Given a prime number p and a polynomial f(x, y) of degree 2 with the only zero in Z/pZ being 0, we can show that f(ka, kb) = k^2f(a, b) using properties of polynomials.
i) To show that [tex]f(ka, kb) = k^2f(a, b)[/tex], we consider the polynomial f(x, y) and apply the properties of polynomials. Since f(x, y) has a degree of 2, we can write it as [tex]f(x, y) = ax^2 + bxy + cy^2,[/tex] where a, b, and c are coefficients. Now, substituting ka for x and kb for y, we get f(ka, kb) = [tex]a(ka)^2 + b(ka)(kb) + c(kb)^2[/tex]. Simplifying this expression, we obtain f(ka, kb) = [tex]k^2(ax^2 + bxy + cy^2) = k^2f(a, b),[/tex] which demonstrates the desired result.
ii) Using the result from part i), we can prove that if a is not congruent to 0 modulo p, then the equation f(x, y) ≡ a (mod p) always has a solution. Suppose f(x, y) ≡ a (mod p) has no solution for some value of a not congruent to 0 modulo p. Therefore, if a is not congruent to 0 modulo p, we can choose an appropriate value of k such that [tex]k^2f(a, b)[/tex] ≡ b (mod p), leading to a solution for f(x, y) ≡ b (mod p). Thus, if a is not congruent to 0 modulo p, f(x, y) ≡ a (mod p) always has a solution.
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To rent a certain meeting room, a college charges a reservation fee of $18 and an additional fee $5 of per hour. The chemistry club wants to spend at most %58 on renting the room. What are the possible numbers of hours the chemistry club could rent the meeting room? Use t for the number of hours. Write your answer as an inequality solved for t.
Answer:
You need the minimum cost for renting the meeting room given the following info
Reservation Fee = $18
Hourly Rate = $5
The renter wants to keep the cost below $58
So the cost will vary with the time, t you use the inequality below
5t + 18 ≤ 58
You can solve this to get that minimum
You can check you answer by graphing it.
I hope this helps!<3
question 1 (2 points) triangle abc is similar to triangle wyz. select all angles whose cosine equals . question 1 options: angle b angle z angle y angle c angle a angle w
In triangle ABC, which is similar to triangle WYZ, the angles whose cosine equals are angle B and angle Z.
To determine the angles whose cosine equals, we need to consider the corresponding angles in similar triangles. In similar triangles, corresponding angles have the same measure, but the side lengths may be different. Since triangle ABC is similar to triangle WYZ, angle B in triangle ABC corresponds to angle Z in triangle WYZ. Similarly, angle Z in triangle ABC corresponds to angle B in triangle WYZ. Therefore, angle B and angle Z are the angles in the respective triangles whose cosine equals.
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What is the equation of the line that passes through the point (-6, -8) and has a slope of 1/3
Answer:
x^2 = (28)^2 + (45)^2
= 784 + 2025
= 2809
x = 53
The equation of the line is y = (1/3)x - 6.
What is an equation of a line?The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
Example:
The slope of the line y = 2x + 3 is 2.
The slope of a line that passes through (1, 2) and (2, 3) is 1.
We have,
The equation of a line is y = mx + c.
Now,
m = 1/3
And,
(-6, -8) = (x, y)
So,
y = mx + c
-8 = 1/3 x -6 + c
-8 = -2 + c
-8 + 2 = c
c = -6
Now,
y = mx + c
y = (1/3)x - 6
Thus,
The equation of the line is y = (1/3)x - 6.
