Let X and Y be independently random variables, with X uniformly distributed on [0, 1] and Y uniformly distributed on (0,2). Find the PDF fz (z) of Z = max{X,Y}.
1. For z < 0 or z > 2: fz (z) =
2. For 0
3. For 1
The probability density function (PDF) of the random variable Z, defined as the maximum of X and Y, can be determined as follows:
For z < 0 or z > 2: fz(z) = 0
For 0 < z < 1: fz(z) = z
For 1 < z < 2: fz(z) = 2 - z
To find the PDF of Z, we need to consider the different regions of the interval [0, 2] and determine the probability density function for each region.
For z < 0 or z > 2:
Since Z cannot be less than 0 or greater than 2, the probability density in these regions is 0. Therefore, fz(z) = 0 for z < 0 or z > 2.
For 0 < z < 1:
In this range, the maximum of X and Y is always Y because Y ranges from 0 to 2. Therefore, we can write fz(z) = P(Z < z) = P(Y < z). Since Y is uniformly distributed on (0, 2), its PDF is constant within this range. The probability of Y being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 2), which is z/2. Therefore, fz(z) = z/2 for 0 < z < 1.
For 1 < z < 2:
In this range, the maximum of X and Y can either be X or Y. To determine the PDF, we need to consider the probability of X being the maximum and the probability of Y being the maximum.
Probability of X being the maximum:
Since X is uniformly distributed on [0, 1], the probability of X being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 1), which is z/1 = z. Therefore, the probability of X being the maximum is z.
Probability of Y being the maximum:
Since Y is uniformly distributed on (0, 2), the probability of Y being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 2), which is z/2. Therefore, the probability of Y being the maximum is z/2.
Since X and Y are independent, we can add their probabilities to find the overall probability of Z being less than z. Therefore, fz(z) = P(Z < z) = P(X < z) + P(Y < z) = z + z/2 = 2z/2 + z/2 = (3z)/2.
To find the PDF fz(z) in this range, we need to calculate the derivative of the cumulative distribution function (CDF). The CDF of Z can be obtained by integrating the PDF fz(z):
Fz(z) = ∫[0,z] fz(u) du
Taking the derivative of Fz(z) with respect to z, we get:
fz(z) = d/dz (Fz(z)) = d/dz ∫[0,z] fz(u) du = d/dz (3z^2/4) = 3z/2.
Therefore, fz(z) = 3z/2 for 1 < z < 2.
For z < 0 or z > 2: fz(z) = 0
For 0 < z < 1: fz(z
To learn more about the cumulative distribution function
Click here brainly.com/question/30402457
#SPJ11
1 (a) Find the Laurent series of the function (22-9)(2+3) centered at z = −3. 1 (b) Evaluate ſc[−3,3] (z²−9)(z+3) dz.
The simplification based on Laurent series of the function (22-9)(2+3) centered at z = −3
[((1/4)(3)⁴ + (2/3)(3)³ + (9/2)(3)² - 27(3))] - [((1/4)(-3)⁴ + (2/3)(-3)³ + (9/2)(-3)² - 27(-3))]
The given problem involves finding the Laurent series of a function centered at z = -3 and evaluating the integral of another function over a specific interval. The Laurent series simplifies to a constant term of 65.
(a) To find the Laurent series of the function (22-9)(2+3) centered at z = −3, we can expand the function in powers of (z + 3):
(22-9)(2+3) = (13)(5) = 65
Since there are no negative powers of (z + 3), the Laurent series of the function is simply the constant term:
f(z) = 65
(b) To evaluate the integral ſc[−3,3] (z²−9)(z+3) dz, we can first simplify the integrand:
(z² - 9)(z + 3) = (z - 3)(z + 3)(z + 3) = (z - 3)(z + 3)²
Now, let's integrate the simplified expression:
∫[(z - 3)(z + 3)²] dz
Expanding the expression:
∫[z³ + 6z² + 9z - 27] dz
Integrating each term:
(1/4)z⁴ + (2/3)z³ + (9/2)z² - 27z
Now, we can evaluate the integral over the given interval [−3, 3]:
∫[−3,3] (z²−9)(z+3) dz = [((1/4)z⁴ + (2/3)z³ + (9/2)z² - 27z)] evaluated from z = -3 to z = 3
Substituting the upper and lower limits into the expression and simplifying, we get:
[((1/4)(3)⁴ + (2/3)(3)³ + (9/2)(3)² - 27(3))] - [((1/4)(-3)⁴ + (2/3)(-3)³ + (9/2)(-3)² - 27(-3))]
To know more about Laurent series:
https://brainly.com/question/32706315
#SPJ11
15 PT PLZ PLZ PLZ PLZ PLZ HELPPPPPPPPPPPPPPPPPP
Answer:
The total distance the students ran in the race would be 5 1/2 km.
