The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 5 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes.

Let [tex]y_2(t)=tv(t)[/tex]. Then

[tex]{y_2}'=tv'+v[/tex]

[tex]{y_2}''=tv''+2v'[/tex]

and substituting these into the ODE gives

[tex]t^2(tv''+2v')+3t(tv'+v)-3tv=0[/tex]

[tex]t^3v''+5t^2v'=0[/tex]

[tex]tv''+5v'=0[/tex]

Let [tex]u(t)=v'(t)[/tex], so that [tex]u'(t)=v''(t)[/tex]. Then the ODE is linear in [tex]u[/tex], with

[tex]tu'+5u=0[/tex]

Multiply both sides by [tex]t^4[/tex], so that the left side can be condensed as the derivative of a product:

[tex]t^5u'+5t^4u=(t^5u)'=0[/tex]

Integrating both sides and solving for [tex]u(t)[/tex] gives

[tex]t^5u=C\implies u=Ct^{-5}[/tex]

Integrate again to solve for [tex]v(t)[/tex]:

[tex]v=C_1t^{-6}+C_2[/tex]

and finally, solve for [tex]y_2(t)[/tex] by multiplying both sides by [tex]t[/tex]:

[tex]tv=y_2=C_1t^{-5}+C_2t[/tex]

[tex]y_1(t)=t[/tex] already accounts for the [tex]t[/tex] term in this solution, so the other independent solution is [tex]y_2(t)=t^{-5}[/tex].

Answer: a) The probability is approximately = 0.5793

b) The probability is approximately=0.8810

Step-by-step explanation:

Given : Mean : [tex]\mu= 62.5\text{ in}[/tex]

Standard deviation : [tex]\sigma = \text{2.5 in}[/tex]

a) The formula for z -score :

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Sample size = 1

For x= 63 in. ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{1}}}=0.2[/tex]

The p-value = [tex]P(z<0.2)=[/tex]

[tex]0.5792597\approx0.5793[/tex]

Thus, the probability is approximately = 0.5793

b)  Sample size = 35

For x= 63 ,

[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{35}}}\approx1.18[/tex]

The p-value = [tex]P(z<1.18)[/tex]

[tex]= 0.8809999\approx0.8810[/tex]

Thus , the probability is approximately=0.8810.

We are to find the surface integral of [tex]\mathbf{\int \int _S \ F.ds}[/tex]

where;

∴

The plane x + y + z = 1

z = 1 - x - y

[tex]\dfrac{\partial z}{\partial x} = -1[/tex]

[tex]\dfrac{\partial z}{\partial y} = -1[/tex]

Since the surface is oriented downward

∴

[tex]dS = \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy[/tex]

[tex]dS = (-i-j-k) dxdy[/tex]

However,

Flux = [tex]\mathbf{\int \int _S \ F.ds}[/tex]

[tex]= \int \int_R F. \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy \\ \\ \\ = \int^1_{ x=0} \int ^{1-x}_{y=0} ( y^3i + 8j -xk)*(-i-j-k) dx dy \\ \\ \\ =\int^1_{ x=0} \int ^{1-x}_{y=0} (-y^3 -8+x) dxdy \\ \\ \\ = \int^1_{ x=0} \Big( \int ^{1-x}_{y=0} \Big(x-y^3 -8\Big) dy \Big) dx \\ \\ \\ = \int^1_{ x=0} \Bigg [xy - \dfrac{y^4}{4}-8y \Bigg]^{1-x}_{y=0} \ dx \\ \\ \\[/tex]

[tex]= \int^1_{ x=0} \Bigg [x(1-x) - \dfrac{1}{4}(1-x)^{4} - 8(1-x) \Bigg ] dx \\ \\ \\ = \int^1_{ x=0} \Bigg [(x-x^2) - \dfrac{1}{4}(x-1)^4+8(x-1)\Bigg] dx[/tex]

[tex]= \Bigg [ \dfrac{x^2}{2}-\dfrac{x^3}{3} - \dfrac{1}{4} \dfrac{(x-1)^5}{5}+(x-1)^2\Bigg] ^1_0[/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} -0+0-0+0+ \dfrac{1}{4}\times \dfrac{(-1)^5}{5}-(0-1)^2\Bigg][/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} - \dfrac{1}{20}-1\Bigg][/tex]

[tex]= \Bigg [ \dfrac{30-20-3-60}{60}\Bigg][/tex]

[tex]= \Bigg [ \dfrac{-53}{60}\Bigg][/tex]

Therefore, we can conclude that the surface integral is [tex]\mathbf{-\dfrac{53}{60}}[/tex]

Learn more about surface integral here:

https://brainly.com/question/1423573?referrer=searchResults

To compute a surface integral, take the antiderivatives of both dimensions defining the area with the surface edges as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

A surface integral over the given oriented surface can be computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

https://brainly.com/question/32088117

#SPJ3

Answer:

2063.9 kg

Step-by-step explanation:

Given,

1 kg = 2.2046 pounds

This can also be written as:

1 pound = 1/2.2046 kg

We have to calculate the value of 4550 pounds in kg up to nearest 10th

Thus,

[tex]4550 pounds= \frac{4550}{2.2046} kg[/tex]

Solving the above equation, we get:

4550 pounds = 2063.8664 kg

Rounding the above result to nearest tenth as:

4550 pounds = 2063.9 kg

[tex]2x\equiv3\pmod{11}[/tex]

Since 3 * 4 = 12 = 1 mod 11, we can multiply both sides by 4 to get

[tex]4\cdot2x\equiv3\cdot4\pmod{11}\implies8x\equiv1\pmod{11}[/tex]

Since 8 * 7 = 56 = 1 mod 11, multiplying both sides by 7 gives

[tex]7\cdot8x\equiv7\cdot1\pmod{11}\implies x\equiv7\pmod{11}[/tex]

Answer:

260 machines for minimum cost.

Step-by-step explanation:

c(x) = 0.3x^2 - 156x + 26.657

Finding the derivative:

c'(x) = 0.6x - 156

0.6x - 156 = 0   for  maxm/minm cost.

x = 156 / 0.6

= 260

The second derivative  is positive (0.6) so this is a minimum.

Answer:

[tex]x=260\ machines[/tex]

Step-by-step explanation:

Note that we have a cudratic function of negative principal coefficient.

The minimum value reached by this function is found in its vertex.

For a quadratic function of the form

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

the x coordinate of the vertex is given by the following expression

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

In this case the function is:

[tex]C(x) = 0.3x^2 - 156x + 26,657[/tex]

So:

[tex]a=0.3\\b=-156\\c=26,657[/tex]

Then the x coordinate of the vertex is:

[tex]x=-\frac{-156}{2(0.3)}[/tex]

[tex]x=260\ machines[/tex]

Then the number of machines that must be made to minimize the cost is:

[tex]x=260\ machines[/tex]

Answer:  The required number of different possible committees is 81172.

Step-by-step explanation:    Given that a department contains 9 men and 15 women.

We are to find the number of different committees of 6 members that are possible if the committee must have strictly more women than men.

Since we need committees of 6 members, so the possible combinations are

(4 women, 2 men), (5 women, 1 men) and (6 women).

Therefore, the number of different committees of 6 members is given by

[tex]n\\\\\\=^{15}C_4\times ^9C_2+^{15}C_5\times ^9C_1+^{15}C_6\\\\\\=\dfrac{15!}{4!(15-4)!}\times \dfrac{9!}{2!(9-2)!}+\dfrac{15!}{5!(15-5)!}\times \dfrac{9!}{1!(9-1)!}+\dfrac{15!}{6!(15-6)!}\\\\\\\\=\dfrac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!}\times\dfrac{9\times8\times7!}{2\times1\times7!}+\dfrac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!}\times\dfrac{9\times8!}{1\times8!}+\dfrac{15\times14\times13\times12\times11\times10\times9!}{6\times5\times4\times3\times2\times1\times9!}\\\\\\=1365\times36+3003\times9+5005\\\\=49140+27027+5005\\\\=81172.[/tex]

Thus, the required number of different possible committees of 6 members is 81172.

1) x-intercept:

x-intercept is the point where the graph of the equation crosses the x-axis. From the given figure, we can see that the line is crossing the x-axis at -10. Thus the x-intercept is -9

2) y-intercept:

y-intercept is the point where the graph of the equation crosses the y-axis. From the given figure, we can see that the line is crossing the y-axis at -10. Thus the y-intercept is -9

3) Slope:

Slope of a line is calculated as:

[tex]slope=m=\frac{\text{Difference in y coordinates}}{\text{Difference in x coordinates}}[/tex]

For calculating the slope we can use both intercepts. x-intercept is ordered pair will be (-9, 0) and y-intercept will be (0, -9). So the slope of the line will be:

[tex]m=\frac{-9-0}{0-(-9)}=-1[/tex]

Therefore, the slope of the line is -1.

4) Slope intercept form of the line:

The slope intercept form of the line is represented as:

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

where,

m = slope of line = -1

c = y-intercept = -9

Using these values, the equation becomes:

[tex]y=-x- 9[/tex]

Answer:

x-intercept: [tex]-9[/tex].

y-intercept: [tex]-9[/tex].

Slope: [tex]-1[/tex]

Equation: [tex]y=-x-9[/tex]

Step-by-step explanation:

We have been given a graph of a line on coordinate plane. We are asked to find the x-intercept, y-intercept, and slope.          

We know that x-intercept of a function is a point, where graph crosses x-axis.

Upon looking at our given graph, we can see that graph crosses x-axis at point [tex](-9,0)[/tex], therefore, x-intercept is [tex]-9[/tex].

We know that y-intercept of a function is a point, where graph crosses y-axis.

Upon looking at our given graph, we can see that graph crosses y-axis at point [tex](0,-9)[/tex], therefore, y-intercept is [tex]-9[/tex].

We have two points on the line. Let us find slope of line using points  [tex](-9,0)[/tex] and  [tex](0,-9)[/tex].

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{-9-0}{0-(-9)}=\frac{-9}{0+9}=\frac{-9}{9}=-1[/tex]

Therefore, the slope of the line is [tex]-1[/tex].

Now, we will substitute [tex]m=-1[/tex] and y-intercept [tex]-9[/tex] in slope form intercept of equation as:

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

m = Slope,

b = The y-intercept.

[tex]y=-1(x-(-9))[/tex]

[tex]y=-1(x+9)[/tex]

[tex]y=-x-9[/tex]

Therefore, the equation of the line would be [tex]y=-x-9[/tex].

First compute the probability that a bill would exceed $115. Let [tex]X[/tex] be the random variable representing the value of a monthly utility bill. Then transforming to the standard normal distribution we have

[tex]Z=\dfrac{X-100}{10}[/tex]

[tex]P(X>115)=P\left(Z>\dfrac{115-100}{10}\right)=P(Z>1.5)\approx0.0668[/tex]

Then out of 300 randomly selected bills, one can expect about 6.68% of them to cost more than $115, or about 20.

The solution of the given differential equation involves rearranging it into the standard form of a first-order linear differential equation, determining the integrating factor, and subsequently solving for the dependent variable y(x) via integration.

To solve the given differential equation, we can rewrite it in the form of dy/dx = f(x, y). That gives us (2(y-4x^2))/x = dy/dx. The resulting equation is a first-order linear differential equation, which can be solved using an integrating factor.

Here, the standard form of the differential equation is dy/dx + P(x)y = Q(x). Comparing this with our equation, we find P(x) = -2/x and Q(x) = -8x. We know that μ(x) = exp(∫P(x) dx) is the integrating factor. On solving we get μ(x) = 1/x2. We then multiply through our differential equation by μ(x) and integrate both sides to solve for y(x).

These steps on how to solve the differential equation involve certain knowledge in differential equation theory, namely about first-order linear differential equations, integrating factors, and the process of integration.

https://brainly.com/question/33814182

#SPJ3

Answer:

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Step-by-step explanation:

We are given the following expression and we are to simplify the given function:

[tex] f ( x ) = x ^ 2 - 8 [/tex]

Applying the function [tex]f(2x)[/tex] on [tex] f ( x ) = x ^ 2 - 8 [/tex] to get:

[tex] f ( 2 x ) = ( 2 x ) ^ 2 - 8 [/tex]

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Answer:

C

Step-by-step explanation:

The melons are 3 for 8 dollars, this means each melon is worth 8/3 dollars, this is the first proportion in the expression.  In this proportion the numerator is the value of money so 25 will be the numerator in the other proportion.

8/3=25/x, C

Answer:

[tex]P(t)=80(10^{0.1t})[/tex]

Step-by-step explanation:

The 'y' axis represent log(P), so it may be modeled as a line (or linear function), where its slope is 0.1:

[tex]log(P)=0.1t+C[/tex]

Pow each part of the equation by 10:

[tex]10^{log(P)}=10^{0.1t+C}\\ P=10^{0.1t+C}[/tex]

Evaluate at t=0, where the population is known.

[tex]P(0)=10^{C}=80[/tex]

Applying logarithmic properties:

[tex]P=10^{0.1t+C}=10^{0.1t}*10^{C}[/tex]

So, the final function is:

[tex]P(t)=80(10^{0.1t})[/tex]

The population size over time follows an exponential growth function described by P(t) = 80 * [tex]e^(0.23026t)[/tex], where the slope on a semilog plot of 0.1 is converted to the natural log base to find the growth rate constant.

When dealing with population growth, plotting the data on a semilog plot where the logarithm of the population size is plotted against time can simplify the analysis. If the resulting plot is a straight line with a slope of 0.1, this indicates exponential growth, because plotting an exponential function on a semilog plot yields a straight line.

The general form of an exponential growth function is:

Where:

Given that the slope of the line on the semi log plot is 0.1, this slope represents the constant k in the context of logs with base 10.

To convert this to a natural logarithm base (e), use the conversion factor ln(10):

k = 0.1 * ln(10)

Since ln(10) ≈ 2.3026, we have:

k = 0.1 * 2.3026

≈ 0.23026

Given the initial population size (P0) at time t = 0 is 80, the function describing the population size over time t is:

P(t) = 80 * [tex]e^(0.23026t)[/tex],

Explanation of probabilities for a worthy research proposal approved by two panels, disapproved by two panels, and approved by one panel.

A worthy proposal is approved by both panels:

Probability = (Probability approved by Panel 1) × (Probability approved by Panel 2) = 0.2 × 0.7 = 0.14

A worthy proposal is disapproved by both panels:

Probability = (Probability disapproved by Panel 1) × (Probability disapproved by Panel 2) = 0.8 × 0.3 = 0.24

A worthy proposal is approved by one panel:

Probability = (Probability approved by Panel 1 and disapproved by Panel 2) + (Probability disapproved by Panel 1 and approved by Panel 2) = (0.2 × 0.3) + (0.8 × 0.7) = 0.06 + 0.56 = 0.62

Answer:

The given expression can be written as -1=19(-1)+18.

Step-by-step explanation:

According to quotient remainder theorem:

For any integer a and a positive integer d, there exist unique integers q and r such that

[tex]a=d\times q+r[/tex]

It can also written as

[tex]a\text{mod }d=r[/tex]

The given expression is

[tex]-1\text{mod }19[/tex]

Here a=-1 and d=19.

[tex]\frac{-1}{19}=19(-1)+18[/tex]

[tex]-1\text{mod }19=18[/tex]

If -1 is divides by 19, then the remainder is 18. The value of r is 18.

Therefore the given expression can be written as -1=19(-1)+18.

-1 mod 19 is equal to 18.

Finding -1 mod 19

To determine -1 mod 19, we use the formula a = dq + r, where a = -1, d = 19, and q and r are the quotient and remainder, respectively.

First, note that any number mod 19 will yield a remainder between 0 and 18.

Since -1 is a negative number, we rewrite it in terms of 19: -1 = -1 + 19k for some integer k.

To find a positive equivalent, we choose k such that -1 + 19k is positive and falls within the range of 0 to 18. When k = 1, we get -1 + 19(1) = 18.

Therefore, -1 mod 19 is equal to 18.

This means that if you divide -1 by 19, the remainder that falls within the range of 0 to 18 is 18.

Answer:

75.

Step-by-step explanation:

Let x be the original price. 20% = 0.2. The price with the discount is 60, that is

x-x*0.2 = 60

x(1-0.2) = 60 using common factor,

x (0.8) = 60

x = 60/0.8

x = 75.

So, the original price is $75.

The original price of an item discounted 20% with a sale price after the discount of $60 is $75.

A discounted price is the marked-down price of an item.

The discounted price represents the selling price after reducing it with the discount.

Discount rate = 20%

Discounted price = $60

Original price = $75 ($60/1-20%)

Thus, the original price of an item discounted 20% with a sale price after the discount of $60 is $75.

Learn more about the discounted price at https://brainly.com/question/7459025

Answer: 0.1310

Step-by-step explanation:

Given : Mean : [tex]\mu = \text{131 millimeters}[/tex]

Standard deviation :  [tex]\sigma = \text{7 millimeters}[/tex]

Sample size : [tex]n=31[/tex]

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters i.e. less than 129.1 milliliters and less than 132.9 milliliters.

The formula for z-score :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x = 129.1 milliliters

[tex]z=\dfrac{129.1-131}{\dfrac{7}{\sqrt{31}}}\approx-1.51[/tex]

For x = 132.9 milliliters

[tex]z=\dfrac{132.9-131}{\dfrac{7}{\sqrt{31}}}\approx1.51[/tex]

The P-value= [tex]P(x<-1.51)+P(x>1.51)[/tex]

[tex]=2P(z>1.51)=2(1-P(z<1.15))\\=2(1-0.9344783)\\=0.1310434\approx0.1310[/tex]

Hence, the required probability = 0.1310

The probability that the sample mean would differ from the population mean by more than 1.9 millimeters is approximately 0.1744.

To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters, we can use the Z-score formula. The Z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. Once we have the Z-score, we can use a Z-table or calculator to find the probability.

In this case, the Z-score is (1.9 - 0) / (7 / sqrt(31)) = 0.935. To find the probability of a Z-score greater than 0.935, we can look up the corresponding area under the normal distribution curve in a Z-table or use a calculator. The probability is approximately 0.1744.

https://brainly.com/question/32117953

#SPJ3

Answer:

See below.

Step-by-step explanation:

1.   Suppose that the sum is rational  then we can write:

a/b + i = c/d     where i is irrational and by definition a/b and c/d are rational.

Rearranging:

i = c/d - a/b

Now the sum on the right is rational  so 'irrational' = 'rational' which is a contradiction.

So  the original supposition is false and the sum must be irrational.

2. Proof of For all integers m if m is even then 3m + 7 is odd:

If m is even then 3m is even.

Suppose 3m + 7 is even, then:

3m + 7 = 2p  where p is an integer.

3m - 2p  = -7

But 3m and 2p are both even so their result is even  and -7 is odd.

Therefore the original supposition is false because it leads to a contradiction,  so 3m + 7 is odd.

The correct statements regarding the sets A and B is:

        II.

[tex]A\bigcap B=\{13,15\}[/tex]

We are given set A as:

[tex]A=\{3,4,13,15\}[/tex]

and set B as:

[tex]B=\{13,15,21,25,30,37\}[/tex]

and the three statements are given by:

I.

[tex]A\bigcup B=\{12,4,21,25,30,37\}[/tex]

II.

[tex]A\bigcap B=\{13,15\}[/tex]

III.

[tex]A\subset B[/tex]

i.e. from the given sets we have:

[tex]A\bigcup B=\{3,4,13,15,21,25,30,37\}[/tex]

i.e.

[tex]A\bigcap B=\{13,15\}[/tex]

All the elements of A are contained in B.

But from the given sets we see that:

3,4 belongs to A but they do not belong to B.

Hence, A is not a subset of B.

Answer:

She earns that day $103.5

Step-by-step explanation:

Lets explain how to solve the question

- She is paid $12.50 per hour between 8 am and 4 pm

- After 4 pm she is paid $14.25 per hour

- She works one particular day from 10 am to 6 pm

* Lets distribute her time of work into two part according to her

 payment above

# 1st part from 10 am to 4 pm

∵ She is paid $12.50 per hour between 8 am to 4 pm

∵ She works from 10 am to 4 pm

- Calculate how many hours between 10 am and 4 pm

∵ am and pm not like terms in time, so change the time to 24-hours

∵ 4 pm in 24-hour time is 4 + 12 = 16

∴ She works form 10 to 16

∴ She works for 6 hours (16  - 10 = 6)

∴ She is paid = 12.50 × 6 = $75

# 2nd part after 4 pm to 6 pm

∵ She is paid $14.25 per hour after 4 pm

∵ She works from 4 pm to 6 pm

- Calculate how many hours between 4 pm and 6 pm

∵ Both times are pm

∴ She works for 2 hours (6 - 4 = 2)

∴ She is paid = 14.25 × 2 = $28.5

- Add the two answers of the 1st part and the 2nd part

∴ She earns = 75 + 28.5 = $103.5

* She earns that day $103.5

Answer:

$103.50.

Step-by-step explanation:

From 10 am until 4 pm ( 6 hours) she earns 6 * 12.50 = $75.00.

Then from 4 pm to 6 pm she earns 2*14.25 = $28.50.

Total for that day = 75 + 28.50

= $103.50.

Answer:

The amount he should borrow is 19.25 hundreds dollars.

Step-by-step explanation:

Given : The cost (in hundreds of dollars) of tuition at the community college is given by [tex]T = 1.25c + 3[/tex], where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits.

To find : Use the model to determine how much money he should borrow?

Solution :

The model is given by [tex]T = 1.25c + 3[/tex]

Where,

c is the number of credits the student has registered.

T is the cost of tuition at the community college.

If a student is planning to take out a loan to cover the cost of 13 credits.

The amount he should borrow will get by putting the value of c in the model,

[tex]T = 1.25c + 3[/tex]

[tex]T = 1.25(13) + 3[/tex]

[tex]T = 16.25 + 3[/tex]

[tex]T =19.25[/tex]

Therefore, The amount he should borrow is 19.25 hundreds dollars.

The fourth point that will complete the quadrilateral is a rhombus is -3.

In a rhombus, two parallel sides must be equal.

            slope(m)=tan∅=y axis/x axis.

Calculation:-

1st coordinate- side=[tex]\sqrt{10}[/tex]

2nd coordinate- side= [tex]\sqrt{18}[/tex]

3rd coordinate  must be [tex]\sqrt{10}[/tex]

so,y=-3

Learn more about coordinate geometry here:-https://brainly.com/question/18269861

#SPJ2

Answer:

Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?

(–1, -5 )

Step-by-step explanation:

No idea what the cited section's method is, but this ODE is linear:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]

[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]

Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:

[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]

[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]

Integrating both sides gives

[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]

[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]

Answer:

[tex]g(4) = 8 \pi h\\\\g(\frac{3}{2}) = 3 \pi h\\\\ g(2c) = 4 \pi ch\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

Step-by-step explanation:

You need to substitute [tex]r=4[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(4) = 2 \pi(4)h\\\\g(4) = 8 \pi h[/tex]

Substitute [tex]r=\frac{3}{2}[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(\frac{3}{2}) = 2 \pi(\frac{3}{2})h\\\\g(\frac{3}{2}) = 3 \pi h[/tex]

Substitute [tex]r=2c[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(2c) = 2 \pi(2c))h\\\\g(2c) = 4 \pi ch[/tex]

Substitute [tex]r=c+3[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(c+3) = 2 \pi (c+3)h\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

For this case we have the following function:

[tex]g (r) = 2 \pi * r * h[/tex]

We must evaluate the function for different values:

[tex]g (4) = 2 \pi * (4) * h = 8 \pi*h\\g (\frac {3} {2}) = 2 \pi * (\frac {3} {2}) * h = 3 \pi*h\\g (2c) = 2 \pi * (2c) * h = 4 \pi * c * h\\g (c + 3) = 2 \pi * (c + 3) * h = 2 \pi * c * h + 6 \pi * h[/tex]

Answer:

[tex]g (4) = 8 \pi*h\\g (\frac {3} {2}) =3 \pi*h\\g (2c) = 4 \pi * c * h\\g (c + 3) = 2 \pi * c * h + 6 \pi * h[/tex]

19 and 42 are coprime, so we can use the CRT right away. Start with

[tex]x=19+42[/tex]

Taken mod 42, we're left with a remainder of 19. Notice that

[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]

so we need to multiply the first term by 3 to get the remainder we want.

[tex]x=19\cdot3+42[/tex]

Next, taken mod 19, we're left with a remainder of 4. Notice that

[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]

so we need to multiply the second term by 6.

Then by the CRT, we have

[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]

so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.

Answer:

433 minutes

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

The nurse is infusing 1.5 ml of solution every x minutes, Since there are a total of 650 ml we can divide the amount the nurse is infusing every x minutes by the total in order to calculate the amount of time it will take the nurse to finish.

[tex]\frac{650}{1.5} = 433.33min[/tex]

So we can see that it will take the nurse 433 min to infuse all of the solution.

Answer:

In 433.33 minutes will it take for the medicine to be given.

Step-by-step explanation:

Given : A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution.

To find : How many minutes will it take for the medicine to be given ?

Solution :

As 1.5 ml of solution infuse in x minutes

i.e. the amount of solution she has = [tex]1.5x[/tex]

Now, she has 650 ml of solution.

which means  [tex]1.5x=650[/tex]

Solving the equation,

[tex]x=\frac{650}{1.5}[/tex]

[tex]x=433.33[/tex]

Therefore, in 433.33 minutes will it take for the medicine to be given.

a. Recall that

[tex]\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C[/tex]

For [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

By integrating both sides, we get

[tex]\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

If [tex]x=0[/tex], then

[tex]\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0[/tex]

so that

[tex]\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

We can shift the index to simplify the sum slightly.

[tex]\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n[/tex]

b. The power series for [tex]x\ln(1-x)[/tex] can be obtained simply by multiplying both sides of the series above by [tex]x[/tex].

[tex]\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n[/tex]

c. We have

[tex]\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)[/tex]

[tex]\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}[/tex]

The power series of f(x) = ln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex], the power series of xln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex] and the value of ln(2) is [tex]\rm \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex].

Given :

f(x) = ln (1−x)

a) The integration of 1/(1 - x) is given by:

[tex]\rm \int \dfrac{1}{1-x}dx=-ln|1-x| + C[/tex]

When |x| >1 :

[tex]\dfrac{1}{1-x} = \sum^{\infty}_{n=0} x^n[/tex]

Now, integrate on both sides in the above equation.

[tex]\rm -ln(1-x) = C+\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]   --- (1)

Now, at (x = 0) the above expression becomes:

[tex]\rm -ln(1-0) = C+\sum^{\infty}_{n=0}\dfrac{0^{n+1} }{n+1}[/tex]

By simplifying the above expression in order to get the value of C.

C = 0

Now, substitute the value of C in expression (1).

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]

Now, by shifting the index the above expression becomes:

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex]

b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).

[tex]\rm xln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex]

c) Now, substitute the value x = 1/2 in the expression (1).

[tex]\rm ln2 = \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex]

For more information, refer to the link given below:

https://brainly.com/question/16407904

Answer:

$333.17

Step-by-step explanation:

Use the compounding formula

[tex]A(t)=P(1+r)^t[/tex]

where A(t) is the amount at the end of the compounding,

P is the initial deposit,

r is the interest rate in decimal form, and

t is the time in years.

Filling in our info:

[tex]A(t)=180(1+.08)^8[/tex]

Simplify a bit to

[tex]A(t)=180(1.08)^8[/tex]

Raise 1.08 to the 8th power and get

A(t) = 180(1.85093021) and then multiply to get

A(t) = $333.17

Answer:

[tex]4\sqrt{17}[/tex]

Step-by-step explanation:

Let's find the answer by using the arc length formula which is:

[tex]\int\limits^a_b {\sqrt{1+(\frac{dy}{dx})^{2} } } \, dx[/tex]

First, let's find dy/dx which is:

y=4x-5

y'=4*(1)-0

y'=4, now let's use the formula:

[tex]\int\limits^3_{-1} {\sqrt{1+4^{2}} } \, dx=\sqrt{17} *(3-(-1))=4\sqrt{17}[/tex]

Now, using the distance formula we have:

[tex]d=\sqrt{(x2-x1)^{2} +(y2-y1)^{2} }[/tex]

[tex]y(-1)=4*(-1)-5=-9 \\y(3)=4*(3)-5=7[/tex]

So we have two points (-1, -9) and (3, 7) so:

[tex]d=\sqrt{(3-(-1))^{2} +(7-(-9))^{2} }=4\sqrt{17}[/tex]

Notice both equations gave the same length [tex]4\sqrt{17}[/tex].

Final answer:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, integrate sqrt(1 + (dy/dx)^2) from x = -1 to x = 3. The length of the curve is 4√17. The result can be confirmed by calculating the length using the distance formula.

Explanation:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, we need to integrate the square root of 1 + (dy/dx)^2 from x = -1 to x = 3. The derivative of y = 4x - 5 is dy/dx = 4. Substituting this into the arc length formula, we have:

L = ∫sqrt(1 + (dy/dx)^2) dx = ∫sqrt(1 + 4^2) dx = ∫sqrt(17) dx = x√17 + C

Now, plugging in the limits of integration, we have:

L = [(3√17) + C] - [(-1√17) + C] = (3√17) - (-1√17) = 4√17

To check our answer, we can use the distance formula. The endpoints of the line segment are (-1, -9) and (3, 7). Using the distance formula:

D = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - (-1))^2 + (7 - (-9))^2) = sqrt(4^2 + 16^2) = sqrt(272) = 16√17

As we can see, the length of the curve obtained using the arc length formula (4√17) matches the length calculated using the distance formula (16√17), confirming our answer.

Answer:

(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

(B) f(x + h) - f(x) = 8xh + 4h² - 6h

(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

Step-by-step explanation:

* Lets explain how to solve the problem

- The function f(x) = 4x² - 6x + 6

- To find f(x + h) substitute x in the function by (x + h)

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) = 4(x + h)² - 6(x + h) + 6

- Lets simplify 4(x + h)²

∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²

∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²

- Lets simplify 6(x + h)

∵ 6(x + h) = 6(x) + 6(h)

∴ 6(x + h) = 6x + 6h

∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6

- Remember (-)(+) = (-)

∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

- Lets find f(x + h) - f(x)

∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6

∵ f(x) = 4x² - 6x + 6

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)

- Remember (-)(-) = (+)

∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6

- Simplify by adding the like terms

∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)

∴ f(x + h) - f(x) = 8xh + 4h² - 6h

* (B) f(x + h) - f(x) = 8xh + 4h² - 6h

- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]

∵ f(x + h) - f(x) = 8xh + 4h² - 6h

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]

- Simplify by separate the three terms

∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]

∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]

* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]