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A pencil holder in the shape of a cylinder has the dimensions shown in the diagram 20. Which of the following is the closest to the volume of the cylinder, in cubio H 907 cm F 209 cm G 3,627 cm 4,773 cm
Answer:
my best guess would be H
Step-by-step explanation:
I hope this helped and if you have more details that could help me get to a more appropriate answer pls comment them and I'll do my best to help but like I said with what you provided my best guess would be H
According to the latest real estate report from a suburban town, the mean number of home purchases during last quarter was 175 homes with a population standard deviation of 6 homes. A real estate agent believes that the recent increase of mortgage interest rates is causing a decline in home purchases. To test this, the real estate agent randomly selects 60 recent home purchases. What is the probability that the mean of this sample of home purchases is between 173 and 174 homes
Answer:
The probability that the mean of this sample of home purchases is between 173 and 174 homes
P(173≤X≤174) = 0.0936
Step-by-step explanation:
Step(i):-
Given that the mean of the Population = 175
Given that the standard deviation of the Population = 6
Let 'x⁻' be the mean of the random sample
Given that x₁⁻ = 173
[tex]Z_{1} = \frac{x^{-}-mean }{\frac{S.D}{\sqrt{n} } } = \frac{173-175}{\frac{6}{\sqrt{60} } } =-2.58[/tex]
Given that x₂⁻ = 174
[tex]Z_{2} = \frac{x^{-}-mean }{\frac{S.D}{\sqrt{n} } } = \frac{174-175}{\frac{6}{\sqrt{60} } } = -1.29[/tex]
Step(ii):-
The probability that the mean of this sample of home purchases is between 173 and 174 homes
P(X₁≤X≤X₂) = P(Z₁≤Z≤Z₂)
= P(Z≤Z₂) - P(Z≤Z₂) ( both values 'Z' values are negative)
= 0.5 -A(Z₁) - (0.5 -A(Z₂))
= |A(Z₂) -A(Z₁)|
P(173≤X≤174) = | A(2.58)-A(1.29)|
= 0.4951 - 0.4015 (∵ from normal table)
= 0.0936
Final answer:-
The probability that the mean of this sample of home purchases is between 173 and 174 homes
P(173≤X≤174) = 0.0936
Need your help answer correctly for 20 points! <3
12.50 dollars per hour
Answer:
$12.50
Step-by-step explanation:
Hi,
Cassie: 250 / 20 = 12.5
Marli: 312.50 / 25 = 12.5
Brad: 350 / 28 = 12.5
Each worker makes $12.50 per hour.
I hope this helps :)
Show that the polynomial p(x) = x² +1 € Z3 [x] is irreducible. Let a be a zero of this polynomial and consider the extension Z3(a) = {0, 1, 2, a, 1+a, 2+a, 2a, 1+ 2a, 2 + 2a} = Z3 [x]/(p(x)) Write out the addition and multiplication tables for this field. What is the multiplicative inverse of 2a + 2?
The multiplicative inverse of 2a + 2 is 2a + 1.
To show that the polynomial p(x) = x² + 1 ∈ Z3[x] is irreducible, we can examine its factors. Since Z3 is a field with three elements {0, 1, 2}, we can substitute each value into p(x) to check for roots. However, none of the elements satisfy p(x) = 0. Therefore, p(x) has no linear factors.
Next, we consider the extension Z3(a) = {0, 1, 2, a, 1+a, 2+a, 2a, 1+2a, 2+2a} = Z3[x]/(p(x)), where a is a zero of p(x). In this field, we can perform addition and multiplication modulo 3.
Addition table:
+ | 0 1 2 a 1+a 2+a 2a 1+2a 2+2a
-----------------------------------------------------
0 | 0 1 2 a 1+a 2+a 2a 1+2a 2+2a
1 | 1 2 a 1+a 2+a 2a 1+2a 2+2a 0
2 | 2 a 1+a 2+a 2a 1+2a 2+2a 0 1
a | a 1+a 2+a 2a 1+2a 2+2a 0 1 2
1+a | 1+a 2+a 2a 1+2a 2+2a 0 1 2 a
2+a | 2+a 2a 1+2a 2+2a 0 1 2 a 1+a
2a | 2a 1+2a 2+2a 0 1 2 a 1+a 2+a
1+2a | 1+2a 2+2a 0 1 2 a 1+a 2+a 2a
2+2a | 2+2a 0 1 2 a 1+a 2+a 2a 1+2a
Multiplication table:
* | 0 1 2 a 1+a 2+a 2a 1+2a 2+2a
-----------------------------------------------------
0 | 0 0 0 0 0 0 0 0 0
1 | 0 1 2 a 1+a 2+a 2a 1+2a 2+2a
2 | 0 2 1 2a 1+2a 2+2a a 2+a 1+a
a | 0 a 2a 1+2a 2+a 1+a 2+2a 2 1
1+a | 0 1+a 1+2a 2+a 2 a 1 2+2a 2a
2+a | 0 2+a 2+2a 1+a a 2 1+2a 1 2a
2a | 0 2a a 2+2a 1 1+2a 2 2+a 1+a
1+2a | 0 1+2a 2+a 2 2+2a 1 2a a 1+a
2+2a | 0 2+2a 1+a 1 2a 2 1+a 1+2a a
To find the multiplicative inverse of 2a + 2, we search for an element in the multiplication table that, when multiplied by 2a + 2, results in 1. In this case, the multiplicative inverse is 2a + 1.