Step-by-step explanation:
1/4 + 1/4 = 2/4 3/41 1/4 + 1 1/4 = 2 2/4 1 3/42/4 + 3/4 + 2 2/4 + 1 3/4 = 3 10/4 = 5 2/4 = 5 1/2 (Simplified)
1
4
2
D
3
In the diagram above, Z4 = 35°.
Find the measure of Z2.
Z2 = [?]
Answer:
35°
Step-by-step explanation:
Due to the parallelism, angle2=angle4
Given an investment of $1,500: which investment would have a larger balance after 5 years? Option 1 - 4% compounded monthly option 2 - 3.9% compounded daily.
Answer:
It is option one
Step-by-step explanation:
I don’t know why but I’m doing a quizzizz with the same question and i picked option 2 but option one was the correct answer
Is it true that for every natural number n, the integer n3 + n2 + 41 is prime? Prove or give a counterexample.
Counterexample: The statement is not true. For n = 41, the expression n^3 + n^2 + 41 equals 41^3 + 41^2 + 41, which is divisible by 41 and therefore not prime.
To prove or disprove the statement, we need to find a counterexample, i.e., a natural number n for which n^3 + n^2 + 41 is not prime. By substituting n = 41 into the expression, we obtain 41^3 + 41^2 + 41. This expression is divisible by 41 since it can be factored as 41(41^2 + 41 + 1). Since a prime number is only divisible by 1 and itself, this means that the expression is not prime and thus disproves the statement. Therefore, the claim that n^3 + n^2 + 41 is prime for every natural number n is false.
To learn more about expression
Click here brainly.com/question/29284305
#SPJ11
Find the interval [μ - z (σ/sqrt(n)), μ + z (σ/sqrt(n))] within which 95 percent of the sample means would be expected to fall, assuming that each sample is from a normal population.
a. μ = 173, σ = 20, n = 42.
b. μ = 874, σ = 12, n = 7.
c. μ = 76, σ = 2, n = 26.
The interval [μ - z (σ/sqrt(n)), μ + z (σ/sqrt(n))] within which 95 percent of the sample means would be expected to fall, assuming that each sample is from a normal population.
a. The interval is [166.01, 179.99].
b. The interval is [849.07, 898.93].
c. The interval is [74.47, 77.53].
a. μ = 173, σ = 20, n = 42.
Here, we have, μ = 173, σ = 20, n = 42.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 173 - 1.96(20/sqrt(42)) , 173 + 1.96(20/sqrt(42)) ]
i.e [ 166.01, 179.99 ]
Therefore, the interval is [166.01, 179.99].
b. μ = 874, σ = 12, n = 7.
Here, we have, μ = 874, σ = 12, n = 7.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 874 - 1.96(12/sqrt(7)) , 874 + 1.96(12/sqrt(7)) ]
i.e [ 849.07, 898.93 ]
Therefore, the interval is [849.07, 898.93].
c. μ = 76, σ = 2, n = 26.
Here, we have, μ = 76, σ = 2, n = 26.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 76 - 1.96(2/sqrt(26)) , 76 + 1.96(2/sqrt(26)) ].
i.e [ 74.47, 77.53 ]
Therefore, the interval is [74.47, 77.53].
To know more about confidence level, visit:
https://brainly.com/question/9386257
#SPJ11
What is the purpose of scientific notation? How is scientific notation represented? Explain.
plz help
Answer:
Step-by-step explanation: The purpose of scientific notation is to make the numbers and quantities used easier to comprehend, to read and to write.
Express both numbers with the same power of ten.
Add the base numbers.
Bring the power of ten down to represent the new power of ten for the sum.
Simplify so that the base number is between 1 and 10.
Answer:
What is the purpose of scientific notation?
"to make the numbers and quantities used easier to comprehend, to read and to write. " <----- Credits to: Yahoo
How is scientific notation represented?
"To start with, scientific notation is a form of expressing very small or large numbers in a simpler form." <--------- Credits to: Yahoo
Jay's hair grows about 8 inches each year. Write a function that describes the length l in inches that Jay's hair will grow for each year k. Which kind of model best describes the function?
