The velocity of the Pacific plate, expressed in centimeters per year (cm/yr), can be determined from the slope of the best-fit line in a geologic study.
In a geologic study, if data points representing the position of the Pacific plate are collected over a period of time, a best-fit line can be calculated to represent the trend of plate movement.
- The slope of this line represents the rate of change of position over time, which corresponds to the velocity of the plate. By examining the slope of the best-fit line and converting it to centimeters per year, we can determine the velocity at which the Pacific plate is moving.
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Which one would result an integer
Answer:
c is the only one that would result in an integer
Step-by-step explanation:
i hope this helps :)
Option c ∛ 27 would result in an integer.
what are integers?An integer is the number zero, a positive natural number or a negative integer with a minus sign. The negative numbers are the additive inverses of the corresponding positive numbers.
Given here, ∛27= 3 while the other options ∛60 is not an integer because 60 is not cubic number similarly for 9 , 18 are not cubic numbers and thus their subsequent cubic roots will not yield an integer.
Hence, Option c ∛ 27 would result in an integer.
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A UMass student is starting their junior year and has accumulated 60 credits so far. Their current cumulative average is a C, or a Grade Point Average (GPA) of 2.0. Their employer has a scholarship program for students who have GPAs of 2.3 or higher. This student wants to get that scholarship to help pay for their senior year. They plan on taking 15 credits each for the fall and spring semesters of their junior year.
a. Can they raise their cumulative average to 2.3 after completing 15 fall semester credits? What semester GPA would they need?
b. What average GPA would they need for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits?
According to the information, we can infer that no, they cannot raise their cumulative average to 2.3 after completing 15 fall semester credits. On the other hand, they would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.
How to calculate the new cumilative GPA?In order to calculate the new cumulative GPA, we need to consider both the current cumulative GPA and the GPA earned in the fall semester. Since the student's current cumulative GPA is 2.0 and they have already accumulated 60 credits, their total grade points earned so far would be 2.0 multiplied by 60, which equals 120 grade points.
To raise the cumulative GPA to 2.3, the student would need a total of 2.3 multiplied by (60 + 15) = 2.3 multiplied by 75 = 172.5 grade points by the end of the fall semester.
Since the student has already accumulated 120 grade points, they would need to earn an additional 52.5 grade points in the fall semester. To calculate the required semester GPA, we divide 52.5 by 15 credits, which gives us a required semester GPA of 3.5.
So, the student would need a semester GPA of 3.5 in order to raise their cumulative average to 2.3 after completing 15 fall semester credits.
What average gpa would they need for their two junior year semesters combined to achieve their goal?They would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.
Explanation: To calculate the average GPA for the two junior year semesters, we need to consider the total grade points earned and the total number of credits taken.
Currently, the student has accumulated 60 credits and 120 grade points. In order to achieve a cumulative GPA of 2.3 after completing 90 credits, they would need a total of 2.3 multiplied by 90 = 207 grade points.
To calculate the required grade points for the two junior year semesters, we subtract the current grade points (120) from the desired total grade points (207), which gives us 207 - 120 = 87 grade points needed for the junior year.
Since the student plans to take 30 credits during their junior year, they would need to earn 87 grade points in those 30 credits. Dividing 87 by 30 gives us an average GPA of approximately 2.9 for the two junior year semesters.
According to the above, the student would need an average GPA of 3.25 (rounded up) for their two junior year semesters combined to achieve their goal of a 2.3 cumulative GPA and 90 credits.
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Function 1 is represented by the equation y = -4/3x-2, and function 2 is represented by the
graph below.
FUNCTION 2
For which of the functions are all the output values less than -1?
A. Both functions
B. Only function 1
C. Only function 2
D. Neither functions
Four minutes is what percent of an hour?
Answer
6 and 2/3 percent of an hour OR 6.666.... hour
I don’t know if you have to round or not but if it does just round
Step-by-step explanation:
Well 4 minutes of an hour is basically 4/60
4/60=1/15
1/15=x/100
solve the proportion by cross multiplying
100=15x
x=6.66666666
That is yoru percent
HELP ME ASAP PLEASE!!!!! LOOK AT SCREENSHOT (10 PTS)
Answer: i would say A and D but i'm not sure of the others
Andrea constructed a triangle. Angle 1 and 3 are the same size and angle 2 has a measurement of 70 degrees. What is the measurement of angle 1 and 3
Answer:
Step-by-step explanation:
By triangle sum theorem,
Sum of all angles of a triangle is 180°.
m∠1 + m∠2 + m∠3 = 180°
(m∠1 + m∠3) + m∠2 = 180°
2(m∠1) + 70° = 180° {Given → m∠1 = m∠3]
2(m∠1) = 110°
m∠1 = 55°
Therefore, m∠1 = m∠3 = 55°
A square has a perimeter of 36 inches and a smaller square has a side length of 4 inches. What is the ratio of the areas of the larger square to the smaller square?
