find the gcd and lcm of 20 and 56

Answer:

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

Step-by-step explanation:

We can write the equation of a circle in the form

(x-h)^2 + (y-k)^2 = r^2

Where (h,k) is the center and r is the radius

(x--7)^2 + (y--4)^2 = 6^2

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

Answer:[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]

Step-by-step explanation:

Given two upward facing parabolas  with equations

[tex]y=6x^2 & y=x^2+2[/tex]

The two intersect at

[tex]6x^2=x^2+2[/tex]

[tex]5x^2=2[/tex]

[tex]x^2[/tex]=[tex]\frac{2}{5}[/tex]

x=[tex]\pm \sqrt{\frac{2}{5}}[/tex]

area  enclosed by them is given by

A=[tex]\int_{-\sqrt{\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left [ \left ( x^2+2\right )-\left ( 6x^2\right ) \right ]dx[/tex]

A=[tex]\int_{\sqrt{-\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left ( 2-5x^2\right )dx[/tex]

A=[tex]4\left [ \sqrt{\frac{2}{5}} \right ]-\frac{5}{3}\left [ \left ( \frac{2}{5}\right )^\frac{3}{2}-\left ( -\frac{2}{5}\right )^\frac{3}{2} \right ][/tex]

A=[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]

The area of the enclosed region is -√2/15 square units.

To find the area of the enclosed region, we need to find the points of intersection between the two equations y=6x^2 and y=x^2+2. Setting them equal to each other:

6x^2 = x^2 + 2

5x^2 = 2

x^2 = 2/5

x = ±√(2/5)

Substituting these values of x back into one of the equations, we can find the corresponding y values:

For x = √(2/5), y = 6(√(2/5))^2 = 6(2/5) = 12/5

For x = -√(2/5), y = 6(-√(2/5))^2 = 6(2/5) = 12/5

Now we can find the area of the enclosed region by calculating the definite integral of y=6x^2 - (x^2+2) from x = -√(2/5) to x = √(2/5). This can be done using the fundamental theorem of calculus:

∫[√(2/5), -√(2/5)] [6x^2 - (x^2+2)] dx = ∫[√(2/5), -√(2/5)] (5x^2 - 2) dx = [5/3x^3 - 2x] [√(2/5), -√(2/5)] = 2(√(2/5))^3 - 5/3(√(2/5))^3 = 4√2/15 - 5√2/15 = -√2/15

Therefore, the area of the enclosed region is -√2/15 square units.

https://brainly.com/question/35257755

#SPJ3

Answer:

Mustapha can arrange 15 bouquets per day.

Step-by-step explanation:

Neneh can arrange 20 bouquets per day.

Together Neneh and Mustapha can arrange 35 bouquets per day.

So,  Mustapha can arrange [tex]35-20=15[/tex] bouquets per day.

Therefore, Mustapha’s marginal product is 15 bouquets.

Answer:

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

Step-by-step explanation:

It is given that a data set lists earthquake depths. The summary statistics are

[tex]n=300[/tex]

[tex]\overline{x}=5.89km[/tex]

[tex]s=4.44km[/tex]

Level of significance = 0.01

We need to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 5.00.

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The formula for z-value is

[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]z=\frac{5.89-5.00}{\frac{4.44}{\sqrt{300}}}[/tex]

[tex]z=\frac{0.89}{0.25634351952}[/tex]

[tex]z=3.4719[/tex]

The p-value for z=3.4719 is 0.000517.

Since the p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

The null and alternative hypotheses are H0: µ = 5.00 km and HA: µ ≠ 5.00 km. A t-test is used to calculate the test statistic, and the p-value is compared to the significance level of 0.01 to either reject or not reject the null hypothesis. The final conclusion is made in consideration of the original claim.

In statistics, hypothesis testing is a tool for inferring whether a particular claim about a population is true. For this question about earthquake depths, we would start by setting our null hypothesis (H0) and our alternative hypothesis (HA).

The null hypothesis would be H0: µ = 5.00 km, and the alternative hypothesis would be HA: µ ≠ 5.00 km.

The test statistic can be calculated using a t test, since we are dealing with a sample mean and we know the sample standard deviation (sequals4.44 km).

