Answer:
U=8
Step-by-step explanation:
48<5U+8
-5U<8-48
-5U<-40
-5U/-5<-40/+5
U>8
Two percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 95% detection rate for carriers and a 3% detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual. A. What is the probability that both tests yield the same result?
The probability that both tests yield the same result is 7.7%.
Simply put, probability is the likelihood that something will occur. When we don't know how an occurrence will turn out, we can discuss the likelihood or likelihood of various outcomes. Statistics is the study of occurrences that follow a probability distribution.
It is predicated on the likelihood that something will occur. The justification for probability serves as the primary foundation for theoretical probability. For instance, the theoretical chance of receiving a head when tossing a coin is 12.
Let's break it down:-
90% don't have of those 99%
5% will be positive
1% positive of those 1%
90% positive
10% negative.
Well we need it to be the same, so 99*(.05*.05+.95*.95)+.01*(.9*.9+.1*.1)= 90.4%.
If both tests are positive, we have:-
0.99*0.05*0.05 and 0.01*0.9*0.9 for being positive, so :-
[tex]\frac{carrier}{positive} = \frac{0.01*0.9*0.9}{(0.99*0.05*0.05+0.01*0.9*0.9)} = 7.7[/tex]
hence, the probability of the two tests yield the same result is 7.7%.
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for each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to 0. explain your choice. (a) weight of a car and gas mileage there is a positive correlation, because heavier cars tend to get lower gas mileage. there is a positive correlation, because heavier cars tend to get higher gas mileage. there is a negative correlation, because heavier cars tend to get lower gas mileage. there is a negative correlation, because heavier cars tend to get higher gas mileage. there is a correlation close to 0, because there is no reason to believe that weight of a car and gas mileage are related to each other. (b) size and selling price of a house there is a positive correlation, because larger houses tend to be more expensive. there is a positive correlation, because larger houses tend to be less expensive. there is a negative correlation, because larger houses tend to be more expensive. there is a negative correlation, because larger houses tend to be less expensive. there is a correlation close to 0, because there is no reason to believe that size and selling price of a house are related to each other. (c) height and weight there is a positive correlation, because taller people tend to be heavier. there is a positive correlation, because taller people tend to be lighter. there is a negative correlation, because taller people tend to be heavier. there is a negative correlation, because taller people tend to be lighter. there is a correlation close to 0, because there is no reason to believe that height and weight are related to each other. (d) height and number of siblings there is a positive correlation, because taller people tend to have more siblings. there is a positive correlation, because taller people tend to have fewer siblings. there is a negative correlation, because taller people tend to have more siblings. there is a negative correlation, because taller people tend to have fewer siblings. there is a correlation close to 0, because there is no reason to believe that height and number of siblings are related to each other.
The correlation close to 0, because there is no reason to believe that height and number of siblings are related to each other.
For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to 0. Explain your choice. (a) Weight of a car and gas mileage: There is a negative correlation, because heavier cars tend to get lower gas mileage. (b) Size and selling price of a house: There is a positive correlation, because larger houses tend to be more expensive. (c) Height and weight: There is a positive correlation, because taller people tend to be heavier. (d) Height and number of siblings: There is a correlation close to 0, because there is no reason to believe that height and number of siblings are related to each other.
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suppose sat writing scores are normally distributed with a mean of 497 and a standard deviation of 114 . a university plans to admit students whose scores are in the top 30% . what is the minimum score required for admission? round your answer to the nearest whole number, if necessary.
Suppose sat writing scores are normally distributed with a mean of 497 and a standard deviation of 114. A university plans to admit students whose scores are in the top 30%, the minimum score required for admission is 434
How we calculate the minimum score required for admission?Given information:
Mean (μ) = 497Standard Deviation (σ) = 114Probability (p) = 0.30 (for the top 30% of the scores)Let X be the random variable which represents the SAT writing scores. Then X ~ N(497, 114)Now we have to find the minimum score required for admission. We can solve the problem using the standard normal distribution table. Here we need to find the z-score.
The formula for z-score is given below:z = (X - μ) / σ z-score corresponding to the probability (p) can be calculated as:z = ZpWhere Zp is the standard normal variable, which gives the area to the left of the z-score. So, Zp = InvNorm(0.30) = - 0.524For the top 30% of the scores, we have Zp = -0.524. Now the z-score is known. So we can calculate the minimum score required for admission as :X = μ + z * σ = 497 + (-0.524) * 114 = 433.584 ≈ 434The minimum score required for admission is 434.
