10. Find the exact value of each expression. b. cos-1 (eln 1-žin2)

Answer:

i think its c

Step-by-step explanation:

Answer:

d = 55.5  

x = 1

c = 11

m = \(\frac{1}{122}\)

k = \(\frac{a}{(c + 5)}\)

Step-by-step explanation:

Sorry, the formatting is slightly hard to understand, but I think this is what you meant.

Q1.

\(\frac{1}{6}\)d - 8 = \(\frac{5}{8}\) x 2

Step 1. Simplify.

\(\frac{5}{8}\) x 2 = \(\frac{5}{8}\) x \(\frac{2}{1}\) = \(\frac{10}{8}\)

Step 2. Cancel out the negative 8.

\(\frac{1}{6}\)d - 8 = \(\frac{10}{8}\)

+ 8 to both sides (do the opposite: \(\frac{1}{6}\)d is subtracting 8 right now, but to cancel that out, we will do the opposite of subtraction, i.e. addition)

\(\frac{1}{6}\)d = \(\frac{10}{8}\) + 8

Step 3. Simplify.

\(\frac{10}{8}\) + 8 = \(\frac{10}{8}\) + \(\frac{8}{1}\) = \(\frac{10}{8}\) + \(\frac{64}{8}\) = \(\frac{74}{8}\) = \(\frac{37}{4}\)

Step 4. Cancel out the \(\frac{1}{6}\).

\(\frac{1}{6}\)d = \(\frac{37}{4}\)

÷ \(\frac{1}{6}\) from both sides (do the opposite: d is multiplied by \(\frac{1}{6}\) right now, but to cancel that out, we will do the opposite of multiplication, i.e. division)

÷ \(\frac{1}{6}\) = x 6

So....

x 6 to both sides

d = \(\frac{37}{4}\) x  6 = \(\frac{37}{4}\) x \(\frac{6}{1}\) = \(\frac{222}{4}\) = \(\frac{111}{2}\) = 55.5

Step 5. Write down your answer.

d = 55.5

Q2.

3x - 4 + 5x = 10 - 2x × 3

Step 1. Simplify

3x - 4 + 5x = 3x + 5x - 4 = 8x - 4

10 - 2x × 3 = 10 - (2x × 3) = 10 - 6x

Step 2. Cancel out the negative 6x

8x - 4 = 10 - 6x

+ 6x to both sides (do the opposite - you're probably tired of reading this now - right now it's 10 subtract 6x, but the opposite of subtraction is addition)

14x - 4 = 10

Step 3. Cancel out the negative 4

14x - 4 = 10

+ 4 to both sides (right now it's 14x subtract 4, but the opposite of subtraction is addition)

14x = 14

Step 4. Divide by 14

14x = 14

÷ 14 from both sides (out of the [14 × x] we only want the [x], so we cancel out the [× 14])

x = 1

Step 5. Write down your answer.

x = 1

Q3.

7(c - 3) = 14 × 4

Step 1. Expand the brackets

7(c - 3) = (7 x c) - (7 x 3) = 7c - 21

Step 2. Simplify

14 x 4 = 56

Step 3. Cancel out the negative 21

7c - 21 = 56

+ 21

7c = 56 + 21

7c = 77

Step 4. Cancel out the ×7

7c = 77

÷ 7

c = 77 ÷ 7

c = 11

Step 5. Write down your answer.

c = 11

Q4.

11(\(\frac{m}{22}\) + \(\frac{3}{44}\)) = 87m + m × 5

Step 1. Expand the brackets

11(\(\frac{m}{22}\) + \(\frac{3}{44}\)) = (11 x \(\frac{m}{22}\)) + (11 x \(\frac{3}{44}\)) = (\(\frac{11}{1}\) x \(\frac{m}{22}\)) + (\(\frac{11}{1}\) x \(\frac{3}{44}\)) = \(\frac{11m}{22}\) + \(\frac{33}{44}\) = \(\frac{m}{2}\) + \(\frac{3}{4}\)

Step 2. Simplify.

