(a) The Inverse Laplace transform is -2[tex]e^{-3t}[/tex] + 2cos(3t) - (1/3)sin(3t) (b) The Inverse Laplace transform is [tex]e^{-t/3} - e^{-t}[/tex] (d) The Inverse Laplace transform is (7/2)t² (e) The Inverse Laplace transform is [tex]3e^{-2t/5}[/tex]
To determine the inverse Laplace transforms of the given functions, we'll use various methods such as partial fraction decomposition and known Laplace transform pairs. Let's calculate the inverse Laplace transforms for each case:
(a) Inverse Laplace transform of (2s² - 5s - 1)/((s + 3)(s² + 9)):
First, we need to perform partial fraction decomposition:
(2s² - 5s - 1)/((s + 3)(s² + 9)) = A/(s + 3) + (Bs + C)/(s² + 9)
Multiplying both sides by (s + 3)(s² + 9), we get:
2s² - 5s - 1 = A(s^2 + 9) + (Bs + C)(s + 3)
Expanding and equating coefficients:
2s² - 5s - 1 = (A + B)s² + (3B + A)s + (9A + 3C)
Comparing coefficients, we find:
A + B = 2
3B + A = -5
9A + 3C = -1
Solving these equations, we get A = -2, B = 4, and C = -1.
Now, we can rewrite the function as:
(2s² - 5s - 1)/((s + 3)(s² + 9)) = -2/(s + 3) + (4s - 1)/(s² + 9)
Taking the inverse Laplace transform of each term using known pairs, we have:
Inverse Laplace transform of -2/(s + 3) = -2[tex]e^{-3t}[/tex]
Inverse Laplace transform of (4s - 1)/(s² + 9) = 2cos(3t) - (1/3)sin(3t)
Therefore, the inverse Laplace transform of (2s² - 5s - 1)/((s + 3)(s²+ 9)) is:
-2[tex]e^{-3t}[/tex] + 2cos(3t) - (1/3)sin(3t)
(b) Inverse Laplace transform of 1/(3s² + 5s + 1):
We can use the quadratic formula to factorize the denominator:
3s² + 5s + 1 = (3s + 1)(s + 1)
Using known pairs, the inverse Laplace transform of 1/(3s + 1) is [tex]e^{-t/3}[/tex] and the inverse Laplace transform of 1/(s + 1) is [tex]e^{-t}.[/tex]
Therefore, the inverse Laplace transform of 1/(3s² + 5s + 1) is:
[tex]e^{-t/3} - e^{-t}[/tex]
(d) Inverse Laplace transform of 7/(s³):
Using known pairs, the inverse Laplace transform of 1/sⁿ is (tⁿ⁻¹)/(n-1)!, where n is a positive integer.
Therefore, the inverse Laplace transform of 7/(s³) is:
7(t³⁻¹)/(3-1)! = 7t²/2 = (7/2)t²
(e) Inverse Laplace transform of 3/(5s + 2):
Using known pairs, the inverse Laplace transform of 1/(s - a) is [tex]e^{at}[/tex].
Therefore, the inverse Laplace transform of 3/(5s + 2) is:
[tex]3e^{-2t/5}[/tex]
The complete question is:
Determine the inverse Laplace transforms of:
(a) (2s² - 5s - 1)/((s + 3)(s² + 9))
(b) 1/(3s² + 5s + 1)
(d) 7/(s³)
(e) 3/(5s + 2)
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PLEASE HELP WITH THIS QUESTION!!
NO LINKS PLEASE
Answer:
Step-by-step explanation:
g(x)=[x-4] h(x)= [x] -5
h= -5
g=[x-4]
x x
g=4
Sonia has two packages of hamburger meat. The first package weighs 1.76 pounds and the second package weighs 2.29 pounds. She mixes the two packages together and forms hamburgers that weigh 0.25 pounds each. What is the greatest number of 0.25- pound hamburgers Sonia can make using the hamburger meat she has?
16 because o.25 times 16 = 4
1.76+2.29=4.05 so 16
Answer:
16.
Step-by-step explanation:
0.25 × 16 = 4
1.76 + 2.29 = 4.05
so the answer is 16
hope this helps!
~mina-san
What is the quotient of 1/6 divided by 5?
