Let S u, v be a linearly independent set. Prove that the {u + v,u - v} is linearly independent

Answer:

Step-by-step explanation:

[tex]y''-4y=cosx[/tex]

The auxialary equation and solutions are as follows:

[tex]m^2-4=0\\m=2,-2[/tex]

General solution is

[tex]y=Ae^{2x} +Be^{-2x}[/tex]

Right side =cosx

Particular Integral

= [tex]\frac{cosx}{-1-4} =\frac{-cosx}{5}[/tex]

Hence full solution is

Answer:

Therefore, the inverse of given matrix is

[tex]=\begin{pmatrix}\frac{2}{5}&-\frac{1}{5}\\ -\frac{1}{5}&\frac{4}{15}\end{pmatrix}[/tex]

Step-by-step explanation:

The inverse of a square matrix [tex]A[/tex] is [tex]A^{-1}[/tex] such that

[tex]A A^{-1}=I[/tex] where I is the identity matrix.

Consider, [tex]A = \left[\begin{array}{ccc}4&3\\3&6\end{array}\right][/tex]

[tex]\mathrm{Matrix\:can\:only\:be\:inverted\:if\:it\:is\:non-singular,\:that\:is:}[/tex]

[tex]\det \begin{pmatrix}4&3 \\3&6\end{pmatrix}\ne 0[/tex]

[tex]\mathrm{Find\:2x2\:matrix\:inverse\:according\:to\:the\:formula}:\quad \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}^{-1}=\frac{1}{\det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}}\begin{pmatrix}d\:&\:-b\:\\ -c\:&\:a\:\end{pmatrix}[/tex]

[tex]=\frac{1}{\det \begin{pmatrix}4&3\\ 3&6\end{pmatrix}}\begin{pmatrix}6&-3\\ -3&4\end{pmatrix}[/tex]

[tex]\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}\:=\:ad-bc[/tex]

[tex]4\cdot \:6-3\cdot \:3=15[/tex]

[tex]=\frac{1}{15}\begin{pmatrix}6&-3\\ -3&4\end{pmatrix}[/tex]

[tex]=\begin{pmatrix}\frac{2}{5}&-\frac{1}{5}\\ -\frac{1}{5}&\frac{4}{15}\end{pmatrix}[/tex]

Therefore, the inverse of given matrix is

[tex]=\begin{pmatrix}\frac{2}{5}&-\frac{1}{5}\\ -\frac{1}{5}&\frac{4}{15}\end{pmatrix}[/tex]

Answer:

(d)

Step-by-step explanation:

Multiple your numbers out first. This always makes it less hectic to look at. Multiply both sides by x to move the x to the right side. Then divide both sides by ((0.4365)(87.9)).  You should get 10.

Answer:(x-3)=-(y-5)=z

Step-by-step explanation:

Given

the point through which line passes is (3,5,0)

so we need a vector along the line to get the equation of line

It is given that line is perpendicular to both i+j & j+k

therefore their cross product will give us the vector perpendicular to both

v=(i+j)\times (j+k)=i-j+k

therefore we get direction vector of line so we can write

[tex]\frac{x-3}{1}=\frac{y-5}{-1} =\frac{z-0}{1}[/tex]=t

i.e.

x=t+3,y=-t+5,z=t

The parametric form of the equation is;

The symmetric form of the equation is [tex]\rm x - 3 = -(y -5) = z[/tex].

Given

The line through (3, 5, 0) and perpendicular to both i + j and j + k

[tex]\rm \dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}=t[/tex]

Where the value of [tex]\rm x_1=3, \ y_1=5\ and \ z_1=0[/tex].

To find a, b, c by evaluating the product of ( i + j) and ( j + k ).

[tex]\rm= (i+j)\times (j+k)\\\\= i-j+k[/tex]

The value of a = 1, b = -1 and c = 1.

Substitute all the values in the equation.

[tex]\rm \dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\\\\\dfrac{x-3}{1}=\dfrac{y-5}{-1}=\dfrac{z-0}{1}\\\\(x-3)=-(y-5)=z[/tex]

Therefore,

The parametric form of the equation is;

[tex]\rm x=x_1+at, \ x=3+1t, \ x=3+t\\\\y=y_1+at, \ y=5+(-1)t, \ y=5-t\\\\z=z_1+at, \ z=0+a(1), \ z=a[/tex]

To know more about symmetric equations click the link given below.

https://brainly.com/question/14701215

Answer: Option 'd' is correct.

Step-by-step explanation:

Since we have given that

Volume of an excavation = 203 cubic feet

Weight of the material = 1200 pounds per cubic yard

So, the weight of the excavation would be

Volume of an excavation × Weight of the material

[tex]=203\times 1200\\\\=243,600\ pounds[/tex]

Hence, the weight of the excavation is 243,600 pounds.

Therefore, Option 'd' is correct.