87m + m x 5 = 87m + 5m = 92m

Step 3. Cancel out the add \(\frac{3}{4}\)

\(\frac{m}{2}\) + \(\frac{3}{4}\) = 92m

- \(\frac{3}{4}\)

\(\frac{m}{2}\) = 92m - \(\frac{3}{4}\)

\(\frac{m}{2}\) = \(\frac{92m}{1}\) - \(\frac{3}{4}\)

\(\frac{m}{2}\) = \(\frac{368m}{4}\) - \(\frac{3}{4}\)

\(\frac{m}{2}\) = \(\frac{368m - 3}{4}\)

Step 4. Cancel out the ÷ 4

\(\frac{m}{2}\) = \(\frac{368m - 3}{4}\)

x 4

2m = 368m - 3

Step 5. Cancel out the 368m

2m = 368m - 3

- 368m

-366m = - 3

Step 6. Cancel out the × -366

-366m = -3

÷ -366

m = \(\frac{-3}{-366}\)

m = \(\frac{1}{122}\)

Step 7. Write down your answer.

m = \(\frac{1}{122}\)

Q5.

ck + 5k = a

Step 1. Factorise

ck + 5k = (c × k) + (5 × k) = (c + 5) x k = k(c + 5)

Step 2. Cancel out the × (c + 5)

k(c + 5) = a

÷ (c + 5)

k = a ÷ (c + 5)

k = \(\frac{a}{(c + 5)}\)

Answer: y = -4/3x + 0. Yes, that point lies on the line

Step-by-step explanation:

Use this equation, Δy/Δx and this one, y = mx + b. Or, change in the y value divided by the change in the x value.

-4 + 8 = 4

3 - 6 = -3

The slope, m, is -4/3. Now solve for b by plugging in the slope.

\(-4 = \frac{-4}3 (3) + b\). B is equal to 0

To see if that point lies on the function, plug it in the equation and see if it is true. Multiply -4 x -3 to get 12, divide by 3 to get 4, then add 0. The equation says that 4 = 4, which is infact true

Answer:

y=-4x/3

the point (-3,4) is on the line y= -4x/3

Step-by-step explanation:

the points (6,-8) and (3,-4)

a) (y- y1)/(y2- y1)=(x- x1)/(x2- x1)

(y+8)/(-4+8)=(x-6)/(3-6)

(y+8)/4=(x-6)/(-3)

-3y-24=4x-24

So, y=-4x/3

b) (-3; 4)

4= -4*(-3)/3

4=4 it is true,

the point (-3,4) is on the line y= -4x/3

The inequality to show the values of [w] that will not exceed the weight limit of the elevator is w + 1643 ≤ 2000. On solving the inequality, we get w ≤ 357. The graph of the inequality is attached.

        ≤ : less than or equal to

        ≥ : greater than or equal to

        < : less than

        > : greater than

        ≠ : not equal to

Given is the maximum load for a certain elevator is 2000 pounds. The total weight of the passengers on the elevator is 1400 pounds. A delivery man who weighs 243 pounds enters the elevator with a crate of weight [w].

        1400 + 243 + w ≤ 2000

        w + 1643 ≤ 2000

        w + 1643 ≤ 2000

        w ≤ 2000 - 1643

        w ≤ 357

Therefore, the inequality to show the values of [w] that will not exceed the weight limit of the elevator is w + 1643 ≤ 2000. On solving the inequality, we get w ≤ 357. The graph of the inequality is attached.

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It is true that the existence of two local maxima does not guarantee the presence of a local minimum. It is possible for a function to have multiple local maxima and no local minimum.

For example, consider the function f(x,y) = x^4 - 4x^2 + y^2. This function has two local maxima at (2,0) and (-2,0), but no local minimum. Therefore, the statement "if f(x,y) has two local maximal, then f must have a local minimum" is false. The presence or absence of local maxima and minima depends on the behavior of the function in the immediate vicinity of a point, and cannot be determined solely based on the number of local maxima. It is possible for a function to have an infinite number of local maxima and minima, or none at all. Therefore, it is important to carefully analyze the behavior of a function in order to determine the presence or absence of local extrema.