Note: The addition and multiplication tables are constructed by performing arithmetic modulo 3 in the field Z3(a) using the given elements.
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Please help me with the questions Please please please ASAP ASAP please ASAP help please please ASAP please please help ASAP ASAP please please help please please ASAP please please help ASAP ASAP please please help please please ASAP please
Answer:
9
Step-by-step explanation:
8/12=6/x
cross multiply to get 9
A school administrator believes that the mean class GPAs for a given course are higher than the preferred mean of 2.75 at a significance level of 0.05, and checks a randomly chosen sample of 50 classes. Use Sheet 3 of the Excel file to create a histogram, and calculate x¯, the t-statistic, and the p-value.
From the histogram, can normality be assumed?
x¯=
t=
p=
Since the p-value is
evidence exists that the mean GPA is greater than 2.75.
than the significance level 0.05, the null hypothesis
.
Pick Yes No
Pick greater less
Pick fails to be rejected is rejected
3.09
2.73
2.58
2.68
2.84
2.68
2.64
2.62
2.35
2.72
2.63
2.77
2.72
3.03
2.71
2.96
2.74
3.01
2.88
2.68
2.83
3.02
2.83
2.71
2.67
2.79
2.65
2.56
2.78
2.89
2.40
2.38
2.41
2.83
2.86
3.01
2.80
2.84
2.95
2.75
2.93
2.48
2.34
2.97
2.85
2.74
2.84
3.10
2.62
2.77
The mean GPA is greater than 2.75.
Here, we have,
A= 4.0, A- = 3.7, B+ = 3.
three, and many others.
Total the first-class factors, and overall the credit hours. Divide the overall satisfactory points by means of the overall credit score hours.
Grade Point Average (GPA):
Grade point average (GPA) is the measure used to summarize your instructional fulfillment at Griffith. After the e-book of final grades every trimester, your software and career.
GPA is calculated and could be utilized by the college to tell choices, which include the subsequent: educational progression.
Divide the total range of grade factors earned with the aid of the total number of letter-graded units undertaken.
Mean GPA = total / 50
= 2.75.
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"
Suppose X is normally distributed with a mean of u = 11.5 and a standard deviation of o = 2. Find the probability of X > 15.14. Show your work.
"
The probability of X > 15.14 is approximately 0.0344 or 3.44%.
To find the probability of X > 15.14, we need to calculate the area under the normal distribution curve to the right of 15.14.
First, we need to standardize the value 15.14 using the formula:
Z = (X - [tex]\mu[/tex]) / o
where Z is the standardized value, X is the given value, [tex]\mu[/tex] is the mean, and o is the standard deviation.
In this case:
Z = (15.14 - 11.5) / 2
= 3.64 / 2
= 1.82
Next, we need to find the area under the standard normal distribution curve to the right of Z = 1.82. This can be done using a standard normal distribution table or a calculator.
Using a standard normal distribution table, we find that the area to the right of Z = 1.82 is approximately 0.0344.
Therefore, the probability of X > 15.14 is approximately 0.0344 or 3.44%.