Answer:
3:2
Step-by-step explanation:
a square has the same side lengths so just √36 = 6 to find the sides of the squares then compare the two sides in ratio form 6:4 then reduce
Solve the following equation for x.
Answer:
If you just solve normally, you will get x=2 and x=-3. But if you plug these in to check your work, you will find that they are wrong. Your answer is no solution
Step-by-step explanation:
ln(2x+3)+ln(x-2)=ln(x^2-2x)
Rule: log(a) + log(b) = log(a*b)
ln( (2x+3)(x-2) ) = ln(x^2-2x)
Rule: If log(a) = log(b) then a = b
(2x+3)(x-2) = x^2 - 2x
2x^2-x-6=x^2-2x
x^2+x-6=0
Using Quadratic Formula:
x = 2 and x = -3
But, plugging these numbers back into the original equation is false!
math because im very bad at it
help please! need an answer asap................
Answer:
x ≈ 121.2 ft
Step-by-step explanation:
Using the tangent ratio in the right triangle
tan60° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{70}[/tex] ( multiply both sides by 70 )
70 × tan60° = x , then
x ≈ 121.2 ft ( to 1 dec. place )
A random sample of nı = 19 securities in Economy A produced mean returns of X 1 = 6.6% with sı = 2.3% while another random sample of n2 = 22 securities in Economy B produced mean returns of # 2 = 5% with s2 = 7.7%. Construct a 98% confidence interval estimate for pl H2 Assume that the samples are independent and randomly selected from normal populations with unequal population variances (012 + 022). T-Distribution Table % % < (H 1 - 2) < Round to two decimal places if necessary
The 98% confidence interval for the distribution of differences is given as follows:
(-2.58%, 5.78%).
How to obtain the confidence interval?The difference of the sample means is given as follows:
6.6 - 5 = 1.6%.
The standard error for each sample is given as follows:
[tex]s_1 = \frac{2.3}{\sqrt{19}} = 0.53[/tex] [tex]s_2 = \frac{7.7}{\sqrt{22}} = 1.64[/tex]The standard error for the distribution of differences is then given as follows:
[tex]s = \sqrt{0.53^2 + 1.64^2}[/tex]
s = 1.72.
The critical value, using a t-distribution calculator, for a two-tailed 98% confidence interval, with 19 + 22 - 2 = 39 df, is t = 2.4286.
The lower bound of the interval is given as follows:
1.6 - 2.4286 x 1.72 = -2.58%.
The upper bound of the interval is given as follows:
1.6 + 2.4286 x 1.72 = 5.78%.
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could any body help me with this
Answer:
1) 1.2 minutes
2) 8.3 laps
3) 25.2 minutes
Which of these shapes is not a parallelogram? HELP ASAP 15 BRAINLY POINTS! TYSM GOD BLESS YOU AND YOUR FAMILY!
Answer:
The answer would be D____________________________________________________________
Why?
Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid can never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals.
____________________________________________________________
What's a parallelogram?
Its a four-sided plane rectilinear figure with opposite sides parallel.
____________________________________________________________
Please don't be afraid to point out errors :)
____________________________________________________________
Cute copy and pastteeeee ☏ ♡ ☆⋆◦★◦⋆°*•°
. * . . ° . ● ° .
¸ . ★ ° :. . • ° . * :. ☆
° :. ° .☆ . ● .° °★
★ ★°★ . * . °☆ . ● . ★ ° . • ○ ● . ☆ ★ ° ☆ ¸. ¸ ★ . • ° . *
¸ . ★ ° :. :. . ¸ . ● ¸ ° ¸. * ● ¸ °☆
☆ °☆ . * ● ¸ . ★¸ .
. * . . ° . ● ° .
° :. ° . ☆ . . • . ● .° °★ Not sure how to copy and paste? Just right click your mouse and choose copy in options, to release repeat the process and just paste it. No mouse? Select the text with your computer pad and use ctrl c to release, ctrl v. On mobile? Press on your screen and select the text, use the option copy and paste wherever you would like!
The correct shape which is not a parallelogram is shown in Option D.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
There are four shapes are shown.
Now, We know that;
In a parallelogram, there are two pair of parallel lines.
But In option D;
There are only one pair of parallel lines.
Hence, The correct shape which is not a parallelogram is shown in Option D.
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a lamina occupies the part of the disk 2 2≤16 in the first quadrant and the density at each point is given by the function (,)=3(2 2).
