Write the terms in standard form. Then state the degree of the polynomial and the leading coefficient (LC).

The general solution is [tex]$$y=c_3 e^x+c_4 e^{-x}+\frac{1}{2} x \sinh x \text {. }$$[/tex]

Consider the following differential equation: y'' − y = cosh x

The given differential equation in the operator form will be: (D^2-1)y=coshx

The auxiliary equation of the corresponding homogeneous equation will be:

m^2-1 = 0

=>m^2=1

=>m=+1,-1

So, the complementary function will be [tex]y_c=c_1 e^x+c_2 e^{-x}.\alpha[/tex]

Now, let [tex]y_1=e^x, y_2=e^{-x}.[/tex]

Compute the Wronskian determinant:

[tex]$$\begin{aligned}W\left(y_1, y_2\right) & =\left|\begin{array}{ll}y_1 & y_2 \\y_1^{\prime} & y_2^{\prime}\end{array}\right| \\& =\left|\begin{array}{ll}e^x & e^{-x} \\e^x & -e^{-x}\end{array}\right| \\& =-e^x e^{-x}-e^x e^{-x} \\& =-2\end{aligned}$$[/tex]

Since the given differential equation is already in the form [tex]$y^{\prime \prime}+P(x) y^{\prime}+Q(x) y=f(x)$[/tex] (that is, the coefficient of y'' is 1 ),

So identify:

f(x)=cosh x

Then, calculate the values of u_1' and u_2', using formulas as follows:

[tex]$$\begin{aligned}u_1^{\prime} & =-\frac{y_2 f(x)}{W} & u_2^{\prime} & =\frac{y_1 f(x)}{W} \\& =-\frac{e^{-x} \cdot \cosh x}{-2} & & =\frac{e^x \cdot \cosh x}{-2} \\& =\frac{e^{-x} \cdot\left(\frac{e^x+e^{-x}}{2}\right)}{2} & \text { and } & =-\frac{e^x \cdot\left(\frac{e^x+e^{-x}}{2}\right)}{2} \\& =\frac{1}{4}\left(1+e^{-2 x}\right) & & =-\frac{1}{4}\left(e^{2 x}+1\right)\end{aligned}$$[/tex]

Integrate on both sides of u_1' and u_2' with respect to ' x', to get:

[tex]$$u_1=\frac{x}{4}-\frac{e^{-2 x}}{8} \text { and } u_2=-\frac{x}{4}-\frac{e^{2 x}}{8} .$$[/tex]

So, the particular solution is,

[tex]$$\begin{aligned}y_p & =u_1 y_1+u_2 y_2 \\& =\left(\frac{x}{4}-\frac{e^{-2 x}}{8}\right) e^x+\left(-\frac{x}{4}-\frac{e^{2 x}}{8}\right) e^{-x} \\& =\frac{1}{4} x\left(e^x-e^{-x}\right)-\frac{1}{8}\left(e^{-x}+e^x\right) \\& =\frac{1}{2} x \sinh x-\frac{1}{8}\left(e^{-x}+e^x\right)\end{aligned}$$[/tex]

Thus, the general solution of the given differential equation is,

[tex]$$\begin{aligned}y & =y_c+y_p \\& =c_1 e^x+c_2 e^{-x}+\frac{1}{2} x \sinh x-\frac{1}{8} e^{-x}-\frac{1}{8} e^x \\& =\left(c_1-\frac{1}{8}\right) e^x+\left(c_2-\frac{1}{8}\right) e^{-x}+\frac{1}{2} x \sinh x \\& =c_3 e^x+c_4 e^{-x}+\frac{1}{2} x \sinh x\end{aligned}$$[/tex]

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The sample space for the experiment, S= {Q, R, S, T}.

A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.

Given:

From the Spinner we can see four portions as

Q(Orange)

R(Green)

S(Blue)

T(Purple)

So, if the spinner spined then it will point Q, R, S or T which is the Sample space for the Spinner.

Hence, the Sample space is {Q, R, S, T}.

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Answer: B: S = {Q, R, S, T}

Step-by-step explanation:

This is the only answer listed with the letters on the spinner.