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Answer:

A + B = 24

16A + 20B = 434

Step-by-step explanation:

To write a system of equations for this scenario, let's say that A represents the number of hours machine A ran, and B represents the number of hours machine B ran.

The first equation will be:

A + B = 24

because the total number of hours ran is 24.

The second equation will be:

16A + 20B = 434

because the total number of items made is 434.

A + B = 24

16A + 20B = 434

First solve for one variable, and let's just do A.

Using the first equation, A + B = 24, A is equal to 24 - B.

Substitute this value to the second equation.

16 (24 - B) + 20B = 434

384 - 16B + 20B = 434

4B = 50

B = 12.5

Now use this value of B to find the value of A.

A + 12.5 = 24

A = 11.5

Machine A ran for 11.5 hours, and Machine B ran for 12 hours.

d. Dummy coding. Categorical variables need to be converted into numerical variables to be used in regression analysis. Dummy coding involves creating binary variables for each category of the categorical variable.

For example, if the categorical variable is "color" with categories "red," "green," and "blue," dummy coding would involve creating three binary variables: "red" (0 or 1), "green" (0 or 1), and "blue" (0 or 1). These binary variables can then be used in the regression analysis. In conclusion, to use categorical variables in regression analysis, dummy coding is necessary.


In order to use categorical variables in a regression analysis, they must be converted into numerical values. This process is called dummy coding (also known as one-hot encoding). Dummy coding involves creating new binary variables (0 or 1) for each category of the categorical variable. This allows the regression model to incorporate the categorical data while maintaining its numerical nature.

To use categorical variables in a regression analysis, you must apply dummy coding to convert them into numerical values.

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The average daily rate is calculated by dividing the total rooms revenue by the total number of rooms sold. In this case, the average daily rate is (option) $114.63.

The average daily rate is a measure of the average amount of money that a hotel is able to generate per room per day. It is calculated by dividing the total rooms revenue by the total number of rooms sold. In this case, the total rooms' revenue was $75,884 and the total number of rooms sold was 662, which gives us an average daily rate of $114.63. This is a useful measure for hotels to gauge the performance of their business and to set pricing accordingly. It also helps hotels identify opportunities for increasing revenue, such as offering discounts or promotions to attract more customers.

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Given the topological spaces X = {D, O, R, K} with the topology Tx and Y = {M, A, T, H} with the topology Jy, we are asked to determine whether the set E = {0, K} is open in X and open in Y.

To determine whether the set E = {0, K} is open in the topological spaces X and Y, we need to check if E belongs to the respective topologies, Tx and Ty.

In X, the topology Tx is given by: Tx = {Ø, {0}, {D, O}, {O, R}, {D, O, R}, X}. We can see that E = {0, K} is not explicitly listed in Tx. Therefore, E is not open in X since it does not belong to the topology.

In Y, the topology Ty is given by: Jy = {0, {M}, {M, A}, {M, A, T}, Y}. Again, E = {0, K} is not explicitly listed in Ty. Hence, E is not open in Y as it does not belong to the topology.

In both cases, the set E = {0, K} is not open in the respective topological spaces X and Y because it is not a member of the defined topologies.

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The solution to the equation for x is x = 1/6[e^2]

from the question, we have the following parameters that can be used in our computation:

ln 9+ln 4x^2=4

Combine the natural logarithm

So, we have the following representation

ln(9 * 4x^2) =4

This gives

ln(36x^2) = 4

Take the exponent of both sides

36x^2 = e^4

Divide both sides by 36

x^2 = 1/36[e^4]

Take the square roots

x = 1/6[e^2]

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Answer:

1) x = 14.14

2) 2

Step-by-step explanation:

1) sin(45°) = x/20

sin(45°) · 20 = x

x = 14.14

2) Since PST are double the size of PQR it would just be a scale factor of 2

Answer:

The no. are 11 and 12

Step-by-step explanation:

n(n+1)=17+5(2n+1)

n^2 +n=17+10n+5

n^2 -9n-22

Splitting the middle term

n^2 -11n+2n -22

n(n-11)+2(n-11)

(n-11)(n+2)

As n cannot ne negative,

n=11 and 12(consecutive no.)

Mark me brainliest!!