The p-value associated with this test statistic would then be calculated, and compared to the significance level of 0.01. If the p-value is less than 0.01, we reject the null hypothesis. If, however, the p-value is greater than 0.01, we cannot reject the null hypothesis.

The final conclusion must be stated in terms of the original claim. If we reject the null hypothesis, we conclude that the evidence supports the claim that the mean earthquake depth is not equal to 5.00 km (supporting the alternative hypothesis). If we do not reject the null hypothesis, we conclude that the evidence does not support the claim that the mean is not 5.00 km. The data does not provide sufficient evidence to support a conclusion that the mean earthquake depth is different than 5.00 km.

https://brainly.com/question/34171008

#SPJ11

Answer:

540

Step-by-step explanation:

we have given E=0.3

σ = 27 hours

100(1-α)%=99%

from here α=0.01

using standard table [tex]Z_\frac{\alpha }{2}=Z_\frac{0.01}{2}=2.58[/tex]

[tex]n=\left ( Z_\frac{\alpha }{2}\times \frac{\sigma }{E} \right )^{2}[/tex] =

[tex]\left ( 2.58\times \frac{27}{3} \right )^{2}[/tex]

n = [tex]23.22^{2}[/tex]

n=539.16

n can not be in fraction so n=540

To obtain a 99% confidence interval with a margin of error of 3 hours, at least 602 components must be sampled.

In order to determine the number of components that must be sampled so that a 99% confidence interval will have a margin of error of 3 hours, we can use the formula:

n = (z * s / E)^2

Where:

n = sample size

z = z-value corresponding to the desired confidence level (in this case, 99% confidence level)

s = population standard deviation

E = margin of error

Plugging in the given values, we have:

n = (2.576 * 27 / 3)^2

n = 601.3696

Rounding up to the nearest whole number, we need to sample at least 602 components.

https://brainly.com/question/34700241

#SPJ3

Answer:

[tex]a) \quad A=\left[\begin{array}{cc}8&5\\5&-1\end{array}\right] \\\\\\b) \quad \{-1+5x, 2+x\}[/tex]

Step-by-step explanation:

To compute the representation matrix A of f with respect the basis {1,x} we first compute

[tex]f(1)=f(1+0x)=(8\cdot 1 + 2 \cdot 0) + (5 \cdot 1 - 0)x=8+5x \\\\f(x)=f(0 + 1\cdot x)=(8 \cdot 0 + 2\cdot 1)+(5 \cdor 0 - 1)x = 2-1[/tex]

The coefficients of the polynomial f(1) gives us the entries of the first column of the matrix A, where the first entry is the coefficient that accompanies the basis element 1 and the second entry is the coefficient that accompanies the basis element x. In a similar way, the coefficients of the polynomial f(x) gives us the the entries of the second column of A. It holds that,

[tex]A=\left[\begin{array}{ccc}8&5\\2&-1\end{array}\right][/tex]

(b) First, note that we are using a one to one correspondence between the basis {1,x} and the basis {(1,0),(0,1)} of [tex]\mathbb{R}^2[/tex].

To compute a basis P1 with respect to which f has a diagonal matrix, we first have to compute the eigenvalues of A. The eigenvalues are the roots of the characteristic polynomial of A, we compute

[tex]0=\det\left[\begin{array}{ccc}8-\lambda & 2\\ 5 & -1-\lambda \end{array}\right]=(8 - \lambda)(-1-\lambda)-18=(\lambda - 9)(\lambda +2)[/tex]

and so the eigenvalues of the matrix A are [tex]\lambda_1=-2 \quad \text{and} \quad \lambda_2=9[/tex].

After we computed the eigenvalues we use the systems of equations

[tex]\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right] = \left[\begin{array}{c}-2x_1\\-2x_2\end{array}\right] \\\\\text{and} \\\\\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1 \\ x_2\end{array}\right]=\left[\begin{array}{cc}9x_1\\2x_1\end{array}\right][/tex]  

to find the basis of the eigenvalues. We find that [tex]v_1=(-1,5 )[/tex] is an eigenvector for the eigenvalue -2 and that [tex]v_2=(2,1)[/tex] is an eigenvector for the eigenvalue 9. Finally, we use the one to one correspondence between the [tex]\mathbb{R}^2[/tex] and the space of liear polynomials to get the basis [tex]P1=\{-1+5x, 2+x\}[/tex] with respect to which f is represented by the diagonal matrix [tex]\left[\begin{array}{ccc}-2&0\\0&9\end{array}\right][/tex]

Answer:

[tex]2x+y=3[/tex]

Step-by-step explanation:

Here we aer given a point (2,-1) and a line [tex]2y=x-4[/tex]. We are supposed to find the equation of the line passing through this point and perpendicular to this line.