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you have a ramp that is 2.5 meters long. its height is 30 centimeters. what is the angle of your ramp?
To determine the angle of the ramp, we can use trigonometry and calculate the inverse tangent (arctan) of the ratio of the height to the length of the ramp.
Given a ramp length of 2.5 meters and a height of 30 centimeters (0.3 meters), the angle of the ramp is approximately 6.87 degrees. The angle of the ramp can be found using the formula: angle = arctan(height/length). By substituting the values, we get angle = arctan(0.3/2.5).
Using a calculator or math software to evaluate the arctan of this ratio, we find that the angle is approximately 6.87 degrees. This means that the ramp has a relatively gentle incline, indicating a low slope or gradient. Knowing the angle of the ramp can be useful in various applications, such as construction, accessibility planning, or determining the safety of the slope for specific purposes.
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Can somebody please help a brother out?:))
Plz help me
The points (6, 10) and (5, 7) fall on a particular line. What is its equation in slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
NO LINKS
Answer:
Finding the equation using the two points is: y=3x-8
The slope is: m=3
The slope and y-intercept is: 3, (0,-8)
Step-by-step explanation:
I put all of these answers because i was basically fully confused by the wording of this question.. Sorry.
WILL GIVE BRAINLIST FOR THE RIGHT ANSWER
plz explain how he got it wrong
Answer: A. Bruno accidentally used the diameter instead of the radius to find the area of the base. B. The correct volume is 602.88
Step-by-step explanation: Area of a cylinder is (r^2)(pi)(h) not (d^2)(pi)(h)
Solve the equation -3x^2+2x+4= -x-3 by writing a linear-quadratic system and solving using the intersection feature of a graphing calculator. Round to the nearest hundredth
A. x= -2.44 and x=3.12
B. x=-1.63 and x=4.44
C. x=-1.11 and x=2.11
D. x=-2.61 and x=0.42
The solution to the equation -3x^2 + 2x + 4 = -x - 3 is x = -2.44 and x = 3.12. Therefore, the correct option is A.
To solve the equation -3x^2 + 2x + 4 = -x - 3, we can rewrite it as a quadratic equation by combining like terms: -3x^2 + 3x + 7 = 0. This equation represents a quadratic function in the form of ax^2 + bx + c = 0, where a = -3, b = 3, and c = 7.
Using a graphing calculator, we can plot the function and find the x-intercepts, which represent the solutions to the equation. The intersection feature of the graphing calculator can help determine the coordinates of the points where the graph intersects the x-axis. Rounding to the nearest hundredth, we find that the solutions to the equation are x = -2.44 and x = 3.12.
Therefore, the correct answer is option A: x = -2.44 and x = 3.12.
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Plz help!! Dont answer if you cant help
Answer:
94.88 cubic units
Step-by-step explanation:
Volume of the prism = Area of base * Height
[tex] = 5( \frac{1}{2} \times 2.75 \times 4) \times 3.45 \\ \\ = 5 \times 2.75 \times 2 \times 3.45 \\ \\ = 10\times 2.75 \times 3.45 \\ \\ = 94.875 \\ \\ \approx 94.88 \: cubic \: units[/tex]
Sally had some marbles. After buying 12 more marbles, she had 25 marbles. How many marbles did Sally have before she bought more?
Answer:
13
Step-by-step explanation:
25 - 12 = 13
Answer:
13
Step-by-step explanation:
12 + 13 = 25 or 25 -12 = 13
sketch a graph of f(t), express the function f(t) in terms of the heaviside function u(t −a), and find the laplace transform l{f(t)}.
(a) f(t) = {sint 0 ≤ t < π {0 π ≤ t
(b) f(t) = {0 0 ≤ t <3
{t² 3 ≤ t
(a) To sketch the graph of f(t) = {sin(t) 0 ≤ t < π, 0 π ≤ t}, we can observe that the function is equal to zero for t greater than or equal to π. Thus, the graph of f(t) consists of a sine wave from 0 to π, and then it becomes zero from π onwards.