Answer:
l=8k
Step-by-step explanation:
Answer:
y=8k
Step-by-step explanation:
In this problem, y = c₁e + c₂e initial conditions. y(1) = 0, y'(1) = e -x-1 y = e X s a two-parameter family of solutions of the second-order DE y" - y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given
The solution to the second-order IVP consisting of the differential equation y" - y = 0 and the initial conditions y(1) = 0, y'(1) = e^(-1).
To find a solution of the second-order initial value problem (IVP) consisting of the differential equation y" - y = 0 and the given initial conditions y(1) = 0, y'(1) = e -x-1, we can follow these steps:
Determine the general solution of the differential equation y" - y = 0:
The characteristic equation is r^2 - 1 = 0. Solving this equation, we find two distinct roots: r = 1 and r = -1.
Therefore, the general solution is y(x) = c₁e^x + c₂e^(-x), where c₁ and c₂ are constants.
Apply the initial condition y(1) = 0:
Substituting x = 1 and y = 0 into the general solution:
0 = c₁e^1 + c₂e^(-1)
Dividing through by e:
0 = c₁ + c₂e^(-2)
Apply the initial condition y'(1) = e -x-1:
Differentiating the general solution:
y'(x) = c₁e^x - c₂e^(-x)
Substituting x = 1 and y' = e^(-1) into the differentiated solution:
e^(-1) = c₁e^1 - c₂e^(-1)
Dividing through by e:
e^(-2) = c₁ - c₂e^(-2)
We now have a system of two equations:
Equation 1: 0 = c₁ + c₂e^(-2)
Equation 2: e^(-2) = c₁ - c₂e^(-2)
Solving this system of equations, we can find the values of c₁ and c₂:
Adding Equation 1 and Equation 2:
0 + e^(-2) = c₁ + c₁ - c₂e^(-2)
e^(-2) = 2c₁ - c₂e^(-2)
Rearranging this equation:
2c₁ = e^(-2)(1 + c₂)
Substituting this value back into Equation 1:
0 = e^(-2)(1 + c₂) + c₂e^(-2)
0 = e^(-2) + c₂e^(-2) + c₂e^(-2)
0 = e^(-2) + 2c₂e^(-2)
-1 = 2c₂e^(-2)
Simplifying:
c₂e^(-2) = -1/2
Substituting this value back into Equation 1:
0 = c₁ - 1/2
c₁ = 1/2
Therefore, the values of c₁ and c₂ are c₁ = 1/2 and c₂ = -1/(2e^2).
Now we can write the particular solution to the IVP:
y(x) = (1/2)e^x - (1/(2e^2))e^(-x)
This is the solution to the second-order IVP consisting of the differential equation y" - y = 0 and the initial conditions y(1) = 0, y'(1) = e^(-1).
To know more about Initial value problem:
https://brainly.com/question/31041139
#SPJ11
Find the difference.
(−x2+9xy)−(x2+6xy−8y2)
Answer:
-2[tex]x^{2}[/tex]+3xy+8[tex]y^{2}[/tex]
Step-by-step explanation:
Find a power series representation for the function. +3 f(x) = (x – 4)2 00 f(x) = Ï ) n = 0 Determine the radius of convergence, R. R=
The power series representation of f(x) = (x - 4)² is f(x) = 9 - 6x + x² + ..., and the radius of convergence, R, is infinity (∞).
To find a power series representation for the function f(x) = (x - 4)², we can expand it using the binomial theorem.
The binomial theorem states that for any real number r and a real number x such that |x| < 1, we have:
[tex](1 + x)^r[/tex] = 1 + rx + (r(r-1))/2! * x² + (r(r-1)(r-2))/3! * x³ + ...
In our case, we have f(x) = (x - 4)², which can be rewritten as (1 + (x - 4))². Using the binomial theorem with r = 2, we get:
f(x) = (1 + (x - 4))² = 1 + 2(x - 4) + (2(2-1))/2! * (x - 4)² + ...
Simplifying this expression, we have:
f(x) = 1 + 2x - 8 + (2/2) * (x² - 8x + 16) + ...
Expanding further, we get:
f(x) = 1 + 2x - 8 + x² - 8x + 16 + ...
Now we can write the power series representation of f(x) as:
f(x) = 9 - 6x + x² + ...
To determine the radius of convergence, R, we need to find the interval of x for which the series converges. In this case, the series converges for all real numbers x since there are no terms involving powers of x that could cause divergence.
Therefore, the radius of convergence, R, is infinity (∞).
To know more about power series, refer here:
https://brainly.com/question/29896893
#SPJ4
Given f(x) and g(x) = kf(x), use the graph to determine the value of k.
Two lines labeled f of x and g of x. Line f of x passes through points negative 4, 0 and negative 3, 1. Line g of x passes through points negative 4, 0 and negative 3, negative 3.
A. 3
B. one third
C. negative one third
D. −3
Answer:
From the given information, we can see that when x = -3, f(x) = 1 and g(x) = -3. Since g(x) = kf(x), we can substitute the values of f(x) and g(x) to solve for k: g(x) = kf(x) -3 = k(1) k = -3 So the value of k is -3, which corresponds to answer choice D.
A cone has a base diameter of 20 centimeters. Its height is 30 centimeters. Calculate the volume in cubic centimeters to the nearest tenth
Answer:
The volume of the cone is 3140cm³
Step-by-step explanation:
Volume of come = 1/3 × πr²h
r = diameter/2 = 20/2 = 10cm
h = 30cm
π = 22/7
Volume = 1/3 × 22/7 × 10² × 30
Volume = 1/3 × 22/7 × 100 × 30
Volume = 3142.86cm³
Volume = 3140cm³
9. A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is 4 minutes. Interarrival times and service times are exponential, and the system may be modeled as an M/M/s/GD/[infinity]/[infinity] queuing system. It costs $20 per hour to operate a cash register, and a cost of 25¢ is assessed for each minute the customer spends in the cash register area. How many registers should the store open? [1 point]
10. On a sunny Spring day MiniGolf has a revenue of $2,000. If the day is cloudy, the revenue drops by 20%. A rainy day reduces the revenue by 80%. If today’s weather is sunny, there is an 80% chance that tomorrow’s weather will be sunny with no chance of rain. If it is cloudy, there is 20% chance that tomorrow will be rainy and 30% chance that it will be sunny. Rain will continue through the next day with a probability of 0.80, but there is 10% chance that it might be sunny. Determine (a) the expected daily revenue for the MiniGolf. (b) the average number of days the weather will not be sunny. [1 point]
The expected number of days with non-sunny weather is given as:
E(x) = P(1) + 2P(2)
= 0.26 + 2 × 0.204
= 0.668.
Hence, this is the required answer.
1 Register should be kept open.
(10)a. The expected daily revenue for the MiniGolf $1,920.
(10)b. The average number of days the weather will not be sunny is 0.668.
9. Calculation of the number of registers:
Given that
λ = 18 customers/hour
µ = 1 / 4 min
= 15 customers/hour
So,
λ / µ = 18 / 15
= 1.2
It is given that s is the number of servers we are looking for.
Let's start solving the queuing system:
For calculating the optimal number of cash registers we have the formula;
rho = λ / (s × µ)
Where,ρ = Traffic Intensity
λ = Arrival Rate
µ = Service Rate
The number of Servers (s) is calculated as:
s = ρλ / µ(If s is fractional, round it up to the nearest integer)
Calculation of ρ;
ρ = λ / (s × µ)
= 18 / (s × 15)
For the above equation let's put some values of s to find the value of ρ;s ρ=λ/(s×µ)
18/(15s)=1.2/1.8
= 0.67, rounded up to 1.1.
The register should be kept open.
10. Calculation of expected daily revenue:
(a) Given that revenue on a sunny day is $2,000.
If it's cloudy, the revenue decreases by 20%, and if it's rainy, the revenue decreases by 80%.
Therefore, Revenue on a cloudy day
= $2,000 - 20% of $2,000
= $2,000 - $400
= $1,600
Revenue on a rainy day = $2,000 - 80% of $2,000= $2,000 - $1,600 = $400
Now, We have to find the expected revenue for the day if it's sunny today.
There is an 80% chance of sunshine tomorrow, which means that there is (a) 20% chance of rain or cloudy weather.
Based on this, Expected Revenue for the day
= 80% of $2,000 + 20% of ($1,600 + $400)
= $1,920
(b) Calculation of the average number of days with non-sunny weather:
Let x be the number of days the weather is not sunny.
Using the probabilities given in the question, the Probability that tomorrow will be sunny if it is sunny today = 80%
The probability that tomorrow will be rainy if it is cloudy today = 20%
The probability that tomorrow will be sunny if it is cloudy today = 30%
The probability that tomorrow will be rainy if it is cloudy today = 20%
The probability that the weather will not be sunny tomorrow if it's rainy today = 80%
Therefore, the Probability of non-sunny weather in a day,
P(x) is given as follows:
P(0) = 0.8 (When it's sunny today, there's an 80% chance it'll be sunny tomorrow)
P(1) = 0.2 × 0.3 + 0.8 × 0.2
= 0.26 (When it's cloudy today, there's a 30% chance it'll be sunny tomorrow and a 20% chance it'll be rainy)
P(2) = 0.2 × 0.7 × 0.1 + 0.8 × 0.2 × 0.2 + 0.2 × 0.3 × 0.2 + 0.8 × 0.8 × 0.1
= 0.204 (When it's rainy today, there's a 10% chance it'll be sunny tomorrow and an 80% chance it'll be rainy tomorrow).
To know more about Probability visit:
https://brainly.com/question/13604758
#SPJ11
PLEASE HELP!!! I’ll mark brainliest
Answer: I think it’s the first choice. Foreign Language, Math/CS, and social studies
Step-by-step explanation:
Number of Jobs A sociologist found that in a sample of 55 retired men, the average number of jobs they had during their lifetimes was 7.1. The population standard deviation is 2.1
.(a) Find the best point estimate of the mean.
The best point estimate of the mean is 7.1
(b) Find the 95% confidence interval of the mean number of jobs. Round intermediate and final answers to one decimal place.
<<μ
The 95% confidence interval of the mean number of jobs is (6.7, 7.5).
Given that, a sociologist found that in a sample of 55 retired men, the average number of jobs they had during their lifetimes was 7.1 and the population standard deviation is 2.1.
The best point estimate of the mean is the sample mean.
Hence the best point estimate of the mean is 7.1.
Therefore, the 95% confidence interval of the mean number of jobs is (6.7, 7.5).
Hence the required solution.
To know more about confidence interval visit:
https://brainly.com/question/20309162
#SPJ11
Solve the equation 6x –3y = 9 for y
Answer:
y= -3
Step-by-step explanation:
Substitute x as 0
6(0)-3y=9
-3y=9
y= -3
Plzz help it do today
Answer:
i think it is A CiNiyah
Step-by-step explanation:
HELPPPP PLSSS I WILL MARK BRAINLYEST
Answer:
B.
Step-by-step explanation:
hope this helps!!
Pls help, question on picture, will do brainliest if right
no links!!!!!
Answer: 24/25
Step-by-step explanation:
Sin = opposite/hypotenuse
Sin = 24/25
PLEASE HELP I DONT UNDERSTAND I WILL GIVE EXTRA POINTS IF ITS RIGHT!!!!!!!!!!!!
Answer:
10.7
Step-by-step explanation:
Hello There!
The image shown below shows the relationship between two chords formed inside of a circle
So the product of the lengths of the lines in the same chord is equal to the product of the other length of the chords lines if that makes sense
So basically 6 * 16 = 9 * x
we can now use this equation that was just made to solve for x
6 * 16 = 96
96 = 9x
*divide each side by 9*
96/9=10.66666667 *round to the nearest tenth* 10.7
9x/x
we're left with x = 10.7
help please important!!!!^click picture
Martin recorded the low temperatures at his house for one week. the temperatures are shown below -
-7, -3, 4, 1, -2, -8. 7
Approximately what was the average low temperature of the week?
Answer:
The average temp was 4
Step-by-step explanation:
Suppose that when your friend was born, your friend's parents deposited $4000 in an account paying 6.3% interest compounded quarterly. What will the account balance be after 13 years
Answer:
$9015.20
Step-by-step explanation:
A = 4000[1 + (.063/4)]^13·4
Answer:
3276
Step-by-step explanation:
4000(6.3*13)÷100
X/8 < -1 please help
Answer:
x<-8
Step-by-step explanation:
x/8<-1
x<-1×8
x<-8
Let 8 denote the minimum degree of any vertex of a given graph, and let A denote the maximum degree of any vertex in the graph. Suppose you know that a certain graph has seven vertices, and that 8 = 3 and Δ= 5. (a) Show that this graph must contain at least 12 edges. (b) What is the largest number of edges possible in this graph?
For a graph with seven vertices, a minimum degree of 3 (8 = 3), and a maximum degree of 5 (Δ = 5), it can be shown that the graph must contain at least 12 edges. The largest number of edges possible in this graph is determined by the Handshaking Lemma, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges.
(a) To show that the graph must contain at least 12 edges, we can use the Handshaking Lemma. The sum of the degrees of all vertices in the graph is equal to twice the number of edges. In this case, with seven vertices and a minimum degree of 3, the sum of the degrees is at least 7 * 3 = 21. Therefore, the minimum number of edges is 21/2 = 10.5, which rounds up to 11. So the graph must contain at least 11 edges, but since the number of edges must be an integer, it must be at least 12.
(b) The largest number of edges possible in this graph can be determined by considering the maximum degree. In this case, the maximum degree is 5. Since the sum of the degrees of all vertices is equal to twice the number of edges, the sum of the degrees is at most 7 * 5 = 35. Therefore, the largest possible number of edges is 35/2 = 17.5, which rounds down to 17. So the largest number of edges possible in this graph is 17.
Learn more about vertices here:
https://brainly.com/question/29154919
#SPJ11
Not a question but need help. Check out the image.
Answer:
well you need to have your friends personal email then login to you email and press new message once your logged in put in you friends email and you are all set
Step-by-step explanation:
What is the product of 2x + 3 and 4x^2 - 5x + 6
Answer:
8
x
3
+
2
x
2
−
3
x
+
18
Step-by-step explanation:
simply the answer thought bc i didn't simplify
An airplane is on a heading of 170 degrees to a vacation island, and is cruising at 250km/hr. It is encountering a wind blowing from the south/west at 50 km/hr.
A. Draw a "logical" vector diagram of "our" flight to the "secret" island.
B. Determine the aircraft’s ground velocity (magnitude and direction and standard bearing). Round your final answer to 1 decimal.
C. If the entire flight took about 5 hours, how far is the vacation island from the airport of departure?
A) Logical vector diagram of the flight is drawn below. B) The aircraft's ground velocity is approximately 260.2 km/hr at a bearing of -153.7°. C) The vacation island is approximately 1301 kilometers from the airport of departure.
A. a logical vector diagram of the flight is given in image.
B. To determine the aircraft's ground velocity, we need to find the resultant vector of the aircraft's velocity and the wind vector. We can use vector addition to calculate this:
Aircraft's velocity = 250 km/hr at a heading of 170°
Wind velocity = 50 km/hr at a heading of 270° (since it's blowing from the south/west)
To add these vectors, we need to resolve them into their horizontal (x) and vertical (y) components:
Aircraft's velocity:
[tex]V_x[/tex] = 250 km/hr * cos(170°)
[tex]V_{y}[/tex] = 250 km/hr * sin(170°)
Wind velocity:
[tex]V_x[/tex]_wind = 50 km/hr * cos(270°)
[tex]V_y[/tex]_wind = 50 km/hr * sin(270°)
Now, we can add the horizontal and vertical components separately:
[tex]V_{x} total = V_x + V_{x}wind\\V_{y} total = V_y + V_{y}wind[/tex]
To find the magnitude and direction of the resultant vector, we can use the Pythagorean theorem and trigonometry:
Magnitude of the resultant vector (ground velocity):
[tex]V_{total} = \sqrt{V_x total^2 + V_y total^2}[/tex]
Direction of the resultant vector:
[tex]\theta = tan^{-1} 2(V_y total, V_xtotal)[/tex]
Let's calculate the values:
[tex]V_x[/tex] = 250 km/hr * cos(170°) ≈ -235.83 km/hr
[tex]V_y[/tex] = 250 km/hr * sin(170°) ≈ -62.85 km/hr
[tex]V_x[/tex]_wind = 50 km/hr * cos(270°) = 0 km/hr
[tex]V_y[/tex]_wind = 50 km/hr * sin(270°) ≈ -50 km/hr
[tex]V_x[/tex]_total = -235.83 km/hr + 0 km/hr = -235.83 km/hr
[tex]V_y[/tex]_total = -62.85 km/hr + (-50 km/hr) = -112.85 km/hr
[tex]V_{total}[/tex] = [tex]\sqrt{((-235.83 km/hr)^2 + (-112.85 km/hr)^2) }[/tex] ≈ 260.2 km/hr
θ = [tex]tan^{-1} 2(-112.85 km/hr, -235.83 km/hr)[/tex] ≈ -153.7°
The aircraft's ground velocity is approximately 260.2 km/hr at a bearing of -153.7°.
C. If the entire flight took about 5 hours, we can calculate the distance traveled by multiplying the ground velocity by the time:
Distance = Velocity * Time
Distance = 260.2 km/hr * 5 hours = 1301 km
The vacation island is approximately 1301 kilometers from the airport of departure.
Read more about vector Diagram at;
brainly.com/question/3184914
#SPJ4