Let us find the slope of the line perpendicular to [tex]2y=x-4[/tex]

Dividing above equation by 2 we get

[tex]y=\frac{1}{2}x-2[/tex]

Hence we have this equation in slope intercept form and comparing it with

[tex]y=mx+c[/tex] , we get Slope [tex]m = \frac{1}{2}[/tex]

We know that product of slopes of two perpendicular lines in -1

Hence if slope of line perpendicular to [tex]y=\frac{1}{2}x-2[/tex] is m' then

[tex]m\times m' =-1[/tex]

[tex]\frac{1}{2} \times m' =-1[/tex]

[tex]m'=-2[/tex]

Hence the slope of the line we have to find is -2

now we have slope and a point

Hence the equation of the line will be

[tex]\frac{y-(-1)}{x-2}=-2[/tex]

[tex]y+1=-2(x-2)[/tex]

[tex]y+1=-2x+4[/tex]

adding 2x and subtracting  on both sides we get

[tex]2x+y=3[/tex]

Which is our equation asked

Answer: This is a combination.

There are 1947792 ways to choose his students.

Step-by-step explanation:

Since we have given that

Number of students in a class = 36

Number of students selected for his English class = 6

We would use "Combination" .

As permutation is used when there is an arrangement.

whereas Combination is used when we have select r from group of n.

So, Number of ways that Prof. Jones can choose his students is given by

[tex]^{36}C_6=1947792[/tex]

Hence, there are 1947792 ways to choose his students.

Answer:

The value of [tex]x_2=-1.6923[/tex].

Step-by-step explanation:

Consider the provided information.

The provided formula is [tex]f(x)=x^3+x+6[/tex]

Substitute [tex]x_1=-2[/tex] in above equation.

[tex]f(x_1)=(-2)^3+(-2)+6[/tex]

[tex]f(x_1)=-8-2+6[/tex]

[tex]f(x_1)=-4[/tex]

Differentiate the provided function and calculate the value of [tex]f'(x_1)[/tex]

[tex]f'(x)=3x^2+1[/tex]

[tex]f'(x)=3(-2)^2+1[/tex]

[tex]f'(x)=13[/tex]

The Newton iteration formula: [tex]x_2=x_1-\frac{f(x_1)}{f'(x_1)}[/tex]

Substitute the respective values in the above formula.

[tex]x_2=-2-\frac{(-4)}{13}[/tex]

[tex]x_2=-2+0.3077[/tex]

[tex]x_2=-1.6923[/tex]

Hence, the value of [tex]x_2=-1.6923[/tex].

To use Newton's method with an initial approximation of -2 on the equation x3 + x + 6 = 0, we identify the function and its derivative. We then substitute into Newton's method's formula, x_2 = x_1 - f(x_1) / f'(x_1). This will provide us with an approximate second root, which can then be refined further.

The subject of the question pertains to Newton's method for root finding, which is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. The function in question is x3 + x + 6 = 0 and the initial approximation provided is -2.

First, in order to use Newton's method, we need to identify the function and its derivative. The function (f(x)) is x3 + x + 6. The derivative (f'(x)) would be 3x2 + 1.

Newton's method follows the formula: x_(n+1) = x_n - f(x_n) / f'(x_n).

With x_1 as -2, we substitute into the Newton method's formula to find x_2. Hence, x_2 = -2 - f(-2) / f'(-2), which results in an approximation of the root that can be further refined. Remember to round your answer to four decimal places after calculations.

https://brainly.com/question/31910767

#SPJ3

Answer:

For any value of C1 and C2, [tex]y = C1 + C2*e^{2x}[/tex] is a solution.