(b) To express f(t) in terms of the Heaviside function u(t - a), we can write f(t) as {0 0 ≤ t < 3, t² 3 ≤ t}. This means that for t less than 3, the function is zero, and for t greater than or equal to 3, the function is t squared. We can rewrite this using the Heaviside function as f(t) = t²u(t - 3).
To find the Laplace transform L{f(t)}, we can apply the Laplace transform to each part of the function separately. Using the Laplace transform properties, we have L{f(t)} = L{t²u(t - 3)}. By applying the time-shifting property, the Laplace transform of t²u(t - 3) is given by L{t²u(t - 3)} = e^(3s)/s^3, where s represents the Laplace variable.
The graph of f(t) consists of a sine wave from 0 to π, and then it becomes zero from π onwards. The function f(t) can be expressed as f(t) = t²u(t - 3) using the Heaviside function. The Laplace transform of f(t) is L{f(t)} = e^(3s)/s^3.
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Let A = {1,3,5,7). B = {5, 6, 7, 8). C = {5, 8} D = (2,5,8), and U = {1,2,3,4,5,6,7,8). Determine whether the expression shown below is true or false. If it is false, then give the reaso DCB . O A. False; the sets must have the same number of elements. B. False; all elements in D are not in B O C. True OD. False; all elements in D are in B O E. None of the above
The statement is False.
Let A = {1,3,5,7). B = {5, 6, 7, 8). C = {5, 8} D = (2,5,8), and U = {1,2,3,4,5,6,7,8).
To determine whether the expression DCB is true or false, we need to know the content of these sets.
To determine the content of DCB:
DCB contains all elements of D, all elements of C, and all elements of B except those that are already in D and C.
D = {2,5,8} C = {5, 8} B = {5,6,7,8}
DCB = {2,5,8,6,7}
Thus, DCB is false because not all elements in D are in B, and all elements in D are not in B.
Therefore, the answer is option B.False; all elements in D are not in B.
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The following chart shows a store?s records of sales of stuffed toys for two months. 2 circle graphs. A circle graph titled September. Teddy bears is 346, rabbits is 297, cats is 199, horses is 245, other is 225. A circle graph titled October. Teddy bears is 308, rabbits is 260, cats is 285, horses is 186, 275 is other. Which of the following are accurate assessments of trends displayed in this graph? I. Sales of teddy bears decreased by about 2.9% between September and October. II. The change in sales of rabbits is roughly equal to the change in sales of all toys. III. Roughly 24.1% fewer horses were sold in October than in September. a. I and II b. II only c. I and III d. III only
Answer:
d
Step-by-step explanation:
..........
Answer:
d
Step-by-step explanation:
e d g e
jada walks at a speed of 3mph. elena walks at a speed of 2.8 mph. if they both begin walkign along a walking trail at the same time, how much father will jada walk adter 3 hours
Given, Jada walks at a speed of 3 mph and Elena walks at a speed of 2.8 mph.Both Jada and Elena start walking along a walking trail at the same time.
Let us determine the distance covered by both of them.Distance travelled by Jada in 3 hours is:Distance = Speed × Time= 3 mph × 3 hours= 9 miles Distance travelled by Elena in 3 hours is:Distance = Speed × Time= 2.8 mph × 3 hours= 8.4 miles Thus, Jada will walk 0.6 miles farther than Elena after 3 hours.
To determine how far Jada will walk after 3 hours, we need to calculate the distance traveled based on her speed.
Jada walks at a speed of 3 miles per hour (mph), so we can calculate her distance using the formula:
Distance = Speed × Time
Plugging in the values, we have:
Distance = 3 mph × 3 hours
Distance = 9 miles
Therefore, Jada will walk a distance of 9 miles after 3 hours.
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The given information is as follows:
Jada walks at a speed of 3 mph and Elena walks at a speed of 2.8 mph. If they both begin walking along a walking trail at the same time.
Therefore, Jada will walk 9 miles after 3 hours.
The distance covered by Jada after 3 hours can be calculated as follows:
Distance = Speed x Time
Since Jada walks at a speed of 3 mph, the distance covered by her in 3 hours can be calculated as:
Distance covered by Jada = 3 mph x 3 hours
= 9 miles.