The prime factor trees for 24 and 84 are attached in the image below. The Lowest Common Multiple of 24 and 84 is 168.

A natural number other than 1 whose only factors are 1 and itself is said to be a prime factor. In actuality, the first few prime numbers are 2, 3, 5, 7, 11, and so forth. The lowest of two or more natural numbers' common multiples is referred to as the Lowest Common Multiple (LCM). A prime factor is a natural integer, other than 1, whose sole factors are itself and 1. Actually, 2, 3, 5, 7, 11, and so forth are the first few prime numbers. The term "Lowest Common Multiple" refers to the common multiple of two or more natural integers that is the lowest (LCM).

The prime factors of 24 are: 2,2,2 and 3.

The prime factors of 84 are: 2,2,3 and 7.

Now, The Lowest Common Multiple (LCM) of 24 and 84 is calculated using the prime factors of both the numbers.

LCM= 2*2*2*3*7= 168

Hence, The Lowest Common Multiple of 24 and 84 is 168.

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Answer:

The restaurant owns 225 forks

Step-by-step explanation:

We can get our answer by displaying as an equality

9 to 4/100

x to 100/100

You can multiply 4 by 25 to get 100 so to get the amount of forks, multiply 9 by 25

9 * 25 = 225

Answer:

225 forks

Step-by-step explanation:

9 to 4/100

x to 100/100

Multiply 4 by 25 to get 100 so to get the number of forks, multiply 9 by 25!

9 * 25 = 225

Hope this helps.

At 2 miles, both the bus and the taxi will cost $6.

What is an equation?

A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.

Given that the fare on bus is $3 per mile.

The taxi fare of taxi is $4. 00 plus $1. 00 per mile.

Assume that at x miles both fare will be the same.

The fare for x miles in bus is $3x.

The fare for x miles in taxi is $4 + $x

Equate to the fares:

3x = 4 + x

Subtract x from both sides:

3x - x = 4

2x = 4

Divide both sides by 2:

x = 4/2

x = 2.

The fare for 2 miles in bus is $3×2 = $6.

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The error interval for the length ,x, of the material is [8.25, 8.35)

We know that error interval is nothing but the limits of accuracy when a number has been rounded.

It is the range of possible values that a number could have been before a number was rounded.

Here, The length L of a some material is 8.3 meters.

The length of the material has been rounded to 8.3 to one decimal place.

The number below 8.3 is 8.2 and the number above is 8.4

Consider a half way between 8.2 and 8.3 for the lower bound and half way between 8.3 and 8.4 for the upper bound.

This means, the length of the material must have been between 8.25 meters and 8.35 meters to round to 8.3 meters to one decimal place.

Therefore, the required error interval would be 8.25 ≤ x < 8.35

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The sine of an acute angle in a right triangle is the ratio of the angle of the triangle to the opposite side of the hypotenuse.

Trigonometric ratios:

Trigonometric ratios are the ratios of the lengths of the sides of a triangle. These ratios in trigonometry relate the aspect ratio of a right triangle to each angle. A basic trigonometric ratio is the ratio of sin, cos, and tan, or the ratio of sin, cos, and tangent. Other important trigonometric ratios cosec, sec, and cot can be derived from sin, cos, and tan, respectively.

Acute Sine Ratio:

The tangent function examines the ratio of his two legs of a right triangle. You can also use the ratio of leg length divided by hypotenuse length.

In a right triangle, each angle has an opposite side and an adjacent side. And there is always a hypotenuse. On the opposite side and the hypotenuse we are dealing with the sine ratio. The sine of an angle is the ratio of the length of the opposite side divided by the length of the hypotenuse. In a right triangle, the sine of an angle is the ratio of the length of the opposite side divided by the length of the hypotenuse.

Let's examine some properties of the sine ratio using the Pythagorean theorem. For a right triangle, let the lengths of the triangle be a, b, and c. where a is the length of the other side of A and b is the length of the other side of B.