The two of the consecutive positive integers whose product is 17 more than 5 times their sum are 11 and 12.

An integer is a complete number, therefore, not including the decimals and fractions. And it that can be either positive, negative, or zero.

Integer examples include -5, 1, 5, 8, 97, and 3,043.

Non-integer numbers include: -1.43, 1 3/4, 3.14, and more.

Let the first of the two consecutive integers be represented by x. Then, the other number can be written as (x+1).

Now, the product and the sum of the two numbers can be written as,

Product = x(x+1) = x² + x

Sum = x + (x+1) = 2x + 1

Further, it is given that the product of two consecutive positive integers is 17 more than 5 times their sum. Therefore, we can write,

x² + x = 17 + 5(2x + 1)

x² + x = 17 + 10x + 5

x² + x - 17 - 10x - 5 = 0

x² - 9x - 22 = 0

x² - 11x + 2x - 22 = 0

x(x - 11) + 2(x - 11) = 0

(x - 11)(x + 2) = 0

x = -2, 11

Thus, there are two numbers that can fulfil the given condition but since it is mentioned that we only need two positive consecutive integers. Therefore, the two numbers are 11 and 12.

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Answer:

15

Step-by-step explanation:

Option 3, (15)

hope it helps!

Thus we only take the positive root:x = 27/8 = 3.375Answer: 3.375 feet.

The problem states that a rectangular garden that measures 50 feet wide and 45 feet long is enclosed by a uniform width of x feet paved path. To solve the problem,

we can use the formula of the combined area of the garden and the paved path and equate it to 2646 square feet. The combined area is computed by adding the area of the garden and the area of the paved path.

Garden area:Length of garden = 45 ftWidth of garden = 50 ftArea of garden = Length x Width= 45 x 50= 2250 square feet

Paved path:

If the garden has a uniform width of x feet paved path, then the width of the paved path would be x + 2x + x= 4x. The width is multiplied by 2 because there are two widths surrounding the garden.

Length of paved path = length of garden + 2 (width of paved path)= 45 + 2 (4x)= 8x + 45Width of paved path = width of garden + 2 (width of paved path)= 50 + 2 (4x)= 8x + 50

The area of the paved path is computed by subtracting the area of the garden from the combined area.Area of paved path = Combined area - Garden area2646 square feet

= (8x + 45) (8x + 50) - 2250= 64x² + 760x + 675

We then solve for the value of x by factoring the quadratic equation.2646 square feet = 64x² + 760x + 6752646 - 2646

= 64x² + 760x + 675 - 264664x² + 760x - 1971

= 0(8x - 27) (8x + 73) = 0

The value of x cannot be negative,

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Answer:

I think the answer is 42.

Step-by-step explanation:

If there is a 50% increase in members then the number of members would go up 14. Adding 14 to 28, you come away with 42.

Sorry if it's wrong.

Answer:

3i√5 is the awnser so it is c

Step-by-step explanation:

i am sure of it

Using Algebraic expression solution ,

The cost of three hats is $18 .

We have given that,

A hat company charge for design fee $4 per hat .

i.e design fee of one hat = $4

total cost of 5 hats = $30

let the cost of one hat without design fee be $x and total cost of one hat is $(x+4) .

using the above statement, we get an algebraic expression,

5( x+ 4)= 30

we solve the above algebra expression,

=> 5x + 20 = 30

=> 5x = 10

=> x = 2

so, cost of a hat without design fee is $2

and total cost of one hat is $6.

we have to calculate cost of 3 hats .

cost of one hat in hat company= $ 6

cost of three hats in hat company= $(6×3)

= $18

Hence, the total cost of 3 hats is $18.

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To begin, let's recall that synthetic division is a method used to divide a polynomial by a linear factor (i.e. a binomial of the form x-a, where a is a constant). The result of synthetic division is the quotient of the division, which is a polynomial of one degree less than the original polynomial.

In this case, we are given that x is a solution of a third-degree polynomial equation. This means that the polynomial can be factored as (x-r)(ax^2+bx+c), where r is the given solution and a, b, and c are constants that we need to determine.