Step-by-step explanation:

Let's verify the solution, but first, let's find the first and second derivatives of the given solution:

[tex]y = C1 + C2*e^{2x}[/tex]

For the first derivative we have:

[tex]y' = 0 + C2*(2x)'*e^{2x}[/tex]

[tex]y' = C2*(2)*e^{2x}[/tex]

For the second derivative we have:

[tex]y'' = C2*(2)*(2x)'*e^{2x}[/tex]

[tex]y'' = C2*(2)*(2)*e^{2x}[/tex]

[tex]y'' = C2*(4)*e^{2x}[/tex]

Let's solve the ODE by the above equations:

[tex]y'' - 2y' = 0[/tex]

[tex]C2*(4)*e^{2x} - 2*C2*(2)*e^{2x} = 0[/tex]

[tex]C2*(4)*e^{2x} - C2*(4)*e^{2x} = 0[/tex]

From the above equation we can observe that for any value of C2 the equation is solved, and because the ODE only involves first (y') and second (y'') derivatives, C1 can be any value as well, because it does not change the final result.  

If [tex]W[/tex] is a random variable representing your winnings from playing the game, then it has support

[tex]W=\begin{cases}43&\text{if you draw something with value at most 3}\\-11&\text{otherwise}\end{cases}[/tex]

There are 52 cards in the deck. Only the 1s, 2s, and 3s fulfill the first condition, so there are 12 ways in which you can win $43. So [tex]W[/tex] has PMF

[tex]P(W=w)=\begin{cases}\frac{12}{52}=\frac3{13}&\text{for }w=43\\1-\frac{12}{52}=\frac{10}{13}&\text{for }w=-11\\0&\text{otherwise}\end{cases}[/tex]

You can expect to win

[tex]E[W]=\displaystyle\sum_ww\,P(W=w)=\frac{43\cdot3}{13}-\frac{11\cdot10}{13}=\boxed{\frac{19}{13}}[/tex]

or about $1.46 per game.

The expected value of the proposition is $7.31.

To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and then sum them up.

The probability of drawing a card with a value of three or less is 12/52 since there are 12 cards with values of three or less in a standard deck of 52 cards. The probability of not drawing a card with a value of three or less is 40/52.

Using these probabilities and the given payoffs, we can calculate the expected value as follows:
Expected Value = (Probability of Winning * Payoff if Win) + (Probability of Losing * Payoff if Lose)
Expected Value = (12/52 * 43) + (40/52 * -11)

Calculating this expression gives us an expected value of $7.31 (rounded to two decimal places).

https://brainly.com/question/35639289

#SPJ2

Answer:

The probability of getting the toy in any given cereal box is [tex]\frac{1}{49}[/tex].

Step-by-step explanation:

Given,

On average, we get a toy in every 49th cereal box,

That is, in every 49 boxes there is a toy,

So, the total outcomes = 49,

Favourable outcomes ( getting a toy ) = 1

Since, we know that,

[tex]\text{Probability}=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}[/tex]

Hence, the probability of getting the toy in any given cereal box = [tex]\frac{1}{49}[/tex]

Answer:

The probability of getting the toy you want in any given cereal box is of 0.0204 = 2.04%.

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected number of trials for r sucesses is:

[tex]E = \frac{r}{p}[/tex]

If, on average, you get the toy you want in every 49th cereal box, what is the probability of getting the toy you want in any given cereal box?

This means that [tex]E = 49, r = 1[/tex]

So

[tex]49 = \frac{1}{p}[/tex]

[tex]49p = 1[/tex]

[tex]p = \frac{1}{49}[/tex]

[tex]p = 0.0204[/tex]

The probability of getting the toy you want in any given cereal box is of 0.0204 = 2.04%.