Therefore, Jada will walk 9 miles after 3 hours.
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One boss decided to increase the salary of an employee by 5%. How much will he get if his salary was $2000?
Answer:
$2100 will be his new salary
Step-by-step explanation:
Well since a 5 percent raise is basically 105% of his pay, we can multiply 105% with 2000
since its a percent in decimal form its 1.05
1.05*2000=2100
$2100 is his new salary
I need to find the diameter of a circle and the line in the middle is 3/5 cms
Answer:
1.2
Step-by-step explanation:
There are 2 lines in the middle of the circle. Im guessing by you saying "The line in the middle" You must mean the radius your trying to find the diameter so, the radius is half of the diameter so to find the diameter you do the radius x 2 in this case 3/5x2 equals 1.2 :)
I Hope this helps!!
Need some help with Algebra!
Today's my last day to get missing work in and I currently have an F in Math and I'm just trying to get it up so I don't fail. Help me out? Heres the picture:
The system represented by the graph is y ≤ 1/2x + 3 and y < -2x + 8
The point (4, 0) is not a solution to the system
Constructing the system represented by the graphFrom the question, we have the following parameters that can be used in our computation:
The graph
On the graph there are two inequalities and they are calculated as follows:
A linear equation is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 3
y = mx + 8
Using the points on the graphs, we have
y = mx + 3 ⇒ -6m + 3 = 0 ⇒ m = 1/2
y = mx + 8 ⇒ 4m + 8 = 0 ⇒ m = -2
So, we have
y = 1/2x + 3
y = -2x + 8
Represent as inequalities
y ≤ 1/2x + 3
y < -2x + 8
Is (4, 0) a solution to the systemFrom the graph the point (4, 0) is out of the shaded region
This means that the point (4, 0) is not a solution to the system
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Solve 2x2 + x − 4 = 0
Answer:
x = 0
Step-by-step explanation:
hi there!
im not sure if the x between the 2's is a x as in a variable or if it is multiplication sign. im going to solve the problem as if it is a multiplication sign but if that is wrong then please let me know!
2 * 2 + x - 4 = 0
is our equation that we are working with ^
first step is, following the pemdas we multiply *2 with its self
( ^ parentheses, exponents, multiplication/ division making sure to solve it left to right, addition and subtraction still making sure to follow the left to right rule )
2 * 2 + x - 4 = 0
2 * 2
= 4 + x - 4 + 0
then we add positive 4 and negative 4 together, canceling eachother out the giving us the answer of x = 0
4 + x - 4 + 0
4 + - 4 = 0
x = 0
i hope this helps you! have a good rest of your day :)
The ratio of cement to gravel in a road
surface is 3:17. Find the percent
of cement in the mixture.
The owner of a tree farm plants pine trees and oak trees in a ratio of 8:3. How many oak trees are planted if 264 pine trees are planted
Answer:
99 oak trees
Step-by-step explanation:
Given data
Ratio= 8:3
Number of pine tree= 264
Let the number of oak trees be x
Let us apply the part to part method
8/264= 3/x
cross multiply
264*3= 8x
792=8x
x= 792/8
x=99 oak trees
The owner of the tree farm planted 99 oak trees.
An equation is an expression used to show the relationship between two or more numbers or variables.
Given that a farmer plants pine trees and oak trees in a ratio of 8:3. If 264 pine trees are planted, let y represent the total number of trees hence:
264 = (8/11) * y
y = 363 trees
Number of oak trees = (3/11) * 363 = 99 trees
Therefore the owner of the tree farm planted 99 oak trees.