Answer:
0.03333333333 = 33/1000 = 3.3%
Step-by-step explanation:
Answer:
30
Step-by-step explanation:
Gigi's Family left their house and drove 14 miles south to a gas station and then 48 miles east to a waterpark. How much shorter with their trip to the waterpark have been if they hadn't stopped at the gas station and had driven along the diagonal path instead?
Answer:
If they hadn't stopped at the gas station and had driven along the diagonal path, their trip would have been 12 miles shorter.
Step-by-step explanation:
Given that Gigi's Family left their house and drove 14 miles south to a gas station and then 48 miles east to a waterpark, to determine how much shorter would their trip to the waterpark have been if they hadn't stopped at the gas station and had driven along the diagonal path instead the following calculation must be performed, using the Pythagorean theorem:
14 ^ 2 + 48 ^ 2 = X ^ 2
196 + 2,304 = X ^ 2
√ 2500 = X
50 = X
14 + 48 - 50 = X
62 - 50 = X
Thus, if they hadn't stopped at the gas station and had driven along the diagonal path, their trip would have been 12 miles shorter.
The average U.S. daily internet use at home is 150 minutes (µ). A sample of 81 (n) homes in Philadelphia showed an average usage of 165 minutes (x¯) with a sample standard deviation of 54 minutes (s). We are interested in determining whether the average usage in Philadelphia city is significantly greater than the U.S. average.
1. State your null and alternative hypotheses:
2. What is the value of the test statistic? Please show all the relevant calculations.
3. What is the rejection criterion based on the critical value approach? Use α = 0.05.
4. What is the Statistical decision (i.e., reject /or do not reject the null hypothesis)? Justify your answer.
(1) The explanation is given below.
(2) The value of the test statistic is 2.5.
(3) The rejection criterion based on the critical value approach 1.664$$.
(4) The explanation is given below.
1. Null Hypothesis:H0: µ = 150Alternative Hypothesis:H1: µ > 1502. Test statistic value:We know that, the sample size is greater than 30, which means the sample mean is approximately normally distributed. Now, we need to calculate the test statistic value. The formula for calculating the test statistic value is given by,$$t = \frac{{\left( {{\bar x} - \mu } \right)}}{{\left( {\frac{s}{{\sqrt n }}} \right)}}$$Substituting the given values, we get,$$t = \frac{{\left( {165 - 150} \right)}}{{\left( {\frac{{54}}{{\sqrt {81} }}} \right)}}$$Simplifying the above expression, we get,$$t = \frac{{15}}{{\frac{{54}}{{9}}}}$$$$t =
2.5$$Therefore, the value of the test statistic is 2.5.
3. Rejection criterion based on the critical value approach: The rejection criterion based on the critical value approach is given by$$t > t_{\alpha ,\,df}$$where$α = 0.05$and$df = n - 1 = 81 - 1 = 80$. Now, we need to find the critical value corresponding to 80 degrees of freedom at a 5% level of significance. Using the t-distribution table with 80 degrees of freedom, we get$$t_{0.05, 80} = 1.664$$Therefore, the rejection criterion is$$t > t_{\alpha, df}$$$$\Rightarrow t > 1.664$$
4. Statistical decision: As the calculated value of the test statistic (t = 2.5) is greater than the critical value (t = 1.664), we reject the null hypothesis i.e., we can conclude that the average internet usage in Philadelphia city is significantly greater than the U.S. average. Hence, we can say that the Statistical decision is to reject the null hypothesis.
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We reject the null hypothesis. Thus, the average usage in Philadelphia city is significantly greater than the U.S. average.
1. The null and alternative hypotheses are given below:
Null hypothesis, H0: µ ≤ 150 (Average daily internet use in Philadelphia city is less than or equal to 150 minutes).
Alternative hypothesis, H1: µ > 150 (Average daily internet use in Philadelphia city is greater than 150 minutes)
2. The value of the test statistic is calculated below: t=(x¯−μ)/(s/√n)
Here, x¯ = 165
µ = 150
s = 54
n = 81t
= (165 - 150)/(54/√81)
= 2.50.
Thus, the value of the test statistic is 2.50.
3. The rejection criterion based on the critical value approach is obtained below:
The critical value for α = 0.05 with 80 degrees of freedom is 1.664.
The rejection criterion is t > 1.664.