[tex]2y^2\,\mathrm dx-(x+y)^2\,\mathrm dy=0[/tex]

Divide both sides by [tex]x^2\,\mathrm dx[/tex] to get

[tex]2\left(\dfrac yx\right)^2-\left(1+\dfrac yx\right)^2\dfrac{\mathrm dy}{\mathrm dx}=0[/tex]

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2\left(\frac yx\right)^2}{\left(1+\frac yx\right)^2}[/tex]

Substitute [tex]v(x)=\dfrac{y(x)}x[/tex], so that [tex]\dfrac{\mathrm dv(x)}{\mathrm dx}=\dfrac{x\frac{\mathrm dy(x)}{\mathrm dx}-y(x)}{x^2}[/tex]. Then

[tex]x\dfrac{\mathrm dv}{\mathrm dx}+v=\dfrac{2v^2}{(1+v)^2}[/tex]

[tex]x\dfrac{\mathrm dv}{\mathrm dx}=\dfrac{2v^2-v(1+v)^2}{(1+v)^2}[/tex]

[tex]x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{v(1+v^2)}{(1+v)^2}[/tex]

The remaining ODE is separable. Separating the variables gives

[tex]\dfrac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=-\dfrac{\mathrm dx}x[/tex]

Integrate both sides. On the left, split up the integrand into partial fractions.

[tex]\dfrac{(1+v)^2}{v(1+v^2)}=\dfrac{v^2+2v+1}{v(v^2+1)}=\dfrac av+\dfrac{bv+c}{v^2+1}[/tex]

[tex]\implies v^2+2v+1=a(v^2+1)+(bv+c)v[/tex]

[tex]\implies v^2+2v+1=(a+b)v^2+cv+a[/tex]

[tex]\implies a=1,b=0,c=2[/tex]

Then

[tex]\displaystyle\int\frac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=\int\left(\frac1v+\frac2{v^2+1}\right)\,\mathrm dv=\ln|v|+2\tan^{-1}v[/tex]

On the right, we have

[tex]\displaystyle-\int\frac{\mathrm dx}x=-\ln|x|+C[/tex]

Solving for [tex]v(x)[/tex] explicitly is unlikely to succeed, so we leave the solution in implicit form,

[tex]\ln|v(x)|+2\tan^{-1}v(x)=-\ln|x|+C[/tex]

and finally solve in terms of [tex]y(x)[/tex] by replacing [tex]v(x)=\dfrac{y(x)}x[/tex]:

[tex]\ln\left|\frac{y(x)}x\right|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C[/tex]

[tex]\ln|y(x)|-\ln|x|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C[/tex]

[tex]\boxed{\ln|y(x)|+2\tan^{-1}\dfrac{y(x)}x=C}[/tex]

Answer: Option 'B' is correct.

Step-by-step explanation:

Let the number of 10 cents pcs be 'x'.

Let the number of 25 cents pcs be 'y'.

Since we have given that

Total number of coins = 32

Sum of money = $5.15

As we know that

$1 = 100 cents

$5.15 = 5.15×100 = 515 cents

According to question, we get that

[tex]x+y=32-----------(1)\\\\10x+25y=\$515------------(2)[/tex]

Using the graphing method, we get that

x = 19

y = 13

So, there are 13 pcs of 25 cents.

Hence, Option 'B' is correct.

By creating a system of equations based on the total number of coins (32) and their total value (P5.15), we calculate that there are 13 pieces of 25-cent coins.

The student's question involves figuring out the number of 25-cent coins among a total of 32 coins which altogether amount to P5.15. This problem can be solved by setting up a system of equations to account for the total number of coins and the total value in pesos.

Let's denote the number of 10-cent coins as t and the number of 25-cent coins as q. We know from the problem that there are 32 coins in total, so:

(1) t + q = 32

We also know that the total value of the coins is P5.15, or 515 cents. Therefore:

(2) 10t + 25q = 515

By solving this system of equations, we can find the value of q, the number of 25-cent coins. First, we can multiply equation (1) by 10 to eliminate t when we subtract the equations:

10t + 10q = 320

Subtracting this from equation (2) gives us:

15q = 195

Dividing both sides by 15, we find that:

q = 13

So, there are 13 pieces of 25-cent coins, which corresponds to option B.

Answer:

B'(3, -8)

Step-by-step explanation:

The image is the mirror image of the trapezoid below the x-axis.

Each x-coordinate remains the same. Each y-coordinate becomes the opposite.

B'(3, -8)

Answer:

(3 , -8)

Step-by-step explanation:

The current coordinates of B. are (3,8). The x-axis is the horizontal line that runs across. This means that if the trapezoid were to be reflected, it would end up upside down. When this happens, only the y value changes its sign.

In short, your Y value would become negative, making 8 change to -8

Answer:

Jayanta needs 2 labor union groups and 8 church groups.

