Which of the following is the Laplace transformation L {f(t)} if f(t)=(sin(-4t) 2 ? 32 33 +645 O None of them s2 $4 + 3252 +256 $+32 S + 64s ) 16 54 + 3252 +256

Geometric series is convergent if the |r|<1 where r is the common ratio.

Let Sn=∑ni=0(−3/4)i then

Sn=(−3/4)n+1−1(−3/4)−1

Now take n→∞ then

Sn→0−1(−3/4)−1=4/7

because |−3/4|<1 and so (−3/4)n→0. Now note that your sum is

lim ∑i=1n+1(−3)i−14i=lim 14∑i=1n+1(−3)i−14i−1=1/4.lim Sn=1/7.

Geometric series: A geometric series is the result of adding together geometric sequences indefinitely. Depending on the sequence given to us, such infinite sums can either be finite or infinite. A series is considered to be convergent if the partial sums gravitate to a certain value, also known as a limit. In contrast, a divergent series is one whose partial sums do not reach a limit. Divergent series frequently reach, reach, or avoid a particular number.

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For the series 3-1+1/3-1/9+..., the sum is 9/4.

To find the sum of the infinite geometric series 3-1+1/3-1/9+..., we need to use the formula for the sum of an infinite geometric series: S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio.

In this series, the first term a is 3, and the common ratio r is -1/3. Plugging these values into the formula, we get:

S = 3/(1-(-1/3)) = 3/(1+1/3) = 3/(4/3) = 9/4

Therefore, the sum of the infinite geometric series 3-1+1/3-1/9+... is 9/4.

In conclusion, the sum of an infinite geometric series can be found using the formula S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio. For the series 3-1+1/3-1/9+..., the sum is 9/4.

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the probability that in a randomly selected hour the number of calls is three is 0.18045, or about 18.045%.

Probability is calculated by dividing the number of favorable outcomes by the number of possible outcomes. Example: When rolling a die, an even number can occur in 3 different ways out of 6 possible. Being 3 the number of favorable outcomes and 6 the number of possible outcomes.

Knowing that:

Let's substitute the values into the formula and calculate the probability:

\(P(X = 3) = (e^(-2) * 2^3) / 3!\\P(X = 3) = (2.71828^(-2) * 2^3) / 3!\\P(X = 3) = (0.13534 * 8) / 6\\P(X = 3) ≈ 0.18045\)

Therefore, the probability that in a randomly selected hour the number of calls is three is approximately 0.18045, or about 18.045%.

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38.

f(3)

= 5(3) - 2

= 15 - 2

= 13

39.

5x - 20 = 0

=> 5x = 0 + 20

=> 5x = 20

=> x = 20/5

=> x = 4

38.)

option H

F(3) = 13

39.)

option C

40.)

{-4 , 8}

The power to which a number or expression is raised is called the exponent.

1. An exponent is a mathematical notation that represents the power to which a number or expression is raised. It is written as a superscript number or variable placed above and to the right of the base number or expression.

2. The base number or expression is the number or expression that is being multiplied repeatedly by itself, raised to the power of the exponent.

3. The exponent tells us how many times the base number or expression should be multiplied by itself. For example, in the expression \(2^3\), the base is 2 and the exponent is 3. This means that 2 should be multiplied by itself three times: 2 * 2 * 2 = 8.

4. The exponent can be a positive whole number, a negative number, zero, or a fraction. Each of these cases has different interpretations:

  - Positive exponent: Indicates repeated multiplication. For example, \(2^4\)means 2 multiplied by itself four times.

  - Negative exponent: Indicates the reciprocal of the base raised to the positive exponent. For example, \(2^{-3\) means 1 divided by \(2^3\).

  - Zero exponent: Always equals 1. For example, \(2^0\) = 1.

  - Fractional exponent: Represents a root. For example, \(4^{(1/2)\)represents the square root of 4.

5. Exponents follow certain mathematical properties, such as the product rule \((a^m * a^n = a^{(m+n)})\), the quotient rule \((a^m / a^n = a^{(m-n)})\), and the power rule \(((a^m)^n = a^{(m*n)})\).

Remember to use these rules and definitions to correctly interpret and evaluate expressions involving exponents.

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Answer:

1300

Step-by-step explanation:

add the S numbers than add the T number than there is your answer

Step-by-step explanation:

m1:90 degrees . .... angles in a perpendicular line

m2: 180-(90+27).....angles in a triangle add to 180 degrees

m3:63 degrees ... angles in an iscolese triangle

The measure of angles 1, 2, 3 are -  \(90^{o} ,\;63^{0} ,\;63^{0}\).

Consider the figure attached.