When you have to repeatedly take the same test, with constant probability of succeeding/failing, you have to use Bernoulli's distribution. It states that, if you take [tex]n[/tex] tests with "succeeding" probability [tex]p[/tex], and you want to "succeed" k of those n times, the probability is

[tex]\displaystyle P(n,k,p) = \binom{n}{k}p^k(1-p)^{n-k}[/tex]

In your case, you have n=18 (the number of tests), and p=0.3 (the probability of succeeding). We want to succeed between 8 and 12 times, which means choosing k=8,9,10,11, or 12. For example, the probability of succeeding 8 times is

[tex]\displaystyle P(18,8,0.3) = \binom{18}{8}(0.3)^8(0.7)^{10}[/tex]

you can plug the different values of k to get the probabilities of succeeding 9, 10, 11 and 12 times, and your final answer will be

[tex]P = P(18,8,0.3) + P(18,9,0.3) + P(18,10,0.3) + P(18,11,0.3) + P(18,12,0.3)[/tex]

Final answer:

In mathematics, calculate the probability of a student guessing between 8 and 12 questions right out of 18 when the probability of guessing correctly is 0.3.

Explanation:

To calculate the probability of getting between 8 and 12 questions right through knowledgeable guessing, we can use the binomial probability formula:

P(k successes in n trials) = nCk × [tex]p^k[/tex] × (1-[tex]p^{(n-k)[/tex])

where:

k = number of successes (correct answers) = 8 to 12

n = total number of questions = 18

p = probability of success (correct answer by guessing) = 0.3

(1-p) = probability of failure (incorrect answer) = 0.7

We need to calculate the probability for each possible number of correct answers between 8 and 12 and then sum them up.

Here's the calculation:

Probability of getting 8 correct:

P(8 successes in 18 trials) = 18C8 ×0.3⁸ × 0.7¹⁰ ≈ 0.076

Probability of getting 9 correct:

P(9 successes in 18 trials) = 18C9 × 0.3⁹ ×0.7⁹ ≈ 0.124

Probability of getting 10 correct:

P(10 successes in 18 trials) = 18C10 ×0.3¹⁰ ×0.7⁸ ≈ 0.122

Probability of getting 11 correct:

P(11 successes in 18 trials) = 18C11 × 0.3¹¹ × 0.7⁷ ≈ 0.072

Probability of getting 12 correct:

P(12 successes in 18 trials) = 18C12 × 0.3¹² × 0.7⁶ ≈ 0.021

Total probability:

P(8 to 12 correct) = 0.076 + 0.124 + 0.122 + 0.072 + 0.021 ≈ 0.415

Therefore, the probability that the student gets between 8 and 12 questions right by knowledgeable guessing is approximately 41.5%.

So we have a set of integers,

[tex]X=\{15,20,25,25,27,28,30,34\}[/tex]

And we need to compute some stats.

First the range,

The range of data set is difference between the maximum and the minimum of the data set.

So the minimum is 15 and maximum is 34 therefore the range is 34 - 15 = 19.

Second the interquartile range,

The interquartile range of data set is the difference of the first and third quartiles.

First quartile is the value separating the lower quarter and higher three - quarters of the set. The first quartile is computed by taking median of the lower half of the sorted set. This is 15, 20, 25, 25 and it's median is 22.5

Third quartile is therefore the median of 27, 28, 30, 34 which is 29.

Next up, variance.

The variance of set measures how much data is spread out. For data set [tex]x_1,\dots, x_n[/tex] with an average [tex]a[/tex],

[tex]\mathsf{var}(X)=\Sigma_{i=1}^{n}\dfrac{(x_i-a)^2}{n-1}[/tex]

So first compute the average value which is actually a mean,

[tex]a=\dfrac{1}{2}\Sigma_{i=1}^{n}a_i[/tex]

The sum [tex]\Sigma_{i=1}^{n}a_i[/tex] of numbers in set is 204.

Divide this by number of elements in the set (8).

[tex]a=\dfrac{204}{8}=25.5[/tex]

Then compute the variance,

[tex]\mathsf{var}(X)=\dfrac{\Sigma_{i=1}^{n}(x_i-a)^2}{n-1}\approx\boxed{34.57}[/tex]

And finally standard deviation,

Since have computed variance the standard deviation is nothing too hard. It's defined as a square root of variance,

[tex]\mathsf{sde}(X)=\sqrt{\dfrac{\Sigma_{i=1}^{n}(x_i-a)^2}{n-1}}=\sqrt{34.57}\approx\boxed{5.88}[/tex]

Hope this helps.

r3t40

The values of the statistical measures for the distribution are :

Given the data :

27, 25, 20, 15, 30, 34, 28, 25.

Rearranging the values :

15, 20, 25, 25, 27, 28, 30, 34

A.)

The range = Maximum - Minimum

Maximum = 34 ; Minimum = 15

Range = 34 - 15 = 19

B.)