Step-by-step explanation:

Let c denotes churches .

Let l denotes labor unions.

We know that Jayanta can only spend 20 hours letter writing and 14 hour of follow-up.

So, equations becomes:

[tex]2c+2l=20[/tex]

[tex]c+3l=14[/tex]

And total money raised can be shown by = [tex]150c+175l[/tex]

We have to maximize [tex]150c+175l[/tex] keeping in mind that [tex]2c+2l \leq 20[/tex] and [tex]c+3l \leq 14[/tex]

We will solve the two equations:  [tex]2c+2l=20[/tex] and [tex]c+3l=14[/tex]

We get l = 2 and c = 8

And total money raised is [tex]150\times8 + 175\times2[/tex] = [tex]1200+350=1550[/tex] dollars.

Hence, Jayanta needs 2 labor union groups and 8 church groups.

Answer:

See below.

Step-by-step explanation:

5/6 is a fraction. The 5 is in the numerator, and the 6 is in the denominator.

The denominator is the number of parts the unit was divided into. In this case, the denominator is 6. That means one unit, 1, was divided into 6 equal parts. Each part is one-sixth.

The numerator is the number of those parts that you use. 5 in the numerator means to use 5 of those parts, each of which is 1/6 of 1.

In other words, 5/6 means divide 1 into 6 equal parts, and take 5 of those parts.

5/6 is the same as 5 divided by 6, so as a decimal it is 0.8333...

As a percent it is 83.333...%

Answer:

see below

Step-by-step explanation:

The horizontal line will have the same y and the y value will be constant

y = -7

The vertical line will have the same x and the x value will be constant

x = -1

Answer:

Solution is [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]

Step-by-step explanation:

the given equation y''-4y[tex]=e^{2x}[/tex] can be written as

[tex]D^{2}y-4y=e^{2x}\\\\(D^{2}-4)y=e^{2x}\\\\[/tex]

The Complementary function thus becomes

y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}

where [tex]m_{1} , m_{2}[/tex] are the roots of the [tex]D^{2}-4[/tex]

The roots of [tex]D^{2}-4[/tex] are +2,-2 Thus the comlementary function becomes

[tex]y=c_{1}e^{2x}+c_{2}e^{-2x}[/tex]

here [tex]c_{1},c_{2}[/tex] are arbitary constants

Now the Particular Integral becomes using standard formula

[tex]y=\frac{e^{ax}}{f(D)}\\\\y=\frac{e^{ax}}{f(a)} (f(a)\neq 0)\\\\y=x\frac{e^{ax}}{f'(a)}(f(a)=0)[/tex]

[tex]y=\frac{e^{2x}}{D^{2}-4}\\\\y=\frac{e^{2x}}{(D+2)(D-2)}\\\\y=\frac{1}{D-2}\times \frac{e^{2x}}{2+2}\\\\y=\frac{1}{4}\times \frac{e^{2x}}{D-2}\\\\y=\frac{1}{4}xe^{2x}[/tex]

Hence the solution is = Complementary function + Particular Integral

Thus Solution becomes [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]

The final general solution is [tex]y(x) = C1e^2x + C2e^-2x + 1/2xe^2x[/tex].

To find the general solution of the given differential equation: y'' - 4y = e2x, we will follow these steps:

1. Solve the Homogeneous Equation

First, solve the homogeneous part: y'' - 4y = 0

The characteristic equation is: r2 - 4 = 0

Solving for r, we get: r = ±2

Thus, the general solution to the homogeneous equation is: yh(x) = C1e2x + C2e-2x

2. Find a Particular Solution

Next, find a particular solution, yp(x), to the non homogeneous equation through the method of undetermined coefficients. Assume a particular solution of the form: yp(x) = Axe2x

Differentiating, we get: yp' = Ae2x + 2Axe2x and yp'' = 4Axe2x + 2Ae2x

Substitute these into the original equation:

4Axe2x + 2Ae2x - 4(Axe2x) = e2x

which simplifies to: 2Ae2x = e2x

Thus, A = 1/2

So, the particular solution is: yp(x) = (1/2)xe2x

3. Form the General Solution

The general solution to the nonhomogeneous differential equation is the sum of the homogeneous solution and the particular solution:

y(x) = yh(x) + yp(x)

Therefore, the general solution is: [tex]y(x) = C1e2x + C2e-2x + (1/2)xe2x[/tex].