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Probability that the Captain does NOT hit the pirate's ship × Probability that the pirate hits the Captain's ship = (2/5) × (1/5) = (2/25) = 0.08

Probability theory, a subfield of mathematics, gauges the likelihood of an occurrence or a claim being true.

The probability that the pirate hits the Captain's ship, but the Captain misses

= 2/25) = 0.08

P(C) = (3/5) is the likelihood that the captain will strike the pirate ship.

P(C') = 1 - (3/5) = (2/5) is the probability that the captain will miss the pirate ship.

P(p) = (1/5) is the likelihood that the pirate will sink the captain's vessel.

P(p') = 1 - (1/5) = (4/5) is the probability that the pirate will miss the captain's ship.

The likelihood that the pirate will hit the captain's ship while the captain misses is increased if they both fire their cannons at the same moment.

P(C' n p) is he needed probability.

P(C' n p) = P(C') P since the two events are independent of one another (p) The likelihood that the pirate will sink the captain's vessel but that the capain will miss = (Chance that the Captain misses the pirate's ship) (Chance that the pirate hits the Captain's ship) = (2/5) × (1/5) = (2/25) = 0.08

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Answer:

Step-by-step explanation:

54×54

The range of the function y = √(x +5)  is y ≥ 0

The domain of a function is the set of all possible values of x

The range of a function is the complete set of all possible resulting values of y, after we have substituted the domain.

we have

y = √(x +5)

Remember that the radicand of the function must be greater than or equal to zero

so (x +5) ≥ 0

solve for x

x ≥ -5

The domain is the interval ----> [-5,∞)

For x= -5

y = √(-5 +5) = 0

so

The range is the interval ----> [0,∞)

y ≥ 0

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The value of each trigonometric ratio is given by :

sin A = 5/13 , cos A = 12/13, tan A = 5 / 12

cosec A = 13 /5 , sec A = 13/12 , cot A = 12 /5.

As given in the question,

Let us consider a triangle be ABC,

Figure is attached.

Side length is given by:

AC = 39

AB = 36

BC = 15

AC represents the longest side

( 36 )² + (15)² = (39)²

⇒AB² + BC² = AC²

By Pythagorean triplets ABC is right angled triangle.

All trigonometric ratio with respect to angle A is given by :

sin A = BC / AC

        = 15/39

        = 5/13

cos A = AB / AC

         = 36 /39

         = 12 /13

tan A = BC / AB

         = 5 /12

cosec A = 1/ sin A

              = 13 /5

sec A = 1/ cos A

         = 13 /12

cot A = 1 / tan A

         = 12 /5

Therefore , the value of all six trigonometric ratio is equal to

sin A = 5/13 , cos A = 12/13, tan A = 5 / 12

cosec A = 13 /5 , sec A = 13/12 , cot A = 12 /5.

The above question is incomplete , the complete question is :

Find the value of each trigonometric ratio tan A

Figure is attached.

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Answer:

Find the value of xx in the equation below.

11=

11=

\,\,x -6

x−6

Step-by-step explanation:

The work would be finished 3 times faster if they had been doing the work at the same time. The correct option is B.

Assumption;

Provided each one of the workers works for an hour which is equivalent to the time it takes to finish the hole.

In essence, the total time required to finish the ditch by 2 workers is; 2 hours.

Hence, for 3 workers working simultaneously, the time required is

1hour  = 60 minutes

However, since they took 1 hour turns on the job, it follows that; The total time used is;

Total time = 3 × 1hr = 3 hours = 180 hours

Hence they could have finished the work in;

R = 180/60 times faster =3 times faster.

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To prove the inverse :

You assemble the functions (that is, insert x into one function, plug that function into the suggested inverse function, then simplify) and check that you end up with simply "x" to establish (or refute) that two functions are the inverses of one another. The functions are inverses if you get x; otherwise, they are not.

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Total distance travelled by the family in the truck is 497.98 miles

An equation is an expression that shows the relationship between two or more numbers.

The rental of the truck has a base cost of $49.95, and $1.19 per mile. The average gas mileage is 18 miles per gallon.