4. The statistical decision is made by comparing the calculated t-value with the critical value.
If the calculated t-value is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Here, the calculated t-value is 2.50, which is greater than the critical value of 1.664.
Therefore, we reject the null hypothesis. Thus, the average usage in Philadelphia city is significantly greater than the U.S. average.
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Rectangle ABCD is congruent to rectangle A″B″C″D″ . Which sequence of transformations could have been used to transform rectangle ABCD to produce rectangle A″B″C″D″ ? Rectangle ABCD was reflected across the y-axis and then across the x-axis. Rectangle ABCD was translated 8 units left and then 7 units down. Rectangle ABCD was translated 2 units left and then 3 units down. Rectangle ABCD was rotated 180° around the origin and then translated 7 units down. A coordinate graph with rectangle A B C D and rectangle A double prime B double primes C double prime and D double prime. Rectangle A B C D has points at A begin ordered pair 2 comma 5 end ordered pair, B begin ordered pair 2 comma 2 end ordered pair, C begin ordered pair 6 comma 2 end ordered pair, D begin ordered pair 6 comma 4 end ordered pair. Rectangle A double prime B double prime C double prime D double prime has points at A double prime begin ordered pair negative 6 comma negative 3 end ordered pair, B double prime begin ordered pair negative 6 comma negative 5 end ordered pair, C double prime begin ordered pair negative 2 comma negative 5 end ordered pair, D double prime begin ordered pair negative 2 comma negative 3 end ordered pair
May someone help me please. Solve for x and then find the indicated length. Pleaseee
Answer:
16
Step-by-step explanation:
3x + 4 = x + 12 [The straight line on the segments AB and BC indicate that they are the same length]
2x = 8 [Subtract x and 8 from both sides]
x = 4 [Divide both sides by two]
Length AB is equal to (3x + 4) as indicated by the diagram. Substituting x = 4, (3 * 4) + 4 = 16
Question 8 of 10
Which of the following are remote interior angles of Z1? Check all that apply.
ole
O A. 25
B. 21
O c. 26
O D. 24
D E. 23
O F. 22
Answer:
D and E
Step-by-step explanation:
remote interior angles are not adjacent to a given angle. or basically 1 is our exterior angle, so the opposites of 1 and the remote interior angle.
The remote interior angles of Z1 are ∠4 and ∠3. The correct options are D and E.
What are remote interior angle?A triangle's remote interior angles are the two angles that are not adjacent to a given exterior angle. An exterior angle of a triangle is formed by extending one of the triangle's sides.
The following theorem can be used to calculate the remote interior angles of a triangle:
A triangle's remote interior angles are equal in size to the exterior angle that is not adjacent to them.
In other words, if we know the measure of one of a triangle's exterior angles, we can find the measure of the distant interior angles by subtracting that measure from 180 degrees.
Thus, the correct options are D and E.
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Two mechanics were working on your car. One can complete the given job
in six hours, but the new guy takes eight hours. They worked together for
the first two hours, but then the first guy left to help another mechanic on
a different job. How long will it take the new guy to finish your car?
PLS EXPLAIN YOUR WORK AND SHOW YOUR STEPS CLEARLY
It will take the new guy approximately 3.33 hours to finish your car.To find out how long it will take the new mechanic to finish the car, we need to consider their individual rates of work.
Let's denote the first mechanic's rate as "M1" (job per hour) and the new guy's rate as "M2" (job per hour).
We are given that M1 can complete the job in six hours, so his rate of work is 1/6 of the job per hour. Similarly, the new guy takes eight hours to complete the job, so his rate of work is 1/8 of the job per hour.
When they worked together for the first two hours, they combined their rates of work. So in the first two hours, they completed (2/6 + 2/8) of the job.
Now, the first mechanic leaves and only the new guy is left to finish the remaining portion of the job. Let's denote the time it takes for the new guy to complete the remaining portion as "T."
Since we know the portion of the job completed in the first two hours, we can set up the equation: (2/6 + 2/8) + (T/8) = 1.
Simplifying the equation, we have (8/24 + 6/24) + (T/8) = 1.
Combining the fractions, we get 14/24 + (T/8) = 1.
Subtracting 14/24 from both sides, we have T/8 = 10/24.
Simplifying further, we find T = (10/24) * 8 = 3.33 hours.
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Factor the expression (s):
7x+63
12x+15
36z+60
WHAT IS THE PROBABILITY OF LANDING ON TAILS ON THE FIRST FLIP AND THEN TAILS AGAIN ON THE SECOND FLIP?
Answer:
25%
Since it's 2 in a row, it is divided by 4
technically it depends on the way you flip the coin, its mostly a 50-50
a more answer-y answer to this would be around (heads) 65-35 (tails) in my opinion.
name a pear of lines that apper to be parallel
Given the data 21, 13, 13, 37, 13, 23, 25, 15: What is the outlier in the data? What is the mean with the outlier? What is the mean without the outlier? A. 13; 21; 17.6 B. 37; 20; 17.6 C. 37; 17.6; 20 D. 13; 17.6; 21
An equation of the cone z = √3x² + 3y2 in spherical coordinates is: P = 3 ¢ = 7 This option None of these This option This option This option Let D be the region bounded by the two paraboloids z = 2x² + 2y² - 4 and z = 5-x² - y² where x 20 and y 20. Which of the following triple integral in cylindrical coordinates allows us to evaluate the volume of D? 2-47² √√√³√²-²³ dzdrdo ar dzdrde This option for r dzdrde This option This option None of these This option Let D be the region in the first octant enclosed by the two spheres x² + y² + z² = 4 and x² + y² + z² = 25. Which of the following triple integral in spherical coordinates allows us to evaluate the volume of D? fofo ₂ p²sinodpdode of p²sinodododo This option p²sinodpode This option This option None of these
An equation of the cone z = √3x² + 3y² in spherical coordinates is: None of these. A cone is formed by rotating a straight line around an axis when one end of the straight line remains fixed.
For example, if the equation of a cone is known, we may use calculus to calculate the cone's volume and surface area. Cone: A cone is a three-dimensional geometric shape that has one circular base at one end and a single point at the other end. A cone's height is the distance from the tip to the base surface's center. The angle formed between the slant height and the base radius is called the cone's half-angle. The vertical plane that passes through the cone's apex and base centerline is called the cone's axis.
Spherical coordinates: In three-dimensional space, spherical coordinates are a coordinate system. Three parameters, the radial distance r, the polar angle θ, and the azimuthal angle φ, are used to specify the position of a point using spherical coordinates. In this case, the radial distance, which is the distance from the origin, is r, and the polar angle and azimuthal angles are represented by θ and φ, respectively. Let D be the region bounded by the two paraboloids z = 2x² + 2y² - 4 and z = 5 - x² - y² where x² + y² ≤ 4.
The cylindrical coordinate system is a three-dimensional coordinate system that uses cylindrical polar coordinates to specify a point in space. Cylindrical coordinates are a generalization of two-dimensional polar coordinates, which are used to define a point in the plane. The integral that allows us to calculate the volume of the region is given by: ar dzdrde. Let D be the region in the first octant enclosed by the two spheres x² + y² + z² = 4 and x² + y² + z² = 25. The integral that allows us to calculate the volume of the region is given by: p²sinodpdode.
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Janet has $40. She gives $25 to Emma. What percent of Janet's money
does Emma have?
Answer:
0.625
Step-by-step explanation:
25 divided by 40= 0.625 percent
Perform the following test of hypothesis. H0: μ = 285, H1: μ < 285, n = 55, x = 266.89, s = H0 is
By Performing the following test of hypothesis, H0 is rejected.
To perform the test of hypothesis, we compare the sample mean (x) to the hypothesized population mean (μ) and consider the sample size (n) and sample standard deviation (s).
Given:
H0: μ = 285 (null hypothesis)
H1: μ < 285 (alternative hypothesis)
n = 55 (sample size)
x = 266.89 (sample mean)
s = ?
To determine whether to reject or fail to reject the null hypothesis, we calculate the test statistic and compare it to the critical value or p-value.
Since the standard deviation (s) is not given, we cannot directly calculate the test statistic. Without the value of s, we cannot proceed with the hypothesis test. Please provide the value of s to continue with the calculation and draw a conclusion.
The test of hypothesis cannot be performed without the value of the sample standard deviation (s). Please provide the necessary information to proceed with the calculation and draw a conclusion.
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find the value of x to the nearest tenth
Answer:
114
Step-by-step explanation:
You would add 46 degree and 20 degrees (66) and subtract (180 - 66 = ?) to get your answer.
Answer:
114
Step-by-step explanation:
help. please. I really need help if its correct i would be happy
Answer:
[tex]-22\frac{1}{3}[/tex]
Step-by-step explanation:
[tex]a=\frac{1}{3} , b = -10, c = -3[/tex]
change [tex]b^{2} - 9a^{2} + c[/tex] to
[tex]-10 -9\frac{1}{3} + (-3)[/tex]
=
[tex]-22\frac{1}{3}[/tex]
Identify the Type I and Type 2 error for the following claim
(a) The average time that customers wait on hold for customer wrvice at a certain company in less than 5 minutes
(b) The number of salespeople employed at a car lot in linearly correlated with the annual profit of the lot.
(a) Type I error: Rejecting the claim , Type II error: Failing to reject the claim. (b) Type I error: Rejecting the claim, Type II error: Failing to reject the claim.
For claim (a):
Type I error: Rejecting the claim that the average time customers wait on hold is less than 5 minutes when, in fact, it is true. This means concluding that the average wait time is longer than 5 minutes when it is not.
Type II error: Failing to reject the claim that the average time customers wait on hold is less than 5 minutes when, in fact, it is false. This means concluding that the average wait time is less than 5 minutes when it is actually longer.
For claim (b):
Type I error: Rejecting the claim that the number of salespeople employed at a car lot is linearly correlated with the annual profit of the lot when, in fact, it is true. This means concluding that there is no linear correlation between the number of salespeople and annual profit when there actually is.
Type II error: Failing to reject the claim that the number of salespeople employed at a car lot is linearly correlated with the annual profit of the lot when, in fact, it is false. This means concluding that there is a linear correlation between the number of salespeople and annual profit when there actually isn't.
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Find the following combinations
Answer:
3
Step-by-step explanation:
Consider f: R2 + R² given by (x,y) → (3x + 2y, 7.x + 5y). . Is f injective? Is f surjective? • Does f have an inverse and if yes what is it?
To determine if f has an inverse, we need to check if it is bijective (both injective and surjective). Since f is not injective, it cannot have an inverse.
To determine whether the function f: R^2 -> R^2 given by (x, y) -> (3x + 2y, 7x + 5y) is injective (one-to-one) and surjective (onto), we need to analyze its properties.
Injectivity:
A function f: A -> B is injective if for every pair of distinct elements in A, their corresponding images in B are also distinct. In other words, if f(x) = f(y), then x = y.
Let's consider two distinct points in R^2: (x1, y1) and (x2, y2). If f(x1, y1) = f(x2, y2), we have:
(3x1 + 2y1, 7x1 + 5y1) = (3x2 + 2y2, 7x2 + 5y2)
By comparing the corresponding components, we get the following system of equations:
3x1 + 2y1 = 3x2 + 2y2 (Equation 1)
7x1 + 5y1 = 7x2 + 5y2 (Equation 2)
We can subtract Equation 1 from Equation 2 to eliminate the variables:
7x1 + 5y1 - (3x1 + 2y1) = 7x2 + 5y2 - (3x2 + 2y2)
4x1 + 3y1 = 4x2 + 3y2
Rearranging the equation gives:
4x1 - 4x2 = 3y2 - 3y1
4(x1 - x2) = 3(y2 - y1)
Since this equation holds for any values of x1, x2, y1, and y2, it implies that the difference in the x-coordinates must be the same as the difference in the y-coordinates for any two distinct points. However, this is not always true, indicating that f is not injective. Therefore, the function is not one-to-one.
Surjectivity:
A function f: A -> B is surjective if every element in the codomain B has a preimage in the domain A. In other words, for every element b in B, there exists an element a in A such that f(a) = b.
To determine surjectivity, we need to find whether there exists an (x, y) in R^2 such that f(x, y) = (a, b) for any (a, b) in R^2.
Let's take an arbitrary point (a, b) in R^2. We need to find an (x, y) such that:
(3x + 2y, 7x + 5y) = (a, b)
This gives us a system of equations:
3x + 2y = a (Equation 3)
7x + 5y = b (Equation 4)
We have two equations with two variables, which can be solved to find the values of x and y. Since the system has a unique solution for every (a, b) in R^2, we can conclude that f is surjective.
Inverse:
To determine if f has an inverse, we need to check if it is bijective (both injective and surjective). Since f is not injective, it cannot have an inverse.