Answer:  No, the given transformation T is NOT a linear transformation.

Step-by-step explanation:  We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :

[tex]T(x,y)=(x,y^2).[/tex]

We know that

a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R

[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]

So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have

[tex]T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)[/tex]

and

[tex]aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).[/tex]

Therefore, we get

[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]

Thus, the given transformation T is NOT a linear transformation.

Answer:

  2

Step-by-step explanation:

The z-score is the number of standard deviations the value is above the mean. If that number is 2, then Z=2. If that number is -2, then Z=-2.

___

Both Z=2 and Z=-2 are values that are two standard deviations from the mean.

Answer:

a) reject null hypothesis since p < 0.01

Step-by-step explanation:

Given that the  average assembly time for a Ford Taurus is

[tex]μ = 38 hrs[/tex]

Sample size [tex]n=36[/tex]

[tex]x bar = 37.5\\s=1.2\\SE = 1.2/6 = 0.2[/tex]

Test statistic t = mean diff/se = 0.5/0.2 = 2.5

(Here population std dev not known hence t test is used)

df = 35

p value = 0.008703

a) reject null hypothesis since p < 0.01

Answer:

56.25 feet.

Step-by-step explanation:

h(t) = 60t - 16t^2

Differentiating to find the velocity:

v(t) = 60 -32t

This  equals zero when  the ball reaches its maximum height, so

60-32t = 0

t = 60/32 = 1.875 seconds

So the maximum height is  h(1.875)

= 60* 1.875 - 16(1.875)^2

= 56.25 feet.

Answer: 56.25 feet.

Step-by-step explanation:

For a Quadratic function in the form [tex]f(x)=ax^2+bx+c[/tex], if [tex]a<0[/tex] then the parabola opens downward.

Rewriting the given function as:

[tex]h(t) = - 16t^2+60t[/tex]

You can identify that [tex]a=-16[/tex]

Since [tex]a<0[/tex] then the parabola opens downward.

Therefore, we need to find the vertex.

Find the x-coordinate of the vertex with this formula:

[tex]x=\frac{-b}{2a}[/tex]

Substitute values:

[tex]x=\frac{-60}{2(-16)}=1.875[/tex]

Substitute the value of "t" into the function to find the height in feet that the ball will reach. Then:

 [tex]h(1.875)=- 16(1.875)^2+60(1.875)=56.25ft[/tex]

Answer:

68%

Step-by-step explanation:

It is given that the diameters of bearing have a normal distribution.

Mean = u = 1.2 cm

Standard deviation = [tex]\sigma[/tex] = 0.03 cm

We have to find the proportion of values which falls in between 1.17 to 1.23

In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:

[tex]z=\frac{x- \mu}{\sigma}[/tex]

For 1.17:

[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]

For 1.23:

[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]

So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question.

According to the empirical rule:

68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1.

Therefore, the answer to this question is 68%

Answer:

Parent function f(x) is inverted, stretched vertically by 1 : 3, shifted 5 units right and 4 units upwards to form new function g(x).

Step-by-step explanation:

The parent function graphed is f(x) = x²

This graph when inverted (parabola opening down)function becomes

g(x) = -x²

Further stretched vertically by a scale factor of 1:3 then new function becomes as g(x) = -3x²

Then we shift this function by 5 units to the right function will be

g(x) = -3(x - 5)²

At last we shift it 4 units vertically up then function becomes as

g(x) = -3(x - 5)² + 4

Answer: 0.064

Step-by-step explanation:

Binomial probability formula :-

[tex]P(X)=^nC_x \ p^x\ (1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.

Given : The proportion of the homes constructed in the Quail Creek area include a security system :  [tex]p=0.40[/tex]

Now, if three homes are selected at random, then the probability all three of the selected homes have a security system is given by :-

[tex]P(3)=^3C_3 \ (0.40)^3\ (1-0.40)^{3-3}\\\\=(0.40)^3=0.064[/tex]

Hence, the probability all three of the selected homes have a security system = 0.064

The probability that all three homes selected at random in the Quail Creek area have a security system is 6.4%.

The probability that all three of the selected homes in the Quail Creek area have a security system, given that 40% of the homes have a security system, can be calculated by using the rule for independent events in probability.

Since each house is selected at random, we can multiply the probability of each house having a security system together:

P(all three homes have a security system) = P(home 1 has security system) × P(home 2 has security system) × P(home 3 has security system)

As every home has a 40% (or 0.40) chance of having a security system:

P(all three) = 0.40 × 0.40 × 0.40 = 0.064

Therefore, there is a 6.4% chance that all three homes selected will have a security system.

Answer: Option 'e' is correct.

Step-by-step explanation:

Qualitative data refers to those data which does not include any numerical value.

Average rainfall on 25 days of the month is a quantitative data as it would represented as numerical value.

Number of accidents occurred in the month of September is quantitative data.

Height of 40 students in a class is quantitative data too.

Weight of 35 students in a class is quantitative data too.

But political affiliation of 2250 randomly selected voters is qualitative data as it has not included any numerical value.

Hence, option 'e' is correct.

Option e, 'Political affiliation of 2,250 randomly selected voters,' represents qualitative data because it categorizes voters based on a non-numeric characteristic, their political affiliation.

The question 'Which of the following is an example of qualitative data? e. Political affiliation of 2,250 randomly selected voters' is asking us to identify a type of data among the given options. Qualitative data refers to information that is categorized based on attributes or qualities rather than numerical values. In contrast, quantitative data involve numbers and can either be discrete, which are countable, or continuous, which can take on any value within a range.

To answer the question, option e, 'Political affiliation of 2,250 randomly selected voters,' represents qualitative data because it describes a characteristic (political affiliation) that is non-numeric and is usually expressed in words or categories. Other options such as average rainfall, number of accidents, weight, and height involve numerical measurements and are therefore examples of quantitative data, which can either be discrete or continuous depending on the nature of the measurement.

Option e is unique as the only example of qualitative data in the options provided, because political affiliation does not result from a measurement or count, but rather, it is a category used to describe a voter's preferred political party or stance.

  f : X → Y  and  g: Y → Z

Now we have to show:

If gof is one-to-one then f must be one-to-one.

Given:

gof is one-to-one

To prove:

f is one-to-one.

Proof:

Let us assume that f(x) is not one-to-one .

This means that there exist x and y such that x≠y but f(x)=f(y)

On applying both side of the function by the function g we get:

g(f(x))=g(f(y))

i.e. gof(x)=gof(y)

This shows that gof is not one-to-one which is a contradiction to the given statement.

Hence, f(x) must be one-to-one.

Let A={1,2,3,4}  B={1,2,3,4,5} C={1,2,3,4,5,6}

Let f: A → B

be defined by f(x)=x

and g: B → C be defined by:

g(1)=1,g(2)=2,g(3)=3,g(4)=g(5)=4

is not a one-to-one function.

since 4≠5 but g(4)=g(5)

Also, gof : A → C

is a one-to-one function.

The answer does not exist.

Note - The statement has typing mistakes. Correct form is presented below:

Let [tex]f(x) = (x-3)^{-2}[/tex]. Find all values of [tex]c[/tex] in (2, 5) such that [tex]f(5) - f(2) = f'(c) \cdot (5-2)[/tex]. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

In this question we should use the Mean Value Theorem, which states that given a secant line between points A and B, there is at least a point C that belongs to the curve whose derivative exists.

We begin by calculating [tex]f(2)[/tex] and [tex]f(5)[/tex]:

[tex]f(2) = (2-3)^{-2}[/tex]

[tex]f(2) = 1[/tex]

[tex]f(5) = (5-3)^{-2}[/tex]

[tex]f(5) = 1[/tex]

And the slope of the derivative is:

[tex]f'(c) = \frac{f(5) - f(2)}{5-2}[/tex]

[tex]f'(c) = 0[/tex]

Now we find the derivative of the function:

[tex]f'(x) = -2\cdot (x-3)^{-3}[/tex]

[tex]-2\cdot (x-3)^{-3} = 0[/tex]

[tex]-2 = 0[/tex] (ABSURD)

Hence, we conclude that the answer does not exist.

We kindly invite to see this question on Mean Value Theorem: https://brainly.com/question/3957181