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The number of members f(x) in a local swimming club increased by 30% every year over a period of x years. The function below shows the relationship between f(x) and x:f(x) = 10(1.3)xWhich of the following graphs best represents the function? (1 point)a Graph of f of x equals 1.3 multiplied by 10 to the power of xb Graph of exponential function going up from left to right in quadrant 1 through the point 0, 0 and continuing towards infinityc Graph of f of x equals 10 multiplied by 1.3 to the power of xd Graph of f of x equals 1.3 to the power of x
The graph of an exponential function with an initial value of 10 and a base of 1.3z. Therefore option D is correct.
The function f(x) is an exponential function with a base of 1.3 and an initial value of 10. The graph of an exponential function with a base greater than 1 increases rapidly as x increases. Therefore, option a can be eliminated.
Option b is not a graph of an exponential function, as the function is not continuous and does not approach any asymptote.
Option c shows an exponential function with an initial value of 10 and a base of 1.3/10, which is less than 1. This means that the function would decrease over time, which is not consistent with the problem statement.
Option d shows an exponential function with an initial value of 10 and a base of 1.3, which is consistent with the problem statement. Therefore, option d is the correct answer.
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The probability that a person in a certain town has brown eyes is 2 out of 5. A survey of 450 people from that same town was taken. How many people would be expected to have
brown eyes?
A. 45
B. 90
C. 180
D. 225
From the given information provided, the number of people having brown eyes in town is 180.
If the probability that a person in the town has brown eyes is 2/5, then we can expect that 2 out of every 5 people have brown eyes.
To find the number of people in the survey who would be expected to have brown eyes, we can use the following proportion:
(2/5) = (x/450)
where x is the number of people expected to have brown eyes.
Solving for x, we can cross-multiply:
5x = 2 × 450
5x = 900
x = 180
Therefore, the expected number of people in the survey who would have brown eyes is 180.
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A company charges $7 for a t-shirt and ships any order for $22. a school principal ordered a number of t-shirts for the school store. the total cost of the order was $1,520. how many t-shirts did the principal order?
Answer:
1,520
Step-by-step explanation:
what is the target domain for a poisson distribution?
The target domain for a Poisson distribution is given the term (0, inf) which can be seen correct in option B.
A Poisson distribution's target domain is (0, inf). This means that the Poisson distribution can only be specified for non-negative integer values of the random variable it is modelling.
The Poisson distribution is a discrete probability function, which indicates that the variable may only take particular values from a finite list of integers. A Poisson distribution estimates how many times an event will occur in "x" amount of time. In other words, it is the probability distribution resulting from the Poisson experiment.
A Poisson experiment is a statistical experiment that categorises the experiment as either successful or unsuccessful. A limiting process of the binomial distribution is the Poisson distribution.
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Complete question:
What is the target domain for a Poisson distribution?
1) (-inf, inf)
2) (0, inf)
3) (-inf, 0]
4) [0, inf)
if you have 25 revolutions in 15 seconds, what is the frequency of rotations in rev/s? (hint: answer should be in two significant figures.)
The frequency of rotations in rev/s for 25 revolutions in 15 seconds is 1.7 rev/s (to two significant figures).
How to find the frequency of rotations:
To determine the frequency of rotations, we divide the number of revolutions by the duration, which is in seconds.
We get a unit of revolutions per second, which is abbreviated as rev/s.
To solve this question, we will use the following formula:
Frequency of rotations (in rev/s) = the number of revolutions ÷ duration
First, we substitute the given values into the formula:
F = 25 ÷ 15
The number of revolutions is 25, and the duration is 15 seconds.
After that, we simplify:
F = 5/3
Next, we convert the fraction to two significant figures, which is 1.7 (rounded to one decimal place).
Therefore, the frequency of rotations in rev/s is 1.7 rev/s (to two significant figures).
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Let a function f be analytic everywhere in a domain D. Prove that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D.
By using the Cauchy-Riemann equations on a real-valued function, it can be proven that the function f(z) is constant in the domain D. This is important for understanding analytic functions in complex analysis.
To prove that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D, let a function f be analytic everywhere in a domain D. We know that a real-valued function is said to be a function whose values lie on the real line. In the case of the complex plane, a function whose values lie on the real line is real-valued.
The Cauchy-Riemann equations, which define the necessary conditions for a function f(z) to be analytic in a domain, say that the imaginary component of f(z) is determined by its real component.