We have a labelled kite similar to mentioned in the question. We have angle ABO = \(27^{o}\).

We have to find the measure of angle 1, 2, 3.

Here are some of the properties of kite -

According to the question -

In ΔAOB,

\(\angle2 + \angle ABO+\angle AOB = 180^{o} \\\angle2+27^{o} +90^{o}=180^{o}\\\angle2+117^{o} =180^{o}\\\angle2 = 63^{o}\)

Th, the two angles opposite to the two equal sides are also equal, since BA and BC are equal, therefore - \(\angle2=\angle3\)\(=63^{o}\).

Since the diagonals of the kite intersect each other perpendicularly, therefore -

\(\angle1=90^{o}\)

Hence the measure of angles 1, 2, 3 are -  \(90^{o} ,\;63^{0} ,\;63^{0}\).

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By using  Fundamental Theorem of Calculus, we find the derivative of the function g(x) = In { sqrt( t + t^3)dt } limit from x to 0 is  ln(sqrt(x + x^3)).  The derivative of the function g(x) = { In (1+t^2) dt} where limit are from x to 1 is  ln(1 + x^2).

The Fundamental Theorem of Calculus, which states that if a function is defined as the definite integral of another function, then its derivative is equal to the integrand evaluated at the upper limit of integration.

So, applying this theorem, we have:

g'(x) = d/dx [∫x_0 ln(sqrt(t + t^3)) dt]

= ln(sqrt(x + x^3)) * d/dx (x) - ln(sqrt(0 + 0^3)) * d/dx (0)

= ln(sqrt(x + x^3))

Therefore, g'(x) = ln(sqrt(x + x^3)).

Using the Fundamental Theorem of Calculus, we have:

g'(x) = d/dx [∫1_x ln(1 + t^2) dt]

= ln(1 + x^2) * d/dx (x) - ln(1 + 1^2) * d/dx (1)

= ln(1 + x^2)

Therefore, g'(x) = ln(1 + x^2).

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____The given question is incomplete, the complete question is given below:

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 7 g(x) = In { sqrt( t + t^3)dt } limit from x to 0.  8. g(x) = { In (1+t^2) dt} where limit are from x to 1.

Answer:

\(x = -1\)

Step-by-step explanation:

-Solve for \(x\) :

\(8x + 1 = -8 - x\)

-Add both side by \(x\) :

\(8x + 1 + x = -8 - x + x\)

\(9x + 1 = -8\)

-Subtract both sides by \(1\) :

\(9x + 1 - 1 = -8 - 1\)

\(9x = -9\)

-Divide both sides by \(9\) :

\(\frac{9x}{9} = \frac{-9}{9}\)

\(x = -1\)

So, the value of \(x\) is \(-1\).

Answer:

\( \boxed{\sf x = - 1} \)

Step-by-step explanation:

\( \sf Solve \: for \: x: \\ \sf \implies 8x + 1 = - 8 - x \\ \\ \sf Add \: x \: to \: both \: sides: \\ \sf \implies 8x + x + 1 = - 8 + (x - x ) \\ \\ \sf x - x = 0 : \\ \sf \implies 8x +x + 1 = - 8 \\ \\ \sf 8x + x = 9x : \\ \sf \implies 9x + 1 = - 8 \\ \\ \sf Subtract \: 1 \: from \: both \: sides: \\ \sf \implies 9x + (1 - 1 )= - 8 - 1 \\ \\ \sf 1 - 1 = 0 : \\ \sf \implies 9x = - 8 - 1 \\ \\ \sf - 8 - 1 : \\ \sf \implies 9x = - 9 \\ \\ \sf Divide \: both \: sides \: of \: 9 x = 9 \: by \: 9: \\ \sf \implies \frac{9x}{9} = - \frac{9}{9} \\ \\ \sf \frac{9}{9} = 1 : \\ \sf \implies x = - 1\)

Answer:

114-135=10

Step-by-step explanation:

class size is 10

Answer:

30

Step-by-step explanation:

60% of what number = 18  is equivalent to:

60% × ? = 18  

18 ÷ 60%  

18 ÷ (60 ÷ 100)  

(100 × 18) ÷ 60  

1,800 ÷ 60  

30

Thomas can pack the items into a maximum of 20 bags, with each bag containing 24 items after calculated with greatest common divisor.

To find the maximum number of bags Thomas can pack, we need to find the greatest common divisor (GCD) of 120, 168, and 192. The GCD will represent the maximum number of items that can be packed into each bag.

To find the GCD, we can use the Euclidean algorithm. First, we find the GCD of 120 and 168:

168 = 1 * 120 + 48

120 = 2 * 48 + 24

48 = 2 * 24 + 0

Therefore, the GCD of 120 and 168 is 24.