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Find the x-intercept(s) Round to nearest hundredth if needed y=-2x^2-8x-4
y=-2x^2-8x-4
-2x^2-8x=4
divide by negative 2
x^2+4x=-2
complete the square
(x+2)^2=-2+4
(x+2)^2=2
x+2=±[tex]\sqrt{2}[/tex]
x=-2±[tex]\sqrt{2}[/tex]
nearest hundreths,
x=-0.59
x=-3.41
Let c(t) be a solution to the system of differential equations: xz(t) *) z'() -52x:(t) + 22x2(t) - 110 x1(t) + 4722(t) -3 Ifr(0) [:) ] find a(t). -3 Put the eigenvalues in ascending order when you enter 31(t), xa(t) below. r(t) = exp( t)+ exp( t) 22(t) exp( t)+ exp( t)
The solution to the system of differential equations with the given initial condition is:
x₁(t) = (-6/17) × exp(-t) + (44/17) × exp(2t)
x₂(t) = (-15/17) × exp(-t) + (108/17) × exp(2t)
The system of differential equations
x₁'(t) = -52x₁(t) + 22x₂(t)
x₂'(t) = -110x₁(t) + 47x₂(t)
Let's find the solution X(t) = [x₁(t), x(t)] with the initial condition x₀ = [-3, -3].
To solve the system, we'll start by finding the eigenvalues and eigenvectors of the coefficient matrix.
The coefficient matrix of the system is
A = [[-52, 22], [-110, 47]]
To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix
| -52 - λ 22 |
| -110 47 - λ | = 0
Expanding the determinant, we have
(-52 - λ)(47 - λ) - (-110)(22) = 0
Simplifying, we get
(λ + 1)(λ - 2) = 0
Solving this quadratic equation, we find two eigenvalues
λ₁ = -1
λ₂ = 2
Now let's find the corresponding eigenvectors for each eigenvalue.
For λ₁ = -1, we solve the equation (A - λ₁I)v = 0
| -51 22 | | v₁ | | 0 |
| -110 48 | | v₂ | = | 0 |
Simplifying, we get the equation
-51v₁ + 22v₂ = 0
-110v₁ + 48v₂ = 0
Solving this system of equations, we find the eigenvector v₁ = [2, 5].
For λ₂ = 2, we solve the equation (A - λ₂I)v = 0
| -54 22 | | v₁ | | 0 |
| -110 45 | | v₂ | = | 0 |
Simplifying, we get the equation
-54v₁ + 22v₂ = 0
-110v₁ + 45v₂ = 0
Solving this system of equations, we find the eigenvector v₂ = [11, 27].
Therefore, the eigenvalues in ascending order are
λ₁ = -1
λ₂ = 2
The corresponding eigenvectors are
v₁ = [2, 5]
v₂ = [11, 27]
To find the solution X(t), we can write it as a linear combination of the eigenvectors:
X(t) = c₁ × v₁ × exp(λ₁ × t) + c₂ × v₂ × exp(λ₂ × t)
Substituting the given values for x₁(t) and x₂(t) into the equation, we can find the coefficients c₁ and c₂:
x₁(t) = c₁ × 2 × exp(-t) + c₂ × 11 × exp(2t)
x₂(t) = c₁ × 5 × exp(-t) + c₂ × 27 × exp(2t)
Using the initial condition x₀ = [-3, -3], we can solve for c₁ and c₂
-3 = c₁ × 2 × exp(0) + c₂ × 11 × exp(0)
-3 = c₁ × 5 × exp(0) + c₂ × 27 × exp(0)
Simplifying, we get:
-3 = 2c₁ + 11c₂
-3 = 5c₁ + 27c₂
Solving this system of equations, we find
c₁ = -3/17
c₂ = 4/17
Substituting these values back into the solution equation, we have
x₁(t) = (-3/17) × 2 × exp(-t) + (4/17) × 11 × exp(2t)
x₂(t) = (-3/17) × 5 × exp(-t) + (4/17) × 27 × exp(2t)
Therefore, the solution to the system of differential equations with the given initial condition is:
x₁(t) = (-6/17) × exp(-t) + (44/17) × exp(2t)
x₂(t) = (-15/17) × exp(-t) + (108/17) × exp(2t)
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The question is incomplete the complete question is :
Kwans beginning account balance was $20. his ending balance is $0. what integer represents the change in his account balance from beginning to end?