The interquartile range = Upper quartile - Lower quartile

Upper quartile (Q3) = 3/4(n+1)th term

n = number of values = 8

Q3 = 3/4(9) = 6.75 th term = (28+30)/2 = 29

Lower quartile (Q1) = 1/4(n+1)th term

Q1 = 1/4(9) = 2.25th term = (20+25)/2 = 22.5

Interquartile range = 29 - 22.5 = 6.5

C.)

Variance :

Σ[(X - mean)²] / (n-1)

ΣX/ n

Mean = 204 / 8 = 25.5

[(15-25.5)² + (20-25.5)² + (25-25.5)² + (25+25.5)² + (27-25.5)² + (28-25.5)² + (30-25.5)² + (34 - 25.5)²] ÷ (8 - 1)

Variance = 242 / 7

Variance = 34.57

Standard deviation = √variance

Standard deviation = √34.57

Standard deviation = 5.879

Standard deviation = 5.88 (2 decimal place).

Learn more : https://brainly.com/question/15528814

For any equation,

[tex]a_ny^(n)+\dots+a_1y'+a_0y=0[/tex]

assume solution of a form, [tex]e^{yt}[/tex]

Which leads to,

[tex](e^{yt})'''-(e^{yt})''+4(e^{yt})'-4e^{yt}=0[/tex]

Simplify to,

[tex]e^{yt}(y^3-y^2+4y-4)=0[/tex]

Then find solutions,

[tex]\underline{y_1=1}, \underline{y_2=2i}, \underline{y_3=-2i}[/tex]

For non repeated real root y, we have a form of,

[tex]y_1=c_1e^t[/tex]

Following up,

For two non repeated complex roots [tex]y_2\neq y_3[/tex] where,

[tex]y_2=a+bi[/tex]

and,

[tex]y_3=a-bi[/tex]

the general solution has a form of,

[tex]y=e^{at}(c_2\cos(bt)+c_3\sin(bt))[/tex]

Or in this case,

[tex]y=e^0(c_2\cos(2t)+c_3\sin(2t))[/tex]

Now we just refine and get,

[tex]\boxed{y=c_1e^t+c_2\cos(2t)+c_3\sin(2t)}[/tex]

Hope this helps.

r3t40

To find the general solution for the differential equation y''' - y'' + 4y' - 4y = 0, we solve the characteristic equation r³ - r² + 4r - 4 = 0, and find the roots r = 1 and r = ±2i. Thus, the general solution is y(x) = C1eˣ + C2cos(2x) + C3sin(2x).

The given differential equation is:

y''' - y'' + 4y' - 4y = 0

This is the general solution to the differential equation y''' - y'' + 4y' - 4y = 0.

Answer:

x=10 and W=12

Step-by-step explanation:

Let's solve the equations. First we need to understand that the problem can be solved because we have two variables (x, W) and two equations.

Now, we have the following equations:

3x+3W-66 making the equation equal to 0:

3x+3W-66=0 which can be express as:

3x=-3W+66

x=(-3W+66)/3

x=-W+22 (equation 1)

The next equation is:

12x+15W-300 making the equation equal to 0 and then divided by 3:

(12x+15W-300)/3=0 which is:

4x+5W-100=0 (equation 2), using equation 1 we can write:

4(-W+22)+5W-100=0

-4W+88+5W-100=0

W-12=0

W=12

Using W=12 in equation 2 we have:

4x+5W-100=0

4x+5*(12)-100=0

4x+(60)-100=0

4x-40=0

4x=40

x=40/4

x=10

In conclusion the solution for the equations are: x=10 and W=12.

Answer:

Matrix A is not invertible.

Step-by-step explanation:

Invertible matrix are those matrix which are square matrix or non singular matrix.

For any matrix to be invertible, matrix should be non singular i.e. det(x)[tex]\neq[/tex] 0.

But for the question given above we cannot find determinant of matrix A as it is not square matrix. so inverse of given matrix does not exist. so it is not possible to have non trival solutions.

Answer:

E = 3P + 3S,

S=(1/3)E-P

Step-by-step explanation:

Given,

A slice of pizza and a can of soda cost $3,

That is, total cost of P slices of pizza and S cans of soda = ( 3P + 3S ) dollars,

According to the question,

Total cost of P slices of pizza and S cans of soda = E dollars,

⇒ 3P + 3S = E

Which is the required equation that correctly describes the amount of money spent,

For rearranging the equation in terms of E and P,

We need to isolate S,

Subtract 3P on both sides,

3S = E - 3P

Divide both sides by 3,

[tex]S=\frac{E-3P}{3}[/tex]

[tex]\implies S=(\frac{1}{3})E-P[/tex]

Which is the required equation.

Answer: 0.5

Step-by-step explanation:

We know that the mean and variance of a binomial distribution with probability of success p is given by :-

[tex]\text{Mean}:\mu=np\\\\\text{Variance}:\sigma^2=np(1-p)[/tex], where n is the total number of trials .

Given : A given binomial distribution has

[tex]\text{Mean}:\mu=np=4.......(1)\\\\\text{Variance}:\sigma^2=np(1-p)=2............(2)[/tex]

Now we substitute , the value of np from (1) in (2), we get

[tex]4(1-p)=2\\\\\Rightarrow\ 1-p=\dfrac{2}{4}\\\\\Rightarrow\ p=1-\dfrac{1}{2}\\\\\Rightarrow\ p=\dfrac{1}{2}=0.5[/tex]

Hence, the probability of success (p) = 0.5

Answer:

If [tex]h(x) = x^2 + 2x + 10[/tex] then [tex](x+1-3i)*(x+1+3i)[/tex].

If [tex]h(x) = x^2 - 2x + 10[/tex] then [tex](x-1-3i)*(x-1+3i)[/tex].

Step-by-step explanation:

In order to write the polynomial as the product of linear factors, we need to find the roots of the polynomial. A quadratic equation is defined as:

[tex]ax^2+bx+c[/tex]

Because the given polynomial expression is a quadratic equation, we can use the following equations for calculating the roots:

[tex]x1=(-b/2a)+\sqrt{b^2-4ac}/2a[/tex]

[tex]x1=(-b/2a)-\sqrt{b^2-4ac}/2a[/tex]

Since the second term sign is not given, then we can write the expression as:

[tex]x^2+s2x+10[/tex], in which a=1, b=s2 where 's' represents a sign (- or +), and c=10.

Using the equation for finding the roots we obtain:

[tex]x1=(-sb/2a)+\sqrt{b^2-4ac}/2a[/tex]

[tex]x1=(-s2/2*1)+\sqrt{2^2-4*1*10}/(2*1)[/tex]; notice that [tex](sb)^{2} = 2^{2}[/tex]

[tex]x1=(-s1)+\sqrt{-36}/2[/tex]

[tex]x1=-(s1)+6i/2[/tex]

[tex]x1=-(s1)+3i[/tex]

[tex]x2=(-sb/2a)-\sqrt{b^2-4ac}/2a[/tex]

[tex]x2=(-s2/2*1)-\sqrt{2^2-4*1*10}/(2*1)[/tex]; notice that (s2)²=2^2

[tex]x2=(-s1)-\sqrt{-36}/2[/tex]

[tex]x2=-(s1)-6i/2[/tex]

[tex]x2=-(s1)-3i[/tex]

If we consider 's' as possitive (+) the roots are:

[tex]x1=-1+3i[/tex] and [tex]x2=-1-3i[/tex]

Whereas if we consider 's' as negative (-) the roots are:

[tex]x1=1+3i[/tex] and [tex]x2=1-3i[/tex]

The above means that if the equation is [tex]h(x) = x^2 + 2x + 10[/tex], then we can express the polynomial as: [tex](x+1-3i)*(x+1+3i)[/tex].

But, if the equation is [tex]h(x) = x^2 - 2x + 10[/tex], then we can express the polynomial as: [tex](x-1-3i)*(x-1+3i)[/tex].

Answer:

The slope is: 7

The y-intercept is: 1

Step-by-step explanation:

The equation of the line in Slope-Intercept form is:

[tex]y=mx+b[/tex]

Where "m" is the slope of the line and "b" is the y-intercept.

To find the slope and the y-intercept of the given line, we can write it in Slope-Intercept form. We can do this by solving for "y".

Then, this is:

[tex]-7x + y = 1\\\\y=7x+1[/tex]

Therefore, you can identify that the slope of this line is:

[tex]m=7[/tex]

And the y-intercept is:

[tex]b=1[/tex]

Answer: 99.73%

Step-by-step explanation:

Given : Mean : [tex]\mu=100[/tex]

Standard deviation : [tex]\sigma=2[/tex]

Let X be the random variable that represents the data values.

Formula for Z-score :  [tex]z=\dfrac{X-\mu}{\sigma}[/tex]

For x=94, we have

[tex]z=\dfrac{94-100}{2}=-3[/tex]

For x=106, we have

[tex]z=\dfrac{106-100}{2}=3[/tex]

The probability that the samples are between 94 and 106:-

[tex]P(-3<x<3)=1-2P(z<-3)\\\\=1-2(0.0013499)=0.9973002\approx0.9973=99.73\%[/tex]

Hence, the percent of the samples are expected to be between 94 and 106 = 99.73%

Approximately 95% of the samples are expected to be between 94 and 106.

The percentage of samples that are expected to be between 94 and 106 can be determined using the empirical rule, also known as the 68-95-99.7 rule. According to this rule, approximately 68% of the samples will fall within one standard deviation of the mean, and approximately 95% of the samples will fall within two standard deviations of the mean.

In this case, the control limits are 94 and 106, which are one standard deviation below and above the mean, respectively. Therefore, 95% of the samples are expected to be between 94 and 106.

Note that this rule applies to data that follow a bell-shaped, symmetric distribution, which is assumed to be the case for this set of sample measurements.

https://brainly.com/question/29309609

#SPJ3

To find the probability of picking an apple with a diameter greater than 3.76 inches from a normally distributed population with a mean of 3 inches and a standard deviation of 0.5 inches, we calculate a z-score and then find the corresponding probability using a z-table or calculator.

The question is about finding the probability in a situation that follows a normal distribution, specifically, the likelihood of picking an apple with a diameter greater than 3.76 inches given a mean of 3 inches and a standard deviation of 0.5 inches.

To solve this, we must first calculate the z-score, which is the number of standard deviations a data point (in this case, the apple size of 3.76 inches) is from the mean. The z-score is calculated as follows: Z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation. Substituting the given values, we get Z = (3.76 - 3) / 0.5, which results in a z-score of 1.52.

Next, we will refer to the z-table to find the probability corresponding to this z-score. However, most z-tables give the probability from the mean to our z-score (the left-hand side). But we want the probability that the apple's diameter is greater than 3.76 inches, i.e., the right-hand side of the distribution curve. As a result, we have to subtract the value we get from the z-table from 1 (because the cumulative probability under the entire normal curve is 1).

If a z-table is not readily available, there are many online calculators available to find this probability.

https://brainly.com/question/34741155

#SPJ3

Answer:

a. [tex]P = \frac{1}{1000}[/tex]

b. [tex]P = \frac{3}{500}[/tex]

Step-by-step explanation:

To win the 'straight play' a ticket must have the exact number. Given that there are 1000 numbers between 000 and 999. The probability of winning in striaght play is:

[tex]P = \frac{1}{1000}[/tex]

To win in a box play, a ticket can have the three digit in daily order.

Given three numbers A, B and C. There are 6 ways to arrange the numbers:

ABC, ACB,  BAC, BCA,  CAB, CBA.

Therefore the probability is: [tex]P = \frac{3}{500}[/tex]

Answer:  The required solution is

[tex]y=(t-\dfrac{t^3}{2}+~~.~~.~~.).[/tex]

Step-by-step explanation:  We are given to find the solution of the following differential equation using power series method :

[tex]y^\prime+ty=0,~~y(0)=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Let [tex]y=\sum_{n=0}^{\infty}a_nq^n[/tex] be the solution of the given equation.

Then, we have

[tex]y^\prime=\sum_{n=1}^{\infty}a_nnt^{n-1}.[/tex]

From equation (i), we get

[tex]\sum_{n=1}^{\infty}a_nnt^{n-1}+t\sum_{n=0}^{\infty}a_nt^n=0\\\\\\\Rightarrow \sum_{n=1}^{\infty}a_nnt^{n-1}+t\sum_{n=-1}^{\infty}a_nt^{n+1}=0.[/tex]

Comparing the coefficients of [tex]t^n,~t^{n+1},~,~~.~~.~~.[/tex] from both sides of the above, we get

[tex]2a_2+a_0\\\\\Rightarrow a_2=-\dfrac{a_0}{2},\\\\\\3a_3+a_1=0\\\\\Rightarrow a_3=-\dfrac{a_1}{3},\\\\\\\vdots~~~\vdots~~~\vdots[/tex]

Therefore, we get

[tex]y=a_0(1-\dfrac{1}{2}t^2+~~.~~.~~.)+a_1(t-\dfrac{t^3}{2}+~~.~~.~~.).[/tex]

The condition y(0) = 1 gives

[tex]a_0=0.[/tex]

So, the required solution is

[tex]y=t-\dfrac{t^3}{2}+~~.~~.~~.[/tex]

Answer with explanation:

The diiferential equation of first order is,

  y'+t y=0

[tex]y=\sum_{n=0}^{\infty}a_{n}x^n\\\\y'=\sum_{n=1}^{\infty}na_{n}x^{n-1}\\\\y'+ty=\sum_{n=0}^{\infty}na_{n-1}x^{n-1}+t\sum_{n=0}^{\infty}a_{n}x^n\\\\=\sum_{n=0}^{\infty}(a_{n-1}\frac{n}{x}+a_{n}t)x^n\\\\\rightarrow y'+t y=0\\\\\rightarrow a_{n-1}\frac{n}{x}+a_{n}t=0\\\\a_{n-1}=\frac{-xta_{n}}{n}[/tex]

For, n=1

[tex]a_{0}=\frac{-xta_{1}}{1}\\\\a_{0}=-txa_{1}\\\\a_{1}=\frac{-a_{0}}{tx}\\\\\text{for}, n=2\\\\a_{1}=\frac{-xta_{2}}{2}\\\\a_{2}=\frac{-2a_{1}}{xt}\\\\=\frac{2a_{0}}{x^2t^2}\\\\\text{for}, n=3\\\\a_{2}=\frac{-xta_{3}}{3}\\\\a_{3}=\frac{-6a_{0}}{x^3t^3}\\\\a_{4}=\frac{24a_{0}}{x^4t^4}\\\\a_{n}=(-1)^n\frac{n!a_{0}}{x^nt^n}[/tex]

So,the series can be written as

    [tex]y_{n}=\sum_{n=1}^{\infty}(-1)^n\frac{n!a_{0}}{x^nt^n}[/tex]

Answer:

  a.  x'(t) = kx(5000-x)

  b.  R_in = 7 lbs/min

     R_out = 5c(t) lbs/min

     c(t) = A(t)/(800 -3t) lbs/gal

     A'(t) = R_in - R_out; A(0) = 200 lbs

Step-by-step explanation:

A. When proportionality is joint between x and y, the expression describing that is kxy, where k is the constant of proportionality. Here, the rate of change is jointly proportional to number who have adopted (x) and number of the community of 5000 who have not (5000-x). The corresponding equation is ...

  x'(t) = kx·(5000-x)

We know the initial number of adopters must be greater than 0, or the equation's only solution would be x=0.

__

B. The rate of influx of salt is ...

  R_in = (3.5 lbs/gal)×(2 gal/min) = 7 lbs/min

  The number of gallons in the tank at time t is ...

  v = 800 gal + (2 gal/min - 5 gal/min)(t min) = (800 -3t) gal

For amount of salt A(t) pounds, the concentration c(t) will be ...

  c(t) = A(t)/v = A(t)/(800 -3t) . . . . lbs/gal

The outflow rate will be the product of outflow volume and concentration:

  R_out = (c(t) lbs/gal)(5 gal/min) = 5c(t) lbs/min

And the differential equation for A(t) is ...

  A'(t) = R_in -R_out . . . with initial condition A(0) = 200 lbs.

__

When you go to solve the equation for A'(t), it will need to be cast in terms of A(t):

  A'(t) = 7 -5A(t)/(800 -3t)

Answer:

The  value of his investment in 3 years = $10007.53

Step-by-step explanation:

Compound interest

A = P[1 +R/n]^nt

Where A - amount

P - principle amount

R = rate of interest

t -  number of years

n - number of times compounded yearly

To find value if investment

Here P = $9500,  R = 1.75% = 0.0175 n = 1 and t = 3

A = P[1 +R/n]^nt

 = 9500[1 + 0.0175/1]^(1 * 3)

 = 10007.53

Answer:

$10,012.07

Step-by-step explanation:

Identify the values of each variable in the formula. Remember to express the percent as a decimal.

APrt=?=$9,500=0.0175=3 years

For compounding continuously, use the formula A=Pert.

Substitute the values into the formula and compute the amount to find

AA=9,500e0.0175⋅3=$10,012.07.

Answer: (16.6%, 21.4%)

Step-by-step explanation:

The confidence interval for proportion is given by :-

[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]

Given : Sample size : n= 708

The proportion of respondents who had​ children = [tex]p=0.19[/tex]

Significance level : [tex]\alpha=1-0.90=0.1[/tex]

Critical value : [tex]z_{\alpha/2}=z_{0.05}=\pm1.645[/tex]

Now, the 90​% confidence interval for proportion will be :-

[tex]0.19\pm (1.645)\sqrt{\dfrac{0.19(1-0.19)}{708}}\approx0.19\pm 0.024\\\\=(0.19-0.024,0.19+0.024)=(0.166,\ 0.214)=(16.6\%,\ 21.4\%)[/tex]

Hence, the 90​% confidence interval for the proportion = (16.6%, 21.4%)

Answer:

[tex]f(x)=(x-2)(x)(x+4)^2[/tex]

Step-by-step explanation:

Since there is a multiplicity at x=-4, that would meant that that part of the factor would have to have an even degree. This means that it would have to be 2.

This would give you [tex]f(x)=(x-2)(x)(x+4)^2[/tex]

Answer:

[tex]f (x) = x (x + 4)^2(x-2)[/tex]

Step-by-step explanation:

The zeros of the polynomial are all the values of x for which the function [tex]f (x) = 0[/tex]

In this case we know that the zeros are:

[tex]x = -4,\ x+4 =0[/tex]

[tex]x = -4,\ x+4 =0[/tex]

[tex]x = 0[/tex]

[tex]x = 2[/tex], [tex]x - 2 = 0[/tex]

Now we can write the polynomial as a product of its factors

[tex]f (x) = x (x + 4)(x+4) (x-2)[/tex]

[tex]f (x) = x (x + 4)^2(x-2)[/tex]

Note that the polynomial is of degree 4 because the greatest exponent of the variable x that results from multiplying the factors of f(x) is 4

Answer with explanation:

Number of Letters from which we have to make four​-letter passwords

   ={A,B,C,D,E,F,G}=7

Since repetition of digit is not allowed.In Permutation order of arrangement is Important,while in combination order of arrangement is not Important.

Out of seven letters we have to select four letters, keeping in mind the order of Alphabets.So, we will use the concept of Permutation.

Number of ways of arrangement,that is to make four letter password from 7 Alphabets

       [tex]_{4}^{7}\textrm{P}\\\\=\frac{7!}{(7-4)!}\\\\=\frac{7!}{3!}\\\\=4\times 5\times 6 \times 7=840[/tex]

So, total number of ways of making password from 7 alphabets=840 ways

Second Method⇒

Since repetition of alphabets is not allowed, first Alphabet can be chosen in 7 ways , second alphabet can be chosen in 6 ways, third alphabet can be chosen in 5 ways, fourth alphabet can be chosen from four alphabets.

So, total number of ways of making four letter password

       =7×6×5×4

    = 840 ways

We can form 840 unique four-letter passwords with no repeats from the letters A, B, C, D, E, F, G. This is a problem in combinatorics, calculating permutations of positions in the password.

We are trying to determine the number of unique four-letter passwords that can be created using the letters A, B, C, D, E, F, G with no repetition of letters. This is a problem of permutations, a concept in combinatorics, a branch of mathematics that deals with counting.

In this case, for the first position of the password, we have 7 choices. Once we've picked this first letter, we then have 6 remaining letters to choose from for the second position. Similarly, there are 5 choices remaining for the third position and then 4 for the last position.

The total number of password combinations will then be a product of the number of choices for each of the positions. Mathematically, this can be represented as 7*6*5*4.

Consequently, there are 840 different four-letter passwords that can be created from the given letters without repeating any letter.

https://brainly.com/question/31293479

#SPJ3

Answer:

The total number of men and women is 42.

So the number of ways of selecting 8 people from 42 is C(42,8) = 118030185.

The number of ways of selecting 4 men from 19 and 4 women from 23 is C(19,4)*C(23,4) = 3876*8855 = 34321980

Therefore, the required probability is 34321980 / 118030185 = 0.2907

Step-by-step explanation:

To determine the probability of exactly four men and four women attending a 60th high school reunion given certain conditions, we use the combination formula. The probability is found by dividing the product of the combinations of 4 men from 18 and 4 women from 22 by the total combinations of 8 people from 40.

The question is asking us to find the probability that exactly four men and exactly four women would attend the high school reunion. Considering that there are 40 (22 women + 18 men) people still alive, we are told that exactly 8 will attend the reunion.

We utilise the formula for combinations, to calculate this problem.

So, the possible combinations of 4 men attending from 18 is calculated as C(18,4).

And the possible combinations of 4 women attending from 22 is calculated as C(22,4).

Finally, the total combinations of 8 people attending from 40 is calculated as C(40,8).

Probability is calculated as the desired outcomes over total possibilities. Therefore, the desired probability will be [C(18,4) * C(22,4)] / C(40,8).

https://brainly.com/question/3901018

#SPJ3

Answer:

  c)  M/40

Step-by-step explanation:

The share of each company was originally M/10. Now it is M/8. The increase is ...

  M/8 -M/10 = M(1/8 -1/10) = M(10-8)/(10·8) = M/40

The increase in the share of each company after two companies dropped out is M/40.

The total amount of money to be paid by the companies is M rupees. Initially, these M rupees were to be divided among 10 companies, but when two dropped out, they had to be divided among 8 companies instead. So, the share of each company initially was M/10 and after the drop out, the share of each company became M/8.

The increase in the amount that each company had to give is given by subtraction of the initial share from the final share, that is, (M/8) - (M/10). To solve this equation, we first must find a common denominator for the fractions, which is 40 in this case. Therefore, the equation becomes (5M/40) - (4M/40) = M/40.

So, the increase in share of each company was M/40. Therefore, the correct answer is (c) M/40.

https://brainly.com/question/34328897

#SPJ11

Answer:

1.2

Step-by-step explanation:

Answer: 18

Step-by-step explanation:

Given : Margin of error = 3 ounces

Significance level : [tex]\alpha = 1-0.99=0.1[/tex]

Critical value : [tex]z_{\alpha/2}=2.576[/tex]

Standard deviation :[tex]\sigma=5\text{ ounces}[/tex]

The formula to calculate the sample size is given by :-

[tex]n=(\dfrac{z_{\alpha/2}\ \sigma}{E})^2[/tex]

[tex]\Rightarrow\ n=(\dfrac{2.576\times5}{3})^2\\\\\Rightarrow\ n=18.4327111111\approx18[/tex]

Hence, the minimum sample size should be 18.

To estimate the birth weight of infants, the doctor must select a sample size of at least 19 infants to be 99% confident that the true mean is within 3 ounces of the sample mean. This calculation is based on the given standard deviation of 5 ounces.

To determine the sample size needed to estimate the birth weight of infants, we will use the formula for a margin of error:

Margin of Error = (Z-value) * (Standard Deviation / Square Root of Sample Size)

Since the doctor wants to be 99% confident that the true mean is within 3 ounces of the sample mean, the Z-value will be 2.576 (corresponding to a 99% confidence level). The given standard deviation is 5 ounces. Plugging these values into the formula, we get:

3 = 2.576 * (5 / Square Root of Sample Size)

Solving for the sample size:

Square Root of Sample Size = 2.576 * 5 / 3

Square Root of Sample Size ≈ 4.293

Sample Size ≈ 18.41

Since the sample size must be a whole number, the doctor must select a sample size of at least 19 infants to be 99% confident that the true mean is within 3 ounces of the sample mean.

https://brainly.com/question/13962027

#SPJ3

Answer:

Elga is 37 years old.

Step-by-step explanation:

Let 'x' be the age of elga.

Alvin's age is: x + 15

Given that the sum of their ages is 89, we have that:

x + 15 + x = 89

2x = 89 - 15

2x = 74

x = 74/2 = 37

Elga is 37 years old.

Answer:

Elga is 37 years old.

Step-by-step explanation:

Alvin (A) is 15 years older than Elga (E).

The sum their anges (A and E) is 89.

So the system we have is:

A=15+E

A+E=89

We are going to input 15+E for A in A+E=89 since first equation says A+15+E.

A+E=89 with A=15+E

(15+E)+E=89

15+E+E=89

15+2E=89

Subtract 15 on both sides:

2E=89-15

2E=74

Divide both sides by 2:

2E/2=74/2

E=74/2

E=37

Elga is 37 years old.

A=15+E=15+37=52 so Alvin is 52 yeard old.

52+37=89.

and

52=15+37.

Answer:

yes. The cost of the insurance is less than the probability cost of the operation

Step-by-step explanation:

yes. The cost of the insurance is less than the probability cost of the operation

The cost of health insurance = $1200

Cost of dramatic injury operation= $500,000

chances of need of operation= 47.3% over a 20 years period

the amount of pay insurance after 20 years= [tex]20\times 1200= 240,00[/tex]

probable  of cost operation= 0.473*500,00= $236,500

clearly the cost of insurance is less than the probable  cost of operation.

Answer:

a) 8259888

b) 34220

c) 45057474

Step-by-step explanation:

Given,

The total number of transistor = 65,

In which, the defective transistor = 6,

So, the number of non defective transistor = 65 - 6 = 59,

Since, out of these transistor 5 are selected,

a) Thus, the number of ways = the total possible combination of 5 transistors = [tex]{65}C_ 5[/tex]

[tex]=\frac{65!}{(65-5)!5!}[/tex]

[tex]=8259888[/tex]

b) The number of samples that contains exactly 3 defective transistors = the possible combination of exactly 3 defective transistors = [tex]{6}C_3\times {59}C_2[/tex]

[tex]=\frac{6!}{(6-3)!3!}\times \frac{59!}{(59-2)!\times 2!}[/tex]

[tex]=20\times 1711[/tex]

[tex]=34220[/tex]

c) The number of sample without any defective transistor = The possible combination of 0 defective transistor = [tex]^6C_0\times ^{59}C_5[/tex]

[tex]=1\times 45057474[/tex]

[tex]=45057474[/tex]

(a) 8259888 ways, (b) 34220 samples and (c) 5006386 samples.

Let's solve each part:

(a) There are 65 total transistors, and we want to select 5 of them. So, the number of ways to select a sample is the number of combinations of 65 things taken 5 at a time. This can be calculated using the combination formula:

Number of combinations = [tex]C(n, k) = \frac{n!}{ k! \times (n-k)!)}[/tex]

where:

n is the total number of items (65 transistors)

k is the number of items to select (5 transistors)

Plugging in the values:

[tex]C(65, 5) = \frac{65!}{ 5! \times (65-5)!)} = 8259888[/tex]

(b) We want samples that contain exactly 3 defective transistors. We can achieve this by selecting 3 defective transistors from the 6 available and 2 non-defective transistors from the remaining 59 (65 total - 6 defective).

So, the number of ways to select this specific scenario is:

Number of ways to choose 3 defective transistors out of 6: C(6, 3)

Number of ways to choose 2 non-defective transistors out of 59: C(59, 2)

Using the combination formula again:

[tex]C(6, 3) \times C(59, 2) = (\frac{6!}{ (3! \times 3!)}) \times (\frac{59!} { (2! \times 57!)}) = 34220[/tex]

(c) Samples with no defective transistors simply means selecting all 5 transistors from the 59 non-defective ones.

Therefore:

[tex]C(59, 5) = \frac{59!}{ (5! \times 54!)} = 5006386[/tex]