To be more precise, if f(z) is real-valued for all z in D, then we can say that:u(x, y) = f(z),v(x, y) = 0
By definition, the Cauchy-Riemann equations can be stated as:
∂u/∂x = ∂v/∂y∂u/∂y = -∂v/∂x
Taking the first equation, we get:
∂u/∂x = ∂v/∂y => ∂v/∂y = 0
Since v is equal to 0 for all values of x and y, the above equation reduces to ∂u/∂x = 0, which implies u is constant with respect to x.
Similarly, taking the second equation, we get:
∂u/∂y = -∂v/∂x => ∂u/∂y = 0
Since u is equal to a constant for all values of x and y, the above equation reduces to ∂v/∂y = 0, which implies v is constant with respect to y. Since u and v are both constant with respect to their respective variables, u + iv = f(z) is a constant with respect to z throughout the domain D. Thus, we have proved that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D.
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Find the missing side of each triangle round your answers to the nearest 10th
when the expression -3x^5+6x^2-9x^3-x is written with terms in descending order (from highest to lowest), which list represents the coefficients of the term?
a. 6, -9, -3, -1
b. -3, 6, -9, -1
c. 1, 6, -9, -3
d. -3, -9, 6, -1
wright fraction as a decimal and as a percent. Divide decimals to the hundredths place and write any remainders as fractions
1/3
The value of 1/3 in decimals is 0.33 and the value in percent is 33 1/3%.
How to covert a fraction into decimal?We must divide a fraction to get it to a decimal form. To convert a fraction to decimal form, divide the fraction's numerator by its denominator. If the division is not accurate, we can round the decimal to the closest desired place value or divide again until the appropriate number of decimal places is obtained.
The given number is 1/3.
Using division the value of 1/3 = 0.33.
Now, to calculate the percentage we multiply the value by 100:
0.33 x 100 = 33.33% = 33 1/3%.
Hence, the value of 1/3 in decimals is 0.33 and the value in percent is 33 1/3%.
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If the block suppported by the pulley system has a weight of 20N what is the input force(effort) on the rope?(Assume the pulley system is frictionless ).
Answer: The input force (effort) on the rope is 20 N.
Step-by-step explanation:
In a frictionless pulley system, the tension in the rope is constant throughout. Therefore, the force applied to the rope on one side of the pulley system is equal to the force exerted on the other side of the system.
In this problem, we know that the weight of the block is 20 N. This means that there is a force of 20 N acting downwards on one side of the pulley system.
Since the pulley system is frictionless, the tension in the rope is also 20 N. Therefore, the force applied to the rope on the other side of the pulley system (i.e., the input force or effort) must also be 20 N in order to balance the weight of the block.
So the input force (effort) on the rope is also 20 N.
3. The population of bees has been increasing by 5% each year since 2010. There were 1,000 bees counted in 2010. a. Create an explicit formula that models the bee population for n years since 2010. (Hint: What function have we studied that can be represented by this situation?)
Answer: 1000 x 1.05^n
Step-by-step explanation:
1.05 is the increase experienced each year and n is the number of years.
Answer:
The population of bees is increasing by 5% each year since 2010. Let P(n) be the population of bees in the nth year since 2010.
The explicit formula for the bee population can be found using the formula for compound interest:
P(n) = P(0) * (1 + r)^n
where P(0) is the initial population in 2010, r is the annual growth rate, and n is the number of years since 2010.
Substituting the given values, we have:
P(n) = 1000 * (1 + 0.05)^n
Simplifying the expression, we get:
P(n) = 1000 * 1.05^n
Therefore, the explicit formula that models the bee population for n years since 2010 is P(n) = 1000 * 1.05^n.
Make a number line and mark the points that represent the following values of x. x²=16
please show with numberline for brainliest :) please help my class is tomorrow and im finishing homework
Answer:
See below.
Step-by-step explanation:
Here's a number line showing the points that represent the values of x that satisfy the equation x² = 16
|---------------------------|---------------------------|
-4 0 4
On this number line, I have marked the points -4, 4, which are the two solutions to the equation x² = 16. These values of x make the left-hand side of the equation equal to 16.
A car is purchased for £8500
In its first year, the value of the car will depreciate
by 10%.
Each year after that, the value of the car will
depreciate by 5%.
What is the value of the car at the end of 3 years?
Answer:
£ 6904.13
Step-by-step explanation:
the final value is given by
[tex]8500 (0.90)[/tex] at the final of the first year
then you need to add (0.95) twice (one from second and one from third year)
Notice if the depreciation is 5% the final value is 0.95 of the initial value at the beginning of the year
finally:
[tex]8500 (0.90) (0.95)^2 = 6904.13[/tex] (rounded to nearest cent!)
Dos veces un número es a lo sumo 24? Ayuda porfavor
After solving the linear inequality, the number can have any value less than or equal to 12 (x ≤ 12). So the maximum possible value of the number is 12.
Let's suppose the number be "x".
From the given statement, we can write the following inequality:
2x ≤ 24
To solve for x, we can divide both sides by 2:
x ≤ 12
Therefore, the number "x" is at most 12. The exact value of "x" could not be found out. As it could be any number less than or equal to 12 that satisfies the inequality.
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The complete question is :
If twice a number is at most 24, find the number.
i have an assignment, its 2n + 10 = 90 our teacher is asking whats the n can someone help, with solutions is okay :)
Answer: n=40
Step-by-step explanation:
let me know if i got this right for you broski
now dance
Katherine is going to invest in an account paying an interest rate of 3.8%
compounded daily. How much would Katherine need to invest, to the nearest dollar,
for the value of the account to reach $1,180 in 17 years?
619 would Katherine need to invest, to the nearest dollar, for the value of the account.
What is interest in simple words?
When you borrow money, you must pay interest, and when you lend money, you must charge interest. The most common way to represent interest is as a percentage of a loan's total amount every year. The interest rate for the loan is denoted by this proportion.
x. (1 + 3.8%/365)¹⁷ˣ³⁶⁵ = 1180
= x * ( 1 + 38/365000)¹⁷ˣ³⁶⁵ = 1180
= x * ( 1 + 19/182500)¹⁷ˣ³⁶⁵ = 1180
= common denominator and write the numerators above common denominator x * ( 182500 + 19/182500)⁶²⁰⁵ = 1180
= x * (182519/182500)⁶²⁰⁵ = 1180
Divide both sides of the equation by the coefficient of variable
x = 1180/(182519/182500)⁶²⁰⁵
x = 1180 * 182500⁶²⁰⁵/182519⁶²⁰⁵
x = 619
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If you have the following data about
a box of bouncy balls, about how
many bouncy balls are estimated to
be in the box?
Mass of Bouncy Balls + Box = 17,342
grams
Mass of 45 Balls = 505 grams
Mass of Box ONLY = 429 grams
Approximately, how many bouncy
balls are in the box?
Bouncy Balls in Box
Enter
there are approximately 1506 bouncy balls in the box.
What is mass ?
Mass is a physical property of matter, which refers to the amount of substance present in an object. It is typically measured in units of grams or kilograms, and can be thought of as a measure of the object's inertia or resistance to acceleration. Mass is often used interchangeably with weight in everyday language, but they are technically different concepts. Weight is a measure of the force of gravity acting on an object, and varies depending on the object's location in the gravitational field.
According to the question:
First, we need to find the mass of the bouncy balls in the box by subtracting the mass of the box only from the mass of the box and bouncy balls.
Mass of bouncy balls in box = Mass of box and bouncy balls - Mass of box only
Mass of bouncy balls in box = 17,342 - 429
Mass of bouncy balls in box = 16,913 grams
Next, we need to find the mass of one bouncy ball by dividing the mass of 45 balls by 45.
Mass of one bouncy ball = Mass of 45 balls / 45
Mass of one bouncy ball = 505 / 45
Mass of one bouncy ball = 11.222 grams
Finally, we can estimate the number of bouncy balls in the box by dividing the mass of the bouncy balls in the box by the mass of one bouncy ball.
Number of bouncy balls in box = Mass of bouncy balls in box / Mass of one bouncy ball
Number of bouncy balls in box = 16,913 / 11.222
Number of bouncy balls in box ≈ 1506
Therefore, there are approximately 1506 bouncy balls in the box.
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Keilantra and Samantha work at a dry cleaners ironing shirts. Keilantra can iron 30 shirts per hour, and Samantha can iron 15 shirts per hour. Keilantra and Samantha worked a combined 11 hours and ironed 240 shirts. Graphically solve a system of equations in order to determine the number of hours Keilantra worked, x, and the number hours Samantha worked, y.
Answer: Kelinatra (x) worked 5 hours and Samantha (y) worked 6 hours
Step-by-step explanation:
We will use the variables x and y the question provides. We know the time worked by each person added together will equal the combined total time. We can write an equation to show this using addition.
x + y = 11 hours
Next, we know that Keilantra ironed 30 per hour, Samantha ironed 15 per hour and that they ironed 240 shirts. We can write another equation to represent this using addition and multiplication.