To use synthetic division, we will divide the polynomial by x-r, where r is the given solution. The result of the division will give us the coefficients of the quadratic factor ax^2+bx+c.

Here's an example of how to do this using synthetic division:

Suppose we are given the polynomial P(x) = x^3 + 2x^2 - 5x - 6 and we know that x=2 is a solution.

1. Write the polynomial in descending order of powers of x:

P(x) = x^3 + 2x^2 - 5x - 6

2. Set up the synthetic division table with the given solution r=2:

2 | 1  2  -5  -6

3. Bring down the leading coefficient:

2 | 1  2  -5  -6
  ---
   1

4. Multiply the divisor (2) by the result in the first row, and write the product in the second row:

2 | 1  2  -5  -6
  ---
   1  2

5. Add the second row to the next coefficient in the first row, and write the sum in the third row:

2 | 1  2  -5  -6
  ---
   1  2 -3

6. Multiply the divisor by the result in the third row, and write the product in the fourth row:

2 | 1  2  -5  -6
  ---
   1  2 -3
       4

7. Add the fourth row to the next coefficient in the first row, and write the sum in the fifth row:

2 | 1  2  -5  -6
  ---
   1  2 -3
       4 -2

The final row gives us the coefficients of the quadratic factor: ax^2+bx+c = x^2 + 2x - 3. Therefore, the factorization of P(x) is

P(x) = (x-2)(x^2+2x-3).

To find the real solutions of the equation, we can use the quadratic formula or factor the quadratic further:

x^2 + 2x - 3 = (x+3)(x-1).

Therefore, the real solutions of the equation are x=2, x=-3, and x=1.

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Given that the correlation between two variables is r=0.23. We need to find out the new correlation that would exist if the following three changes are made to the existing variables: All values of the x-variable are added by 0.14. All values of the y-variable are doubled Interchanging the two variables.  the correct option is B. 0.37.

The effect of changing the variables on the correlation coefficient between the two variables can be determined using the following formula: `r' = (r * s_x * s_y) / s_u where r' is the new correlation coefficient, r is the original correlation coefficient, s_x and s_y are the standard deviations of the two variables, and s_u is the standard deviation of the composite variable obtained by adding the two variables after weighting them by their respective standard deviations.

If we assume that the x-variable is the original variable, then the new values of x and y variables would be as follows:x' = x + 0.14 (since all values of the x-variable are added by 0.14)y' = 2y (since every value of the y-variable is doubled)Now, the two variables are interchanged. So, the new values of x and y variables would be as follows:x" = y'y" = using these values, we can find the new correlation coefficient, r'`r' = (r * s_x * s_y) / s_u.

To find the new value of the standard deviation of the composite variable, s_u, we first need to find the values of s_x and s_y for the original and transformed variables respectively. The standard deviation is given by the formula `s = sqrt(sum((x_i - mu)^2) / (n - 1))where x_i is the ith value of the variable, mu is the mean value of the variable, and n is the total number of values in the variable.

For the original variables, we have:r = 0.23s_x = standard deviation of x variable = s_y = standard deviation of y variable = We do not have any information about the values of x and y variables, so we cannot calculate their standard deviations. For the transformed variables, we have:x' = x + 0.14y' = 2ys_x' = sqrt(sum((x_i' - mu_x')^2) / (n - 1)) = s_x = standard deviation of transformed x variable` = sqrt(sum(((x_i + 0.14) - mu_x')^2) / (n - 1)) = s_x'y' = 2ys_y' = sqrt(sum((y_i' - mu_y')^2) / (n - 1)) = 2s_y = standard deviation of transformed y variable` = sqrt(sum((2y_i - mu_y')^2) / (n - 1)) = 2s_yNow, we can substitute all the values in the formula for the new correlation coefficient and simplify:

r' = (r * s_x * s_y) / s_ur' = (0.23 * s_x' * s_y') / sqrt(s_x'^2 + s_y'^2)r' = (0.23 * s_x * 2s_y) / sqrt((s_x^2 + 2 * 0.14 * s_x + 0.14^2) + (4 * s_y^2))r' = (0.46 * s_x * s_y) / sqrt(s_x^2 + 0.0396 + 4 * s_y^2)Now, we can substitute the value of s_x = s_y = in the above formula:r' = (0.46 * * ) / sqrt( + 0.0396 + 4 * )r' = (0.46 * ) / sqrt( + 0.1584 + )r' = (0.46 * ) / sqrt(r' = (0.46 * ) / sqrt(r' = (0.46 * ) / sqrt(r' = r' = Therefore, the new correlation coefficient, r', would be approximately equal to.