Next, we find the GCD of 24 and 192:192 = 8 * 24 + 0

Therefore, the GCD of 120, 168, and 192 is 24.

So, Thomas can pack 24 items into each bag. To find the maximum number of bags he can get, we divide the total number of items by 24:

Total number of items = 120 + 168 + 192 = 480

Number of bags = 480 / 24 = 20

Therefore, Thomas can get a maximum of 20 bags.

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Different possibilities for the given case are, 267540

4 railroads chosen from 15 class I railroads

2 railroads chosen from 8 class II railroads

1 railroad chosen from 7 class III railroads

So total combinations    = ¹⁵C₄ ˣ ⁸C₂ ˣ ⁷C₁

   = (15!)/(4!*11!) x (8!)/(2!6!) x (7!)/(1!6!)

   = 15 x 7 x 13 x 4 x 7 x 7 = 267540

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In the Mediterranean, the Mycenaeans rose to prominence by a mix of military prowess, economic connections, and cultural sway.

Thus, Military Might: The Mycenaeans had a strong military force and were expert warriors. They used sophisticated bronze weapons and chariot warfare as a tactic.

They were able to increase their influence and take control of neighbouring territories thanks to their military might.

Control over Trade Routes: The Mycenaeans had complete control over key trade routes in the Mediterranean, particularly those that extended from the Aegean to the eastern Mediterranean and beyond.

Thus, In the Mediterranean, the Mycenaeans rose to prominence by a mix of military prowess, economic connections, and cultural sway.

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Answer:

Domain:  (-infinity, infinity)

Range:  [0, infinity)

Step-by-step explanation:

y = 4x^2 is a quadratic function whose parabolic graph goes through (0, 0) and opens upward.  (0, 0) is both the vertex and the minimum of this function and graph.  Here x can take on any value, but y is limited to [0, infinity).

Answer: Yes.

Step-by-step explanation:

600 x 0.41 or 41% = 246, which yes is different from 0.41.

Answer/Step-by-step explanation:

Given:

m<12 = 121°

m<6 = 75°

a. m<1 = m<6 (vertical angles)

m<1 = 75° (substitution)

b. m<12 = m<1 + m2 (alternate exterior angles)

121° = 75° + m<2 (substitution)

121° - 75° = m<2 (subtraction property of equality)

46° = m<2

m<2 = 46°

c. m<1 + m<2 + m<3 = 180° (angles on a straight line)

75° + 46° + m<3 = 180° (substitution)

121° + m<3 = 180°

m<3 = 180° - 121° (subtraction property of equality)

m<3 = 59°

d. m<4 = m<3 (vertical angles)

m<4 = 59° (substitution)

e. m<5 + m<4 + m<6 = 180° (angles on a straight line)

m<5 + 59° + 75° = 180° (substitution)

m<5 + 134° = 180°

m<5 = 180° - 134° (Subtraction property of equality)

m<5 = 46°

f. m<7 = m<12 (vertical angles)

m<7 = 121° (substitution)

g. m<8 = m<4 (vertical angles)

m<8 = 59° (substitution)

h. m<9 = m<6 (Alternate Interior Angles)

m<9 = 75° (substitution)

i. m<10 + m<9 = 180° (Linear Pair)

m<10 + 75° = 180° (substitution)

m<10 = 180° - 75° (Subtraction property of equality)

m<10 = 105°

j. m<11 = m<8 (vertical angles)

m<11 = 59° (substitution)

k. m<13 = m<10 (vertical angles)

m<13 = 105° (substitution)

l. m<14 = m<9 (vertical angles)

m<14 = 75° (substitution)

A check or quadrangle is defined by the intersection of pairs of diagonals of a parallelogram.

A parallelogram is a quadrilateral with two pairs of parallel sides. In a parallelogram, the opposite sides are parallel and have the same length. The opposite angles of a parallelogram are equal. A parallelogram is a unique type of quadrilateral with specific characteristics.

A quadrangle or check is defined as the intersection of pairs of diagonals of a parallelogram. In other words, it is the area inside the parallelogram that is divided into four triangles, each of which shares a common vertex in the center of the parallelogram.

The diagonals of a parallelogram are the line segments that connect the opposite vertices of a parallelogram. When these diagonals intersect, they form four triangles, which are also known as the parallelogram's "sub-triangles." The point where the diagonals intersect is called the center of the parallelogram, and it divides each diagonal into two equal parts.

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Answer:

638 miles

Step-by-step explanation:

Key skills needed: Proportions, Unit Rates

1) The car can go 464 miles with 16 gallons of gasoline. We need to find how many miles each gallon gasoline results in.

To find the miles per gallon, we would do 464 miles/16 gallons --> 29 mpg

This means each gallon lets the car run 29 miles.

2) To find out how many miles 22 gallons is, we then do 29 x 22, which is 638 miles.

Therefore, your answer is 638.

Hope you understood and have a nice day!!

Answer:

Perimeter = 105 + 15√3 in

Step-by-step explanation:

✔️Find the height of the trapezoid using trigonometric ratio:

Reference angle = 60°

opposite = height

Adj = 45 - 30 = 15 in

Thus:

Tan 60 = opp/adj

Tan 60 = height/15

height = 15*tan 60

height = 15√3 (tan 60 = √3)

✔️Find the slant height (hypotenuse) using Pythagorean Theorem:

Hypotenuse = √(height² + adj²)

Hypotenuse = √((15√3)² + 15²)

= √(675 + 225)

Hyp = √900

Hyp = 30

✔️Perimeter = 30 + 45 + 30 + 15√3

Perimeter = 105 + 15√3 in

Answer:

1/6 a cup

Step-by-step explanation:

a) Hypothesis test:

Null hypothesis (H0): The percentage of graduate students looking for housing on campus is equal to 15%.

Alternative hypothesis (Ha): The percentage of graduate students looking for housing on campus is greater than 15%.

To test the hypothesis, we can use a one-sample proportion test. We will calculate the test statistic and compare it to the critical value or p-value to make a decision.

The observed proportion of graduate students looking for housing on campus is 78/433 = 0.1804.

Using a significance level (α) of 0.05, we will conduct the test and calculate the test statistic and p-value.

Plan:

Test statistic: z = (p - p) / sqrt(p(1-p)/n)

where p is the observed proportion, p is the hypothesized proportion (0.15), and n is the sample size (433).

Do:

Calculating the test statistic:

z = (0.1804 - 0.15) / sqrt(0.15 * 0.85 / 433)

z ≈ 2.07

Conclude:

Since the test statistic is 2.07, we compare it to the critical value or calculate the p-value.

The critical value for a one-sided test with a significance level of 0.05 is approximately 1.645. Since 2.07 > 1.645, the test statistic falls in the rejection region.

The p-value associated with the test statistic of 2.07 is less than 0.05. Therefore, we reject the null hypothesis.

Based on the survey data, there is evidence to suggest that the percentage of graduate students looking for housing on campus is greater than 15%. The housing office should consider increasing the amount of housing available to graduate students.

b) The p-value obtained in part a) represents the probability of obtaining a test statistic as extreme as the one observed (or more extreme), assuming the null hypothesis is true.

In this case, the p-value is less than 0.05, which suggests strong evidence against the null hypothesis. It indicates that the observed proportion of graduate students looking for housing on campus is significantly higher than the hypothesized proportion of 15%.

c) Including the 21 non-respondents would change the sample size and potentially affect the estimated proportion. If these additional respondents had similar characteristics to the 433 who responded, it is possible that the recommendation might still remain the same.

However, the exact impact depends on the responses of the non-respondents, so it is difficult to determine the precise effect without their data.

d) Type II error in this study would occur if the housing office fails to increase the amount of housing on campus when it is actually necessary (i.e., the percentage of graduate students looking for housing on campus is higher than 15%).

This means the null hypothesis would not be rejected when it should have been. A consequence of this type of error would be the unmet demand for housing, potentially causing dissatisfaction among graduate students and a shortage of available housing options.

e) Type I error in this study would occur if the housing office increases the amount of housing on campus when it is not necessary (i.e., the percentage of graduate students looking for housing on campus is not higher than 15%). This means the null hypothesis would be rejected incorrectly.

A consequence of this type of error would be allocating resources and efforts towards increasing housing capacity unnecessarily, which could result in wastage of resources and potentially impact other areas of the university's operations.

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To gain 1/4 mile over the four days, the hiker must gain 1/16 mile of elevation on the fourth day.

Since the hiker needs to gain a total of 1/4 mile of elevation over four days, the elevation gain must be evenly distributed across each day.

To find the elevation gain on the fourth day, we divide 1/4 mile by 4 (the number of days). This yields 1/16 mile. Therefore, the hiker must gain 1/16 mile of elevation on the fourth day to achieve a total elevation gain of 1/4 mile over the four days.

If we assume that the hiker gains an equal amount of elevation each day, we can divide the total desired elevation gain (1/4 mile) by the number of days (4) to determine the elevation gain needed on each day.