Let us assume the total distance travelled was x miles, hence:

Total gas consumed = x miles / 18 miles per gallon = x/18 gallons

The cost is $3.89 per gallon, hence:

Total cost of gas = $3.89 per gallon * x/18 gallons = 3.89x/18

Total cost of journey = 1.19x + 3.89x/18 + 49.95 = 2531x/1800 + 49.95

If $724.85 was charged, hence:

724.85 = 2531x/1800 + 49.95

x = 497.98

Total distance is 497.98 miles

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Reason:

The tickmarks tell us that the pair of hypotenuses are congruent.

The shared horizontal sides at the bottom are the pair of congruent legs of the right triangles. Therefore, we can use the hypotenuse leg (HL) theorem. This theorem only works for right triangles.

On solving the provided question, we can say that by  linear equation 10/3 pounds of peanuts and how many pounds of raisins should you use.

A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept.When an equation has the formula y=mx+b, with m denoting the slope and b the y-intercept, it is referred to as being linear.

here,

the linear equation that is formed is  

10p +7 (10 - p) = 80

10p + 70 - 7p = 80

3p = 70

p = 10/3

10/3 pounds of peanuts and how many pounds of raisins should you use

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Answer:

To find out how many cups a single jug of water fills, you can divide the total number of cups filled by the number of jugs:

184 cups / 23 jugs = <<184/23=8>>8 cups/jug

This means that one jug of water fills 8 cups.

Step-by-step explanation:

The equation 1.6: n = 0.16 can be written as: [tex]1.6 = 2.511^n[/tex]

And we know that this is correct because we can check that it is the equivalent by multiplying [tex]2.511^n[/tex] by itself n times which results in 1.6.

What is an exponent?

An exponent is a mathematical operation that represents the number of times a value is multiplied by itself. It is usually written as a small number raised to the power of a larger number, such as 2^3, which means 2 multiplied by itself 3 times, or 2 * 2 * 2 = 8.

To write the equation 1.6: n=0.16 using an exponent, we can divide both sides of the equation by 1.6 to get: n = 0.16/1.6 = 0.1

And then we can raise the base 10 to the power of n: [tex]10^n = 10^{0.1} = 2.511[/tex]

Hence, the equation 1.6: n = 0.16 can be written as: [tex]1.6 = 2.511^n[/tex]

And we know that this is correct because we can check that it is the equivalent by multiplying [tex]2.511^n[/tex] by itself n times which results in 1.6.

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The equation 1.6: n = 0.16 can be written as: 1.6 = [tex]2.511^n[/tex]

And we know that this is correct because we can check that it is the equivalent by multiplying [tex]2.511^n[/tex] by itself n times which results in 1.6.

What is an exponent?

An exponent is a mathematical operation that represents the number of times a value is multiplied by itself. It is usually written as a small number raised to the power of a larger number, such as 2^3, which means 2 multiplied by itself 3 times, or 2 * 2 * 2 = 8.

To write the equation 1.6: n=0.16 using an exponent, we can divide both sides of the equation by 1.6 to get: n = 0.16/1.6 = 0.1

And then we can raise the base 10 to the power of n: [tex]10^n=10^{0.1}=2.511[/tex]

Hence, the equation 1.6: n = 0.16 can be written as: 1.6 = [tex]2.511^n[/tex]

And we know that this is correct because we can check that it is the equivalent by multiplying [tex]2.511^n[/tex] by itself n times which results in 1.6.

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The numeric value for each trigonometric expression is given as follows:

3. [tex]\sin{\left(-\frac{5\pi}{6}\right)}\cos{\left(\frac{2\pi}{3}\right)} - \cos{(-\pi)} = \frac{5}{4}[/tex]

4. [tex]\frac{\sin{\left(\frac{\pi}{4}\right)}}{\cos^2{\left(\frac{\pi}{3}\right)}} = 2\sqrt{2}[/tex]

The first trigonometric expression for this problem is defined as follows:

[tex]\sin{\left(-\frac{5\pi}{6}\right)}\cos{\left(\frac{2\pi}{3}\right)} - \cos{(-\pi)}[/tex]

The numeric value of each term of the expression is given as follows, using equivalent angles on the trigonometric circle:

Hence:

[tex]\sin{\left(-\frac{5\pi}{6}\right)}\cos{\left(\frac{2\pi}{3}\right)} - \cos{(-\pi)} = \left(-\frac{1}{2}\right)^2 + 1 = \frac{1}[4} + 1 = \frac{5}{4}[/tex]

For item 4, the angles are on the first quadrant, hence:

Hence the numeric value of the expression is given as follows:

[tex]\frac{\frac{\sqrt{2}}{2}}{\left(\frac{1}{2}\right)^2} = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}[/tex]

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The two statements that are true about the table are:

A) "The range is {15.80, 31.60, 43.45, 57.25} ."

B) "The set of independent values is {4, 8, 11, 15}."

Here we have the table for number of gallons and the costs of these:

g          c

4         15.80

8         31.60

11         43.45

15         57.25

The statements that are true about that table are:

"The range is {15.80, 31.60, 43.45, 57.25} ."

This is true, the cost is the output of the table, thus, the range is that.

"The set of independent values is {4, 8, 11, 15}."

This is also called the domain, which is the input, in this case, the number of gallons of gas.

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Answer: 12

Step-by-step explanation:

The smaller triangles on the right can be proven congruent, so the other side also equals 6. 6+6=12

Isolating for a variable: rearranging an equation so that a variable is on its own. This is done by performing inverse operations until a variable is isolated.

[tex]P-9=-3w+9-9[/tex]

[tex]p-9=-3w[/tex]

[tex]\frac{p-9}{-3}=\frac{-3w}{-3}[/tex]

[tex]-\frac{P-9}{3} =w[/tex]

The measure of the central tendency is 67 as the data in the set are integers.

A single number that seeks to characterize a set of data by pinpointing the center position within that set of data is referred to as a measure of central tendency.

As a result, measures of central location are occasionally used to refer to measures of central tendency. They also fit within the category of summary statistics. You are probably most familiar with the mean (sometimes known as the average), but there are additional central tendency measures, including the median and the mode.

Given, A data set 70, 67, 75, 60, 67, 65.

Arranging the data set in ascending order,

60, 65, 67, 67, 70, 75.

Now, The mean of the set is (60 + 65 + 67 + 67 + 70 + 75)/6.

= 67.33.

Mode of the data set 67.

So, The central tendency of the data set is 67 as the data in the set are all integers.

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Answer:

69%

Step-by-step explanation:

There are 42 teachers who work in a school. On Monday, 29 teachers were present. The percentage of the teachers that showed up for work will be:

= 29 / 42 × 100

= 69%

Answer:

Step-by-step explanation:

In any case where you're asked, "what percentage of A is B?" you can simply divide the number B by the A. The decimal you get is the answer. for example, if you're being asked what percentage 12 is of 25, you just divide 12 by 25 to get 0.48, which translates to 48 percent. in your case, 29/42 is 0.69 (rounded down from 0.6904761905) which equals 69%.

In a case where B is larger than A, you still do the same, only with the addition of the whole number. Ex: what percentage of 25 is 86?

86/25 is 3.44, so your answer is 344%.

Hope this helped :)

Answer:

(-3, -2)

Step-by-step explanation:

Notice how the lines intersect? that exact point is the solution.

Completing the square – can be used to solve any quadratic equation. It is a very important method for rewriting a quadratic function in vertex form. Quadratic formula – is the method that is used most often for solving a quadratic equation.

In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.

Here,

Any quadratic problem may be solved by completing the square. It is a critical way for expressing a quadratic function in vertex form. The quadratic formula is the most often used method for solving a quadratic problem.

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The list of the sample space is s = {Q, R, S, T}. Option B is the correct option.

What is the sample space?

A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results. Discrete or finite sample spaces are those that have a finite number of outcomes.

The random experiment's sample spaces are written in curly brackets, "{ } ".

Given that a spinner wheel has four parts. It is labeled with the names Q, R, S, and T.

The elements of sample space are Q, R, S, and T.

The sample space is a set, thus write the elements in curly brackets.

The sample space is s = {Q, R, S, T}.

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