In summary:
The function f: R^2 -> R^2 given by (x, y) -> (3x + 2y, 7x.
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IVE BEEN STUCK ON THIS FOR AN HOUR PLEASE HELP WHAT IS 9.85714285714 X 7
Answer:
69
Step-by-step explanation:
lol
find the angles marked with a letter
Answer:
Step-by-step explanation:
O;B
HELPPPP ASAPPPP PLEASE
A postal service offers flat-rate shipping for mail in special boxes. Today, Ahmed shipped 1
small box and 6 medium boxes, which cost him 360 dirhams to ship. Meanwhile, Sarah
shipped 3 small boxes and 6 medium boxes, and paid 420 dirhams.
a) Write a system of equations that represents Ahmed and Sara's purchases.
Answer:
Ahmed:
x + 6y = 360
Sarah:
3x + 6y = 420
Step-by-step explanation:
Let
Cost of the small box = x
Cost of the medium box = y
Ahmed:
x + 6y = 360
Sarah:
3x + 6y = 420
x + 6y = 360 (1)
3x + 6y = 420 (2)
Eliminate y by subtracting (1) from (2)
3x - x + 6y - 6y = 420 - 360
2x = 60
x = 60/2
x = 30
Substitute x = 30 into (1) to find y
x + 6y = 360
30 + 6y = 360
6y = 360 - 30
6y = 330
y = 330/6
y = 55
Cost of the small box = x = 30 dirhams
Cost of the medium box = y = 55 dirhams
probablity question: A bag contains 4 red marbles, 3 blue marbles and 7 green marbles. If three marbles are drawn out of the bag, what is the probability, to the nearest 10th of a percent, that all three marbles drawn will be green?
Answer: 12.5%
Step-by-step explanation:
i dunno for sure so probably use someone elses answer
A computer tablet is 0.24 meter long. How long is the computer tablet in centimeters?
Answer:
24 cm long
Step-by-step explanation:
Which expression is equivalent to (8x + 14) + (9x - 5)? + A. 22x - 4 B. 31x - 5 C. 17x + 19 D. 17x + 9
Answer: D
Step-by-step explanation: =8x+14+9x+−5
=(8x+9x)+(14+−5)
=17x+9
In 1945, what concern did the Allies have about invading the Japanese mainland with ground forces? They were worried they did not have enough soldiers. They were worried there would be too great a loss of life. They were worried the Japanese were developing an atomic bomb. They were worried the Japanese had a stronger fleet in mainland ports.
Answer:
b
Step-by-step explanation:
i need points
In 1945, the concern which the Allies had about invading the Japanese mainland with ground forces was:
B. They were worried there would be too great a loss of life.What is an Invasion?This refers to a military tactic where the armed forces of a country or a place go into enemy territory with the aim of defeating the troops and claiming the territory.
With this in mind and from the events in history, we can see that after the bombing of Pearl Harbor by the Japanese Navy, there were concern by the Allies of invading the Japanese mainland with ground forces because they feared the great loss of life.
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Need to get this one right to pass lesson
Answer:
not sure search it up
Step-by-step explanation:
Answer:
33ft²
Step-by-step explanation:
b= base (3)
l= slant height (4)
Surface area = b²+2bl
=3²+2(3)(4)
=9+2(12)
=33
Describe the error Sadie made, and explain how to find the correct answer. (Refer to image)
Step 1: Explain the error made
Step 2: Explain how to find the correct answer.
a)Error:multiplied the numerator and denominator by 3 instead of 2.
b)The correct answer to the given expression is 8/15.
In the given image, Sadie made an error in the simplification of the expression.
The error is that she multiplied the numerator and denominator by 3 instead of 2.
She simplified the numerator and the denominator before carrying out multiplication by 2.
This resulted in the final answer being incorrect.
The correct answer would be 8/5.
The correct way to simplify the expression is as follows:
[tex]$$\frac{4}{3} \div \frac{5}{6} = \frac{4}{3} \times \frac{6}{5}$$[/tex]
Now, cross-cancelling can be performed because the numerator of the first fraction and the denominator of the second fraction have a common factor of 2.
[tex]$$=\frac{4 \times 2}{3 \times 5} = \frac{8}{15}$$[/tex]
Therefore, the correct answer to the given expression is 8/15.
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