30x + 15y = 240
Next, we will graph these two equations. See attached. The solution is the point of intersection written as (x, y). This is (5, 6) meaning that Kelinatra (x) worked 5 hours and Samantha (y) worked 6 hours.
The sides of a triangle are 5p^2 − q^2 ; 6p^2 − 8 + 5q^2
− 5p^2+ 8 + 9q^2 . Find its perimeter.
Answer:
Step-by-step explanation:
To find the perimeter of the triangle, we need to add the lengths of all three sides. So,
Perimeter = (5p^2 − q^2) + (6p^2 − 8 + 5q^2) + (−5p^2+ 8 + 9q^2)
Simplifying the above expression, we get:
Perimeter = 6p^2 + 14q^2 - 8
Therefore, the perimeter of the triangle is 6p^2 + 14q^2 - 8.
a study on students drinking habits asks a random sample of 60 male uf students how many alcoholic beverages they have consumed in the past week. the sample reveals an average of 5.84 alcoholic drinks, with a standard deviation of 4.98. construct a 95% confidence interval for the true average number of alcoholic drinks all uf male students have in a one week period.
The 95% confidence interval for the true average number of alcoholic drinks all UF male students have in a one week period is (4.58, 7.10) that is option C.
Using the following formulas the lower and upper limits of the Interval are calculated,
n = 60
x = 5.84
s = 4.981 = 95% = 0.95
Because the population standard deviation is unknown, the Student T-distribution should be used. Yet, because the sample is huge, some books will utilise the normal distribution. I'll provide solutions for both techniques.
Error = z x s/√n
= 1.96 x 4.98/√60
Error margin ≈1.26
Lower limit = 5.84 - Error
= 5.54 - 1.2601
= 4.58
Upper limit = 5.84 + Error
= 5.84 + 1.2601
= 7.10
Therefore, the upper limit and lower limit is 4.58 and 7.10.
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Complete question:
A study on students drinking habits asks a random sample of 60 male UF students how many alcoholic beverages they have consumed in the past week. The sample reveals an average of 5.84 alcoholic drinks, with a standard deviation of 4.98. Construct a 95% confidence interval for the true average number of alcoholic drinks all UF male students have in a one week period.
A. (4.78, 6.90)
B. (0, 15.60)
C.(4.58, 7.10)
D. (-3.92, 15.60)
determine the smallest integer value of x in -2x+1< -9
Answer: The smallest integer value of x that satisfies the inequality -2x+1<-9 is x=5.
Step-by-step explanation:
Answer: The smallest integer value of x that satisfies the inequality -2x+1<-9 is x=5.
Explanation:
To solve the inequality, we need to isolate the variable x on one side of the inequality symbol. Here are the steps:
Subtract 1 from both sides of the inequality:
-2x < -10
Divide both sides of the inequality by -2, remembering to reverse the direction of the inequality symbol:
x > 5
Therefore, the smallest integer value of x that satisfies the inequality is x = 5, since any value less than 5 would make the inequality false.
Answer:
x = 6
Step-by-step explanation:
- 2x + 1 < - 9 ( subtract 1 from both sides )
- 2x < - 10
divide both sides by - 2, reversing the symbol as a result of dividing by a negative quantity.
x > 5
since x must be greater than 5, it cannot equal 5
then the smallest integer value of x is x = 6
The city plans a new road that will be parallel to Village Way and pass through the intersection of Gray Dr and Canon Rd. What is the equation of the road in slope-intercept form?
The equation line of the of the road in slope-intercept form is; y = 2·x - 10
What is the standard form of the equation of a line?The standard form of the equation of a line is Ax + Bx + C, where A, B, and C are constants and A and B are nonzero numbers.
The parameters for the new road are;
The road will be parallel to village way with points (0, 5), and (-4, -3)
The road will pass through the intersection of Gray Dr and Canon Rd., which is the point with coordinates (3, -4)
Required; The equation of the road
Since the new road is parallel to Village Way, which has slope;
m = (5 - (-3))/(0 - (-4)) = 8/4 = 2
The slope of the new road will also be 2.
Let the equation of the new road be y = m·x + c, where m = 2 is the slope we just found. To find c, we use the fact that the road passes through the point (3, -4);
y - (-4) = 2 × (x - 3)
y = 2·x - 6 - 4 = 2·x - 10
y = 2·x - 10
Therefore, c = -10
Therefore, the equation of the new road in slope-intercept form, therefore is; y = 2·x - 10
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The product of two numbers is 19,200 and HCF is 40 find the LCM
If product of two numbers is 19,200 and HCF is 40, then the LCM is 480
Given that the product of two numbers is 19,200 and the highest common factor (HCF) is 40.