1/4 mile divided by 4 equals 1/16 mile. This means that the hiker must gain 1/16 mile of elevation on the fourth day to achieve a cumulative elevation gain of 1/4 mile over the four days.

By spreading out the elevation gain evenly across the four days, the hiker ensures a steady and consistent progress towards the total desired gain.

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Answer:

x = 7.2 units

Step-by-step explanation:

Because this is a right triangle, we can use trigonometric functions to solve for variable x. We are given an adjacent leg to our triangle, an acute angle, and the hypotenuse so we are going to take the cosine of that angle.

Cosine of an angle equals the adjacent leg divided by the hypotenuse so our equation looks like:
cos 37° = \(\frac{x}{9}\)

To isolate variable x we are going to multiply both sides by 9:
9(cos 37°) = 9(\(\frac{x}{9}\))

Multiply and simplify:
9 cos 37° = 9x / 9
9 cos 37° = 1x
9 cos 37° = x

Break out a calculator and solve, making sure to round to the nearest tenth as the directions say:
x = 7.2

With a credit card balance of $17,426, a minimum monthly payment of $461, and an interest rate of 15.5%, we need to calculate the number of years it will take to pay off the card without adding any additional debt.

To determine the time required to pay off the credit card, we consider the monthly payment and the interest rate. Each month, a portion of the payment goes towards reducing the balance, while the remaining balance accrues interest.

To calculate the time needed for repayment, we track the decreasing balance each month. First, we determine the interest accrued on the remaining balance by multiplying it by the monthly interest rate (15.5% divided by 12).

We continue making monthly payments until the remaining balance reaches zero. By dividing the initial balance by the monthly payment minus the portion allocated to interest, we obtain the number of months needed for repayment. Finally, we divide the result by 12 to convert it into years.

In this scenario, it will take approximately 3.81 years to pay off the credit card (17,426 / (461 - (17,426 * (15.5% / 12))) / 12).

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The first digit must be 1. The remaining seven ones must be either 0 or 1.

Therefore, there can be formed \(2^7=128\) different numbers.

The mean of a observed values approaches the population mean more and more as the number of data points gathered rises.

The average calculated from the group members, distribution, or population is known as the population mean.

Thus, choose a population with a finite mean and a random sample of observations. The mean of a observed values approaches the population mean to a greater extent as the number of data gathered rises.

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The given statement: n(n+1)(2n+1) is divisible by 6 for all positive integers n.

Step-by-step explanation: We need to prove that n(n+1)(2n+1) is divisible by 6 for all positive integers n using mathematical induction.

Step 1: First, lets prove the statement for the smallest possible value of n. The smallest value of n is 1. So, let's check whether the given statement is true for n = 1 or not.

\(LHS = 1(1+1)(2*1+1) = 6\)

It is divisible by 6. Hence, the statement is true for n=1.

Step 2: Let's assume that the statement is true for n=k. So, we can say that n(n+1)(2n+1) is divisible by 6 for n=k. i.e., n(n+1)(2n+1) = 6a (for some integer a).

Step 3: We need to prove the statement is true for n=k+1.

So, we need to prove that (k+1){(k+1)+1}[2(k+1)+1] is divisible by 6

Using the assumption we made in step 2.

\(LHS = (k+1){(k+2)}[2k+3] = (k+1)(k+2)(2k+3)LHS = (2k+3)k(k+1)(k+2)LHS = 2[k(k+1)(k+2)] + 3[k(k+1)]\)

Now, k(k+1)(k+2) is divisible by 6 using the assumption made in step 2.

So, LHS = 2[k(k+1)(k+2)] + 3[k(k+1)] is also divisible by 6.

Hence, the statement is true for n=k+1.

Step 4: Using steps 1 to 3, we can say that the given statement is true for n=1 and if it is true for n=k, then it is also true for n=k+1.

Hence, by mathematical induction, we can say that the statement is true for all positive integers n.

Hence, it is proved that n(n+1)(2n+1) is divisible by 6 for all positive integers n.

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The interest rate for the savings account is 5% after 4 years, $20,000 deposited in a savings account with simple interest earned $800 in interest.

We can use the formula for simple interest to solve the problem:

Simple interest = (Principal * Rate * Time) / 100

where Principal is the initial amount deposited, Rate is the interest rate, and Time is the time period for which the interest is calculated.

We know that the Principal is $20,000 and the time period is 4 years. We are also given that the interest earned is $800. So we can plug in these values and solve for the interest rate:

$800 = (20,000 * Rate * 4) / 100

Multiplying both sides by 100 and dividing by 20,000 * 4, we get:

Rate = $800 / (20,000 * 4 / 100) = 0.05 or 5%

Therefore, the interest rate for the savings account is 5%.

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