A lamina occupies the region of a disk in the first quadrant where 2 ≤ r ≤ 16, and the density at each point is given by the function ρ(r, θ) = 3[tex](r^2).[/tex] Further analysis is required to determine the mass and other properties of the lamina.
The given information describes a lamina occupying a region in the first quadrant of a disk. The radial distance from the origin is limited to the range 2 ≤ r ≤ 16. The density of the lamina at any point within this region is determined by the function ρ(r, θ) = 3[tex](r^2)[/tex], where r represents the radial distance and θ represents the angle in the polar coordinate system.
To fully analyze the lamina, additional calculations are necessary. One important calculation is determining the mass of the lamina, which involves integrating the density function over the given region. By integrating the function ρ(r, θ) = 3[tex](r^2)[/tex] over the appropriate range of r and θ, we can find the total mass of the lamina. Additionally, other properties such as the center of mass or moment of inertia of the lamina could be determined by using appropriate formulas and integration techniques.
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HELP PLEASE‼️
There are 148 legs in a farm yard full of goats and chickens. There are 62 total animals. How many of each are there.
It is 12 goats and 50 chickens to make those animals and that exact number of legs
Solve the system of differential equations S x1 = – 5x1 + 0x2 – 16x1 + 322 X2' x1(0) = 1, X2(0) = 5 21(t) = = 22(t) - = X2
The solution to the system of differential equations is x₁(t) = e⁻⁵ˣ + 3e³ˣ and x₂(t) = 2e⁻⁵ˣ + 5e³ˣ
Let's solve the given system of differential equations: x₁' = -5x₁ + 0x₂ ...(1) x₂' = -16x₁ + 3x₂ ...(2)
To solve this system, we can rewrite it in matrix form. Let's define the vector X = [x₁, x₂] and the matrix A as:
A = [[-5, 0], [-16, 3]]
The system can then be written as X' = AX, where X' is the derivative of X with respect to time.
Now, let's find the eigenvalues and eigenvectors of matrix A. The eigenvalues are obtained by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix.
A - λI = [[-5 - λ, 0], [-16, 3 - λ]]
det(A - λI) = (-5 - λ)(3 - λ) - 0(-16) = λ² + 2λ - 15 = (λ + 5)(λ - 3)
Setting the characteristic equation equal to zero, we find the eigenvalues: λ₁ = -5 λ₂ = 3
To find the corresponding eigenvectors, we substitute each eigenvalue back into the matrix A - λI and solve the system of equations (A - λI)v = 0, where v is the eigenvector.
For λ₁ = -5: A - (-5)I = [[0, 0], [-16, 8]]
Using Gaussian elimination, we can solve the system of equations to find the eigenvector corresponding to λ₁: -16v₁ + 8v₂ = 0 => -2v₁ + v₂ = 0 => v₁ = (1/2)v₂
Let v₂ = 2, then v₁ = 1. Therefore, the eigenvector corresponding to λ₁ is v₁ = [1, 2].
For λ₂ = 3: A - 3I = [[-8, 0], [-16, 0]]
Solving the system of equations, we find: -8v₁ = 0 => v₁ = 0
Thus, the eigenvector corresponding to λ₂ is v₂ = [0, 1].
Now, let's express the solution of the system in terms of the eigenvalues and eigenvectors.
X(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂
Substituting the eigenvalues and eigenvectors we found earlier, we have: X(t) = c₁e⁻⁵ˣ[1, 2] + c₂e³ˣ[0, 1]
Using the initial conditions, x₁(0) = 1 and x₂(0) = 5, we can find the values of c₁ and c₂.
At t = 0: [1, 5] = c₁[1, 2] + c₂[0, 1] 1 = c₁ 5 = 2c₁ + c₂
Solving these equations, we find: c₁ = 1 c₂ = 3
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Complete Question:
Solve the system of differential equations
x₁' = – 5x₁ + 0x₂
x₂' = – 16x₁ + 3x₂
x₁(0) = 1, x₂(0) = 5
Gina and Rhonda work for different real estate agencies. Gina earns a monthly salary of $5,000 plus a 6% commission on her sales. Rhonda earns a monthly salary of $6,500 plus a 4% commission on her sales. How much must each sell to earn the same amount in a month?
Answer: Gina and Rhonda work for different real estate agencies. Gina earns a monthly salary of $5,000 plus a 6% commission on her sales. Rhonda earns a monthly salary of $6,500 plus a 4% commission on her sales. How much must each sell to earn the same amount in a month?
Step-by-step explanation:
$5,000 + 6% + $6,500 + 4% = 11500.1 or 11500