The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean 0.73 . Please give your answers to two decimal places. Part a) What is the probability that the time between consecutive customers is less than 15 seconds?

The hole is at (-5/2, 0), the vertical asymptotes at x = -3/2 and x = 5, the x-intercepts are (-5, 0) and (5/2, 0) and  y-intercept is (0, -20/3), and the horizontal asymptote will be y = 2.

To find the hole, vertical asymptote, x and y intercepts, and horizontal asymptote of the function;

f(x) = (4x² - 100) / (2x² - 7x - 15)

Hole; Factor the numerator and denominator to simplify the function.

f(x) = [(2x + 10)(2x - 10)] / [(2x + 3)(x - 5)]

The function has a hole at x = -5/2 because this value makes the denominator zero but not the numerator. To find the y-coordinate of the hole, substitute x = -5/2 into the simplified function;

f(-5/2) = [(2(-5/2) + 10)(2(-5/2) - 10)] / [(2(-5/2) + 3)(-5/2 - 5)]

= 0

Therefore, the hole is at (-5/2, 0).

Vertical asymptotes; The function has vertical asymptotes at x = -3/2 and x = 5 because these values make the denominator zero but not the numerator.

X-intercepts; To find the x-intercepts, set the numerator equal to zero and solve for x

(2x + 10)(2x - 10) = 0

x = -5 or x = 5/2

Therefore, the x-intercepts are (-5, 0) and (5/2, 0).

Y-intercept; To find the y-intercept, set x = 0.

f(0) = (4(0)² - 100) / (2(0)² - 7(0) - 15)

= -20/3

Therefore, the y-intercept is (0, -20/3).

Horizontal asymptote; To find the horizontal asymptote, divide the leading term of the numerator by the leading term of the denominator.

f(x) ≈ 4x² / 2x² = 2

Therefore, the horizontal asymptote is y = 2.

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The answer is [B] 0.58.

For the first question, we look at the table and see that the probability of an offender having 10 or more years of education and not being convicted is 0.30.

Therefore, the answer is [B] 0.30. For the second question, we use conditional probability. We want to find the probability that an offender is not convicted given that they have less than 10 years of education. This can be represented as P(not convicted | less than 10 years of education). Using Bayes' theorem, we have:

P(not convicted | less than 10 years of education) = P(less than 10 years of education | not convicted) * P(not convicted) / P(less than 10 years of education)

We can find each of these probabilities from the table: P(less than 10 years of education | not convicted) = 0.35 / 0.65 = 0.5385 P(not convicted) = 0.65 P(less than 10 years of education) = 0.60

Plugging these values into the formula, we get: P(not convicted | less than 10 years of education) = 0.5385 * 0.65 / 0.60 = 0.58

Therefore, the answer is [B] 0.58.

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Answer:

10.4 miles

Step-by-step explanation:

We can model the Cost of Lupita's trip using the formula

C(m) = 3.20m + 6.75, where C is the cost in dollars and m is the number of miles she travels.  We can allow C(m) to equal 40.03 and we will need to solve for m:

40.03 = 3.20m + 6.75

33.28 = 3.20m

m = 10.4

The probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018. The probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003.

To calculate the probability of winning $100 by matching exactly three of the first five and the sixth numbers, we first need to determine the total number of possible combinations for the first five numbers. Since each of the five numbers can be any number between 1 and 69, there are 69 choose 5 (written as 69C5) possible combinations, which is equal to 11,238,513. Out of these 11,238,513 possible combinations, we need to choose three numbers that will match the drawn numbers and two numbers that will not match. The probability of matching three numbers is calculated as 5C3/69C5, which is equal to 0.0018. The probability of not matching the remaining two numbers is 64C2/64C2, which is equal to 1.

Therefore, the probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018 x 1, which is equal to 0.0018. To calculate the probability of winning $100 by matching four of the first five numbers but not the sixth number, we need to determine the total number of possible combinations for four of the first five numbers. Since each of the four numbers can be any number between 1 and 69, there are 69 choose 4 (written as 69C4) possible combinations, which is equal to 4,782,487.

Out of these 4,782,487 possible combinations, we need to choose four numbers that will match with the drawn numbers and one number that will not match. The probability of matching four numbers is calculated as 5C4/69C4, which is equal to 0.0003. The probability of not matching the remaining number is 64/64, which is equal to 1. Therefore, the probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003 x 1, which is equal to 0.0003.

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The variable X represent the value of one coin.

The unitary method is a method of solving problems by finding the value of one unit and then using it to find the value of any number of units. In this problem, we can use the unitary method to find the value of X.

We know that 10 coins have a total value of 10X cents. Therefore, the value of one coin is X cents. To find the value of 2 coins, we can use the unitary method as follows:

Value of 2 coins = 2 * X cents

Similarly, we can find the value of any number of coins using the unitary method.

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Answer:

D !!

Step-by-step explanation:

The two angles are 18 degrees and 162 degrees.

What are supplementary angles?

If the addition of the measures of two angles is 180 degrees, then they are supplementary angles.

Let x be the measure of the smaller angle in degrees.

Then the measure of the larger angle in degrees is:

x + 144

The two angles are supplementary, so their sum is 180 degrees:

x + (x + 144) = 180

Simplifying the left side:

2x + 144 = 180

Subtracting 144 from both sides:

2x = 36

Dividing both sides by 2:

x = 18

So the smaller angle measures 18 degrees, and the larger angle measures:

x + 144 = 18 + 144 = 162

Therefore, the two angles are 18 degrees and 162 degrees.

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Answer: 19 degrees

Step-by-step explanation:

If two angles are complementary, they form a 90° angle. So the angle that is complementary to 71° is 19° because 90-71=19.

Perimeter = 72cm

Area = 374.1cm²

To determine the perimeter of a regular hexagon the formula given below is used;

Perimeter of hexagon = 6a

where a = 12 cm

Perimeter = 6×12 = 72cm

Area = 3√3/2(a²)

where a = 12cm

area = 3√3/2×144

= 3√3× 72

= 374.1cm²

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Answer:

f(1)= -2

g(4)=6

-8× -2 -4×6=-8

There are twenty five groups of 1/5 in 5.

1/5 is equal to 0.2. 5 divided by 0.2 equals 25.

Or here's another way: It takes five 1/5 to make 1. So multiply it by five and you get twenty five.

Using the limit comparison test, it is determined that the integral ∫(3 to 20) dx / (√(3-x)) is convergent. The evaluated value is approximately 7.98.

To determine whether the integral ∫(3 to 20) dx / (√(3-x)) is convergent or divergent, we can use the limit comparison test. Let's compare it with the integral ∫(3 to 20) dx / x^(1/2):

lim x->3+ (√(x-3)) / (√x) = lim x->3+ (√(1+(x-4))) / (√x) = 1

Since the limit of the ratio is a positive finite number, and the integral ∫(3 to 20) dx / x^(1/2) is convergent (it is the integral of the p-series with p=1/2), we conclude that ∫(3 to 20) dx / (√(3-x)) is also convergent. Therefore, we need to evaluate it:

∫(3 to 20) dx / (√(3-x)) = 2(√17 - √2) ≈ 7.98

So the integral converges to 2(√17 - √2).

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Differentiate the functions

a) y' = 5*2⁵ˣ*ln(2) + (3/(3x-5))*(1/ln(2))

b) f'(x) = (8x-1+10)/(4x²-x+10)

c) g'(t) = 1

d) h'(x) = (1/(1+x) + 1/(1-x)) / 2.

a) y = 2⁵ˣ + log₂(3x - 5)
Differentiate using the chain rule and properties of logarithms:
y' = 5*2⁵ˣ*ln(2) + (3/(3x-5))*(1/ln(2))

b) f(x) = log(4x² - x + 10ˣ)
Apply the chain rule and properties of logarithms:
f'(x) = (8x-1+10)/(4x²-x+10)

c) g(t) = ln([tex]e^t^+^1[/tex] / (1+6t+t²))
Using properties of logarithms, we can simplify this to:
g(t) = (t+1) - ln(1+6t+t²)
Differentiate using the chain rule:
g'(t) = 1

d) h(x) = ln(4√(1+x) / (1-x))
Using properties of logarithms, we can simplify this to:
h(x) = (1/2) * ln((1+x)/(1-x))
Differentiate using the chain rule:
h'(x) = (1/(1+x) + 1/(1-x)) / 2.

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Using the law of cosines, the length of side b of triangle ABC is approximately 47.20 feet.

To find the length of side b of triangle ABC using the Law of Cosines, you can apply the following formula:

b² = a² + c² - 2ac * cos(B)

Given the information, angle B = 73.5 degrees, side a = 28.2 feet, and side c = 46.7 feet. Plug these values into the formula:

b² = (28.2)² + (46.7)² - 2(28.2)(46.7) * cos(73.5)

Calculate the values and solve for b:

b² ≈ 795.24 + 2180.89 - 2633.88 * 0.2840
b² ≈ 2228.07

Now, take the square root to find the length of side b:

b ≈ √2228.07
b ≈ 47.20 feet

So, the length of side b of triangle ABC is approximately 47.20 feet.

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If the period of repayment is extended to 20 years and 25 years, the monthly payment amount will be $1,097.83 and $948.43 respectively.

Principal amount is the initial amount borrowed or invested in a loan, investment, or deposit. It is the amount that is used to calculate interest payments, and it is distinct from the interest or any other fees associated with the loan.

If the period of repayment is extended to 20 years, the monthly payment amount will be $1,097.83. This is calculated as follows:

P = Principal Amount

r = Interest rate (4.5% p.a.)

n = Number of years (20)

Monthly Payment Amount (P) = P x (r / (1 - (1 + r)⁻ⁿ))

= 160,000 x (4.5 / (1 - (1 + 4.5)⁻²⁰))

= 160,000 x (4.5 / (1 - 0.375))

= $1,097.83

Similarly, if the period of repayment is extended to 25 years, the monthly payment amount will be $948.43. This is calculated as follows:

P = Principal Amount

r = Interest rate (4.5% p.a.)

n = Number of years (25)

Monthly PaymentAmount (P) = P x (r / (1 - (1 + r)⁻ⁿ))

= 160,000 x (4.5 / (1 - (1 + 4.5)⁻²⁵))

= 160,000 x (4.5 / (1 - 0.242))

= $948.43

This is happening because when the loan period is extended, the number of payments increases, leading to a lower monthly payment amount.

This is because the total amount to be repaid remains the same, but is spread over a longer period of time, resulting in lower monthly payments. In this case, extending the loan period from 15 years to 20 years and 25 years reduces the monthly payment amount from $1,395.87 to $1,097.83 and $948.43 respectively.

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The minimum distance from (-2,-2,0) to the surface z = √(1-2x-2y) is |-1/3(x+y+5)| / 6, where x and y are the coordinates of the closest point on the surface to (-2,-2,0).

To find the minimum distance from a point to a surface, we need to first find the normal vector to the surface at that point. Then, we can use the dot product to find the projection of the vector connecting the point and the surface onto the normal vector, which gives us the minimum distance.

In this problem, the surface is given by z = √(1-2x-2y). Taking partial derivatives with respect to x and y, we get the gradient vector:

grad(z) = (-1/√(1-2x-2y), -1/√(1-2x-2y), 1)

At the point (-2,-2,0), the gradient vector is

grad(-2,-2,0) = (-1/3, -1/3, 1)

Next, we find the vector connecting the point (-2,-2,0) to a general point on the surface (x,y,z):

v = (x+2, y+2, z)

Then, we find the projection of v onto the gradient vector:

proj(grad(z)) = (v · grad(z)) / ||grad(z)||^2 * grad(z)

= -(x+y+5)/6 * (-1/3, -1/3, 1)

Finally, we can calculate the minimum distance as the magnitude of the projection vector:

dist = ||proj(grad(z))||

= |-1/3(x+y+5)| / 6

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The double integral in polar coordinates is equal to (9π/2).

To solve the double integral of √(x² + y²) over the triangular region R with vertices (0,0), (3,0), and (3,3), we first convert the Cartesian coordinates to polar coordinates using x = rcosθ and y = rsinθ. The given integral becomes:

∬_R r dr dθ

Next, we determine the bounds for r and θ. Since R is a right triangle, the bounds for θ are from 0 to π/4. The bounds for r are from 0 to 3secθ, as it starts at the origin and goes to the hypotenuse of the triangle, which can be represented by y = x or rcosθ = rsinθ. Thus, the integral becomes:

∫(θ=0 to π/4) ∫(r=0 to 3secθ) r dr dθ

Solving the integral gives us:

∫(θ=0 to π/4) [(1/2)r²]_0^(3secθ) dθ = ∫(θ=0 to π/4) (9/2)sec²θ dθ = (9/2)[tanθ]_0^(π/4) = (9π/2).

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x² + 2 ln|x| - 1/x + C is the most general antiderivative of the function.

How to find the antiderivative of function?

To find the antiderivative of the given function, we need to find a function F(x) such that F'(x) = f(x).

We can start by separating the function into three terms f(x) = 2x⁴/x³ + 2x³/x³ - x/x³

Simplifying each term,

f(x) = 2x + 2/x - 1/x²

Now we can find the antiderivative of each term separately,

∫ 2x dx = x² + A

∫ 2/x dx = 2 ln|x| + B

∫ -1/x^2 dx = 1/x + D

Putting it all together,

∫ f(x) dx = x² + 2 ln|x| - 1/x + C

where C = A + B + C is the constant of integration.

To check our answer, we can differentiate it and see if we get back the original function (d/dx) [x² + 2 ln|x| - 1/x + C] = 2x + 2/x + 1/x²

= 2x⁴/x³ + 2x³/x³ - x/x³

= f(x)

So our antiderivative is correct.

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Let's use the letters OD, OE, and OF to represent the distances between points O and D, O and E, and O and F, respectively.

We know that OD = OE since O is located on the perpendicular bisector of line segment DE. Similarly, we know that OE = OF because O is located on the perpendicular bisector of line segment EF.

These two equations are combined to provide OD = OE = OF. This demonstrates that points D, E, and F are equally distant from point O.

By definition, all points on a circle are equally distant from the circle's center. Point O is the circle's center since it is equidistant from points D, E, and F.

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a. The probability of 3 accidents is P(X=3) = 0.15.

b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents is  0.90.

c. The expected number of traffic accidents reported in a day in Corvallis is 1.95.

d. The variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.

e. The standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.

a. The probability of 3 accidents is P(X=3) = 0.15.

b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents, which is P(X≥1) = 1 - P(X=0) = 1 - 0.10 = 0.90.

c. The expected value (mean) of the number of accidents is calculated as the sum of the products of the possible values of X and their probabilities, which is:

E(X) = 0(0.10) + 1(0.20) + 2(0.45) + 3(0.15) + 4(0.05) + 5(0.05) = 1.95.

Therefore, the expected number of traffic accidents reported in a day in Corvallis is 1.95.

d. The variance of the number of accidents is calculated as the sum of the squares of the differences between each possible value of X and the expected value, weighted by their probabilities, which is:

Var(X) = [ (0-1.95)²(0.10) + (1-1.95)²(0.20) + (2-1.95)²(0.45) + (3-1.95)²(0.15) + (4-1.95)²(0.05) + (5-1.95)²(0.05) ]

= 1.6525.

Therefore, the variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.

e. The standard deviation of the number of accidents is the square root of the variance, which is:

SD(X) = √(1.6525) = 1.284.

Therefore, the standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.

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Answer:

C

Step-by-step explanation:

[tex](3x-2)(x^2-2x+3)[/tex]

[tex]3x(x^2)-3x(2x)+3x(3)-2(x^2)-2(-2x)-2(3)[/tex]

[tex]3x^3-6x^2+9x-2x^2+4x-6\\[/tex]

Now, combine like terms

[tex]3x^3-6x^2-2x^2+9x+4x-6[/tex]

[tex]-6x^2-2x^2=-8x^2\\\\9x+4x=13x[/tex]

Thus, we have:

[tex]3x^3-8x^2+13x-6[/tex]

So the answer is C. Hope this helps.

Answer:

3rd option

Step-by-step explanation:

(3x - 2)(x² - 2x + 3)

each term in the second factor is multiplied by each term in the first factor , that is

3x(x² - 2x + 3) - 2(x² - 2x + 3) ← distribute parenthesis

= 3x³ - 6x² + 9x - 2x² + 4x - 6 ← collect like terms

= 3x³ - 8x² + 13x - 6

The type II error for the hypothesis test in this scenario would be failing to reject the null hypothesis, which means accepting the owner's claim that the average attendance at games is over 60,000, even though it may not be true.

A type II error, also known as a false negative, occurs when the null hypothesis is actually false, but the hypothesis test fails to reject it. In this case, the null hypothesis would be that the average attendance at games is 60,000 or less, while the alternative hypothesis would be that the average attendance is over 60,000, as claimed by the owner.

If the hypothesis test fails to reject the null hypothesis, it means that there is not enough evidence to conclude that the average attendance is indeed over 60,000, even though it may be true. As a result, the owner's claim would be accepted, and the team may be moved to a city with a larger stadium based on an incorrect conclusion.

Therefore, the type II error in this scenario would be failing to reject the null hypothesis, which may result in accepting the owner's claim that the average attendance at games is over 60,000, even if it is not supported by the data

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These questions are considered statistical because they involve collecting and analyzing numerical data in average.

The statistical questions among the options are:

e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?

These questions are considered statistical because they involve collecting and analyzing numerical data. The average, or mean, is a statistical measure that summarizes a set of data by determining its central tendency. Therefore, questions that ask for the average or mean of a certain variable are considered statistical questions.
Here, we need to identify the statistical questions among the given options.

Statistical questions are those that can be answered by collecting data and using that data to analyze, compare, or summarize certain characteristics. Average are commonly used in statistical analysis.

From the options given, these are the statistical questions:

e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?

These questions involve collecting data and calculating an average, which are characteristics of statistical questions.

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Answer:

No, the litter box will overflow.

Step-by-step explanation:

first you need to find the volume of the litter box.

FORMULA:

cat litter box: V=lxWxH

V= 1 x 2 x 1/3

V= 2/3

Since the bag of cat litter has more cat litter than the litter box can hold, the answer is no. 1 ft 3 is more than 2/3 ft.

IM SORRY IF THIS DOESN'T MAKE SENSE I WAS CONFUSED BY THE 1 ft 3.

The value of  (3x^2 - 5x + 1) dx is x^3 - (5/2)x^2 + x + C, the integrate of  2/(3x^4) dx is 3πx - (1/2)e^(2x) + C and value of 2/(5x) dx is (2/5) ln|x| + C.

a) To integrate (3x^2 - 5x + 1) dx, we need to use the power rule of integration, which states that the integral of x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get:

∫ (3x^2 - 5x + 1) dx

= (3x^3/3) - (5x^2/2) + x + C

= x^3 - (5/2)x^2 + x + C

b) To integrate 2/(3x^4) dx, we can rewrite the expression as 2x^(-4)/3 and then use the power rule of integration again:

∫ 2/(3x^4) dx

= 2/3 ∫ x^(-4) dx

= 2/3 * (-x^(-3))/3 + C

= -2/(9x^3) + C

c) To integrate (3π - e^(2x)) dx, we can use the constant multiple rule of integration and the rule for integrating e^x, which states that the integral of e^x dx = e^x + C:

∫ (3π - e^(2x)) dx

= 3πx - ∫ e^(2x) dx

= 3πx - (1/2)e^(2x) + C

d) To integrate 2/(5x) dx, we can use the power rule of integration and then simplify:

∫ 2/(5x) dx

= (2/5) ∫ x^(-1) dx

= (2/5) ln|x| + C

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The complete question is:

Evaluate the following:

a) integrate (3x ^ 2 - 5x + 1) dx

b) integrate 2/(3x ^ 4) dx

d) integrate (3pi - e ^ (2x)) dx

integrate 2/(5x) dx dx

The integral of 1/ (x √(x²-a²)) is (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C, where C is a constant of integration.

To find the integral of 1/ (x √(x²-a²)), we can use a trigonometric substitution.

First, let's rewrite the denominator as:

√(x² - av) = a sin(θ)

where θ is an angle in the right triangle formed by a, x, and √(x² - a²).

Differentiating both sides with respect to x, we get:

(x / √(x² - a²)) dx = a cos(θ) dθ

Solving for dx, we get:

dx = (a cos(θ) / √(x² - a²)) dθ

Substituting this into our integral, we get:

∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a² sin(θ) cos(θ))] (a cos(θ) / √(x² - a²)) dθ

Simplifying, we get:

∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a sin(θ) cos(θ))] dθ

We can use the trigonometric identity:

1 / (sin(θ) cos(θ)) = 1 / (2 sin(θ) cos(θ)) + 1 / 2

to rewrite the integral as:

∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (sin(θ) cos(θ))] dθ + (1/2) ∫ dθ

Using the substitution u = sin(θ), we get:

∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (u(1-u²[tex])^{0.5}[/tex])] du + (1/2) θ + C

where C is the constant of integration.

We can solve the first integral using a substitution of v = u^2, and then use the natural logarithm to obtain:

∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(u + (1-u²[tex])^{0.5}[/tex]) / u] + (1/2) θ + C

Substituting back in terms of x, we get:

∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C

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The gradient vector of f(x,y) is (-18x - 4y, -4x - 8y).

Using Lagrange multipliers, we find the extreme value(s) by solving the system of equations: -18x - 4y = 3λ, -4x - 8y = -λ, and -3x + y + 6 = 0. The only solution is (x, y) = (2, -4), and f(2, -4) = 32. This shows that f has a maximum and no minimum.


1. Find the gradient vector of f(x,y) = -9x² - 4xy - 4y² - 8: ∇f(x,y) = (-18x - 4y, -4x - 8y).
2. Define the constraint function g(x,y) = -3x + y + 6, and set ∇f(x,y) = λ∇g(x,y), where λ is the Lagrange multiplier.
3. Solve the system of equations: -18x - 4y = 3λ, -4x - 8y = -λ, and -3x + y + 6 = 0.
4. The only solution to the system is (x, y) = (2, -4), and λ = 2.
5. Plug the solution into f(x,y) to find the extreme value: f(2, -4) = 32.
6. Since there is only one extreme value, f has a maximum and no minimum.

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Answer:

The answer is A

-1/2

Step-by-step explanation:

y=sin330

270≤x≥360

cosine=posite

sin=negative

tan=negative

y=sin330°= -1/2

y= -1/2

If in the school consisting of 30 students, 10% are boys , then the number of boys in the school are 3.

The "Percent" is defined as a unit of measurement which expresses a proportion or ratio as a fraction of 100. It is commonly used to represent relative quantities or comparisons.

In a school with 30 students, if 10% of them are boys, we can calculate the number of boys by finding 10% of 30.

The 10% can be written as a decimal by dividing it by 100,

So, 10% is equivalent to 0.10.

Multiplying 0.10 by 30,

We get,

⇒ 0.10 × 30 = 3,

Therefore, the  number of boys are 3.

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The  Laplace transform of the following functions

1. f(p) = 2p/ (p² + 4)²

2. f(p) = -54/ (p² + 9)

1. f(t)=sin(2t)cos (2t)

Using Laplace Transform

sin 2t = 2/ p² + 2² = 2/ p² + 4

and, cos 2t = p/ p² + 2² = p/p² + 4

So, f(p)= 2/ p² + 4 x p/ p² + 4

f(p) = 2p/ (p² + 4)²

2. f(t)= cos² (3t)

Using Laplace Transform

cos² (3t) = (-1)² d/dt(-6 sin (3t)) = -18 cos(3t)

and, -18 cos (3t)= -18 x 3/p² + 9 = -54/ (p²+9)

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The Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

Given are the functions, we need to find the Laplace transformations of the function,

a) f(t) = sin(2t) cos(2t)

[tex]L\left\{\sin \left(2t\right)\cos \left(2t\right)\right\}[/tex]

Use the following identity : cos (2) sin (x) = sin (2x)1/2

[tex]=L\left\{\sin \left(2\cdot \:2t\right)\frac{1}{2}\right\}[/tex]

[tex]=L\left\{\frac{1}{2}\sin \left(4t\right)\right\}[/tex]

Use the constant multiplication property of Laplace Transform:

[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]

[tex]=\frac{1}{2}L\left\{\sin \left(4t\right)\right\}[/tex]

[tex]=\frac{1}{2}\cdot \frac{4}{s^2+16}[/tex]

[tex]=\frac{2}{s^2+16}[/tex]

b) f(t) = cos2 (3t)

[tex]L\left\{\cos \left(2\right)\left(3t\right)\right\}[/tex]

Use the constant multiplication property of Laplace Transform:

[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]

[tex]=\cos \left(2\right)\cdot \:3L\left\{t\right\}[/tex]

[tex]=\cos \left(2\right)\cdot \:3\cdot \frac{1}{s^2}[/tex]

[tex]=\frac{3\cos \left(2\right)}{s^2}[/tex]

c) f(t) = [tex]te^{2t}sin (3t)[/tex]

[tex]L\left\{e^{2t}t\sin \left(3t\right)\right\}[/tex]

Use the Laplace Transformation table:

[tex]L\left\{t^kf\left(t\right)\right\}=\left(-1\right)^k\frac{d^k}{ds^k}\left(L\left\{f\left(t\right)\right\}\right)[/tex]

[tex]\mathrm{For\:}te^{2t}\sin \left(3t\right):\quad f\left(t\right)=e^{2t}\sin \left(3t\right),\:\quad \:k=1[/tex]

[tex]=\left(-1\right)^1\frac{d}{ds}\left(L\left\{e^{2t}\sin \left(3t\right)\right\}\right)[/tex]

[tex]=\left(-1\right)^1\left(-\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}\right)[/tex]

[tex]=\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

Hence, the Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

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