Hence, the correct option is B. 0.37.

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Answer:

homedogxD: 74 + x + 62 + 55 = 180. So x = -11

Step-by-step explanation:

This is from homedogxD but no one put it as an answer so..................

Answer:

independent

Step-by-step explanation:

The two events are independent because the seat that Martin selects does not affect the seats that are left for Eric to pick.

An Independent Event is defined as if the outcome of one event has no bearing on the outcome of the other, the two events are said to be independent events. Or, we may say that an event is considered independent if it does not affect the probability of another event. Probability-independent events mirror actual occurrences.

Martin and Eric choose their seats as they get ready to ride a roller coaster together. Eric chooses a seat from the various options after Martin does.

Because Martin's choice of seat has no impact on the seats that Eric has left to choose from, the two events are independent of one another.

Hence, the two events are independent because the seat that Martin selects does not affect the seats that are left for Eric to pick.

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1. The coordinate of P are (a, b).

2. The coordinate of Q are (3a, b).

3. The coordinate of R are (2a, 0).

4. The length of PR = √a² + b²

5. The length of QR = √a² + b²

What is Coordinates?

A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

Given that;

Andrew is writing a coordinate proof to show that the triangle formed by connecting the midpoints of the sides of an isosceles triangle is itself an isosceles triangle.

Now,

Since, P is the midpoint of DE.

And, The coordinate of D and E are (2a, 2b) and (0, 0) respectively.

Hence, By the definition of midpoint we get;

The coordinate of P = ( (2a + 0) / 2, (2b + 0) / 2)

                             = ( a , b)

Since, Q is the midpoint of DF.

And, The coordinate of D and F are (2a, 2b) and (4a, 0) respectively.

Hence, By the definition of midpoint we get;

The coordinate of Q = ( (2a + 4a) / 2, (2b + 0) / 2)

                             = ( 3a , b)

Since, R is the midpoint of EF.

And, The coordinate of E and F are (0, 0) and (4a, 0) respectively.

Hence, By the definition of midpoint we get;

The coordinate of R = ( (4a + 0) / 2, (0 + 0) / 2)

                             = ( 2a , 0)

Since, The coordinate of P and R are (a, b) and (2a , 0).

So, The length of PR = √(2a - a)² + (0 - b)²

The length of PR = √a² + b²

And, The coordinate of Q and R are (3a, b) and (2a , 0).

So, The length of QR = √(2a - 3a)² + (0 - b)²

The length of QR = √a² + b²

Hence, The expression for the length of PR and QR are same.

So, The △PQR is isosceles.

Therefore,

1. The coordinate of P are (a, b).

2. The coordinate of Q are (3a, b).

3. The coordinate of R are (2a, 0).

4. The length of PR = √a² + b²

5. The length of QR = √a² + b²

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Answer:

Remove the radical by raising each side to the index of the radical.

x=−6+8i,−6−8i

Step-by-step explanation:

As we have proved that the Fourier transform F(ω), with the only difference being the sign of ω

Firstly, let us recall what the Fourier transform is. It is a mathematical technique that decomposes a function into its frequency components. The Fourier transform is defined as:

F(ω) = ∫ f(t) \(e^{(-iwt)}\)dt

where f(t) is the function being transformed, ω is the frequency, i is the imaginary unit, and e^(-iωt) is a complex exponential function. The inverse Fourier transform is defined as:

f(t) = (1/2π) ∫ F(ω) \(e^{(iwt)}\) dω

where F(ω) is the Fourier transform of f(t).

Now, let's move on to the symmetry properties of the Fourier transform.

If f(t) is an even function, meaning f(-t) = f(t), then F(ω) is also even, meaning F(-ω) = F(ω).