Let us assume the two numbers to be x and y.
Therefore, x × y = 19,200
Also, HCF of x and y is 40.
We can write x = 40a and y = 40b, where a and b are co-prime.
Then, x × y = 40a × 40b = 1600ab = 19,200
Therefore, ab = 12
Now, we need to find the LCM of x and y.
LCM(x, y) = (x × y)/HCF(x, y) --- (1)
Substituting the values of x, y and HCF in the above equation, we get,
LCM(x, y) = (40a × 40b)/40 = 40ab
= 40 × 12 (as ab = 12)
= 480
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Due tomorrow!!! Pls help
Answer:
Step-by-step explanation:
[tex]P=\frac{\theta}{360} \pi d[/tex]
[tex]= \frac{120}{360} \pi \times 10[/tex] (diameter = 2 x radius)
[tex]=\frac{10\pi}{3}[/tex]
[tex]=10.5cm[/tex]
42 Développer chaque produit, puis réduire les expres-
sions obtenues.
A= 5x-3(x+12)
C = 2x² + x(4x - 5)
B=3x-6+7(2x+4)
D=4x²-x+x(5x-9)
The image shows triangle ABC.
1. Sketch the result of dilating triangle ABC using a scale factor of 2 and a center of A. Label it AB'C'.
2. Sketch the result of dilating triangle ABC using a scale factor of -2 and a center of A. Label it AB"C".
3. Find a transformation that would take triangle AB'C' to AB"C".
The triangle ΔA'B'C' formed following the dilation of ΔABC is a similar
triangle to ΔABC.
What are the correct responses?. a. Please find attached the drawing of the dilated triangle ΔA'B'C', created with MS Excel
b. The properties of dilations indicate that ∠B = ∠B'
Reasons:
a. With the assumption that the vertices of the triangle are;
A(0, -3), C(0, 5), and B(6, 3)
Let point P = (0, 0)
A' = 2/3 *(0,-3) = (0, -9/2) (0, -4.5)
C' = 2/3 *(0,5) = (0, 15/2) (0, 7.5)
B' = 2/3 *(6,3) = (9, 9/2) (9, 4.5)
We have;
b. From the attached diagram, and from the properties of dilation, given
that the image of ΔABC is larger than the image of ΔA'B'C' by a scale
factor of 1.5, we have that the ratio of the corresponding sides of ΔABC
and ΔA'B'C' are equal and therefore the angle formed by segment BC and BA which is ∠B and the angle formed by segment B'C' and B'A' which is ∠B'. are equal.
AC/AB = A'C'/A'B'
AC/Sin(B) = AB/Sin (C)
AC/AB = Sin(B)/Sin(C)
Similarly, we have;
A'C'/A'B' = Sin(B')/Sin(C')
Therefore;
Sin(B)/Sin(C) = Sin(B')/Sin(C')
According to the properties of dilation, ∠B = ∠B'
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1 point) Consider the linear system -3-21→ a. Find the eigenvalues and eigenvectors for the coefficient matrix. 0 and 42 b. Find the real-valued solution to the initial value problem yj 5y1 +3y2, y2(0) = 15. = Use t as the independent variable in your answers. y (t) = y(t) =
(a) The eigenvalues of the coefficient matrix is [-1,3] and for λ=42, we get the eigenvector [1,5].
Itcan be found by solving the characteristic equation |A-λI|=0, where A is the coefficient matrix and λ is the eigenvalue. Solving for λ, we get λ=0 and λ=42.
o find the eigenvectors, we substitute each eigenvalue into the equation (A-λI)x=0 and solve for x. For λ=0, we get the eigenvector [-1,3]. For λ=42, we get the eigenvector [1,5].
(b) The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5].
To find the real-valued solution to the initial value problem, we can use the eigenvectors and eigenvalues to diagonalize the coefficient matrix. We have A = PDP^-1, where P is the matrix whose columns are the eigenvectors and D is the diagonal matrix with the eigenvalues on the diagonal.
Using the initial condition y2(0) = 15, we can solve for the constants c1 and c2.
The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5]. Solving for c1 and c2 using the initial condition, we get
y(t) = [-15e^(42t) + 3e^(0t), 15e^(42t) + 5e^(0t)].
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