To prove this, we start with the definition of the Fourier transform and substitute -t for t:

F(-ω) = ∫ f(-t)\(e^{(iwt)}\) dt

Then, using the even symmetry of f(t), we can replace f(-t) with f(t):

F(-ω) = ∫ f(t) \(e^{(iwt)}\) dt

Therefore, F(-ω) = F(ω) if f(t) is even.

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Given that we have to find the probability of each of the following when we roll a pair of fair dice:

Probability of getting a sum of 1 = 0

Probability of getting a sum of 5= {4}/{36}=0.1111

Probability of getting a sum of 12= {1}/{36}= 0.0278

Explanation: When we roll a pair of dice, there are 36 possible outcomes or events. When we roll the dice, the number on the dice will be an integer from 1 to 6.

The following table represents the possible outcome when we roll a pair of dice.

There is only one way to obtain the sum of 1, i.e., when both dice show 1. As there is only one way, the probability is 0.

There are 4 possible ways to obtain the sum of 5. They are (1,4),(2,3),(3,2),(4,1). The probability is 4/36.

There is only one way to obtain the sum of 12, i.e., when both dice show 6. As there is only one way, the probability is 1/36.

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Using Lagrange multipliers, the largest possible volume of a rectangular box can be found with a given diagonal length l.

Let's denote the dimensions of the rectangular box as length (L), width (W), and height (H). The volume (V) of the box is given by V = LWH. The constraint equation is the Pythagorean theorem: L² + W² + H² = l², where l is the given diagonal length.

To find the largest possible volume, we can set up the following optimization problem: maximize the volume function V = LWH subject to the constraint equation L² + W² + H² = l².

Using Lagrange multipliers, we introduce a new variable λ (lambda) and set up the Lagrangian function:

L = V + λ(L² + W² + H² - l²).

Next, we take partial derivatives of L with respect to L, W, H, and λ, and set them equal to zero to find critical points. Solving these equations simultaneously, we obtain the values of L, W, H, and λ.

By analyzing these critical points, we can determine whether they correspond to a maximum or minimum volume. The critical point that maximizes the volume will give us the largest possible volume of the rectangular box with a diagonal length l.

By utilizing Lagrange multipliers, we can optimize the volume function while satisfying the constraint equation, enabling us to determine the dimensions of the rectangular box that yield the maximum volume for a given diagonal length.

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the dimensions of the rectangle are L = 15 inches and W = 15 inches, and the minimum perimeter is:    P = 2L + 2W = 60 inches.

Let the length and width of the rectangle be L and W, respectively, so that the area of the rectangle is A = LW = 225. We want to find the dimensions of the rectangle with minimum perimeter P = 2L + 2W, and then find the minimum perimeter.

Using the given area, we can solve for one of the variables in terms of the other:

L = 225/W

Substituting this expression for L into the expression for the perimeter, we get:

P = 2(225/W) + 2W

Taking the derivative of P with respect to W and setting it equal to zero to find the minimum, we get:

\(dP/dW = -450/W^2 + 2 = 0\)

Solving for W, we get:

W^2 = 225

Since W must be positive (it is a length), we take the positive square root:

W = 15

Substituting this value of W back into the expression for L, we get:

L = 225/W = 15

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Answer:

12 miles

Step-by-step explanation:

Total miles = Length of trail ×

Number of times she rode

Total miles = 3 miles × 4 times

Total miles = 12 miles

Mika traveled a total of 12 miles.

The correct statement is Most of the songs were between 3 minutes and 3.8 minutes long.

Based on the given information from the graph, we can determine the following:

The graph shows an upward trend from 2.8 to 3.4 on the horizontal axis.

Then, there is a downward trend from 3.4 to 4 on the horizontal axis.

From this, we can conclude that most of the songs played during the one-hour interval were between 3 minutes and 3.8 minutes long. This is because the upward trend indicates an increase in length from 2.8 to 3.4, and the subsequent downward trend suggests a decrease in length from 3.4 to 4.

Therefore, the correct statement is:

Most of the songs were between 3 minutes and 3.8 minutes long.

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Answer:

A

Step-by-step explanation: