A rumor spreads among a group of 400 people. The number of people, N(t), who have heard the rumor by the time t in hours since the rumor started to spread can be approximated by a function of the form
N(t) = 400 / (1+399e^(-0.4t))
Approximately how long will it take until half the people have heard the rumor?

Answer:

C) It is a linear function because the graph contains the points (0, 8), (1, 12), (2, 16), which are on a straight line.  

Explanation:

The missing options for this question are:

A) It is a linear function because the graph contains the points (8, 0), (12, 1), (16, 2), which are on a straight line.  

B) It is a nonlinear function because the graph contains the points (8, 0), (12, 1), (16, 2), which are not on a straight line.

C) It is a linear function because the graph contains the points (0, 8), (1, 12), (2, 16), which are on a straight line.  

D) It is a nonlinear function because the graph contains the points (0, 8), (1, 12), (2, 16), which are not on a straight line.

The given equation is:

y = 4x + 8

Replacing x by 0, we get y = 8. This means point (0, 8) lies on the graph of the function.

Replacing x by 1, we get y = 12. This means point (1, 12) lies on the graph of the function.

Replacing x by 2, we get y = 16. This means point (2, 16) lies on the graph of the function.

If we plot these three points on a graph we can draw a straight line through these. Hence, based on this we can conclude that:

C) It is a linear function because the graph contains the points (0, 8), (1, 12), (2, 16), which are on a straight line.  

The line passing through these 3 points would actually be the given equation y = 4x + 8 as one and only one line can pass through 3 distinct points.

(a)

          [tex]f(x+ h)=8x+8h+3[/tex]  

(b)

            [tex]f(x+ h)-f(x)=8h[/tex]          

(c)

             [tex]\dfrac{f(x+ h)-f(x)}{h}=8[/tex]

We are given a function f(x) as :

              [tex]f(x)=8x+3[/tex]

(a)

           [tex]f(x+ h)[/tex]

We will substitute (x+h) in place of x in the function f(x) as follows:

[tex]f(x+h)=8(x+h)+3\\\\i.e.\\\\f(x+h)=8x+8h+3[/tex]

(b)

       [tex]f(x+ h)-f(x)[/tex]              

Now on subtracting the f(x+h) obtained in part (a) with the function f(x) we have:

[tex]f(x+h)-f(x)=8x+8h+3-(8x+3)\\\\i.e.\\\\f(x+h)-f(x)=8x+8h+3-8x-3\\\\i.e.\\\\f(x+h)-f(x)=8h[/tex]

(c)

           [tex]\dfrac{f(x+ h)-f(x)}{h}[/tex]            

In this part we will divide the numerator expression which is obtained in part (b) by h to get:

           [tex]\dfrac{f(x+ h)-f(x)}{h}=\dfrac{8h}{h}\\\\i.e.\\\\\dfrac{f(x+h)-f(x)}{h}=8[/tex]    

f(-2)=g(-2)

Think of the number in parentheses as your x value. f(x)=y. In this case the line hit at (-2,4) so when f(x) = 4, g(x)= 4 and 4=4 so you then have to find the x which in this case is -2. I’m pretty bad at explaining but there’s your answer

Answer:

f(-2)=g(-2)

Think of the number in parentheses as your x value. f(x)=y. In this case the line hit at (-2,4) so when f(x) = 4, g(x)= 4 and 4=4 so you then have to find the x which in this case is -2. I’m pretty bad at explaining but there’s your answer

Step-by-step explanation:

Answer:

B. price

Step-by-step explanation:

The equation is linear and looks like this:

C(x) = 5.99x

where C(x) is the cost of x number of movies.  The cost is the dependent variable, since it is dependent upon how many movies you rent at 5.99 each.

Answer:

Step-by-step explanation:

Follow these steps.  First let's look at the problem:

[tex]1.04=2^{\frac{1}{j}}[/tex]

Multiply each side by j/1:

[tex]1.04^{\frac{j}{1}} =2^{\frac{1}{j} *\frac{j}{1}}[/tex]

which simplifies to

[tex]1.04^j=2[/tex]

Now take the natural log of both sides:

[tex]ln(1.04)^j=ln(2)[/tex]

The power rule says that we can bring the exponent down in front of the natural log:

(j) ln(1.04) = ln(2) and then divide both sides by ln(1.04)

Do the division on your calculator to get that j = 17.7

Answer:  The required solution of the given system is

(x, y) = (3, 1)  and  (4, 0).

Step-by-step explanation:  We are given to solve the following system of equations :

[tex]x+y=4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\y=x^2-8x+16~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]

From equation (i), we have

[tex]x+y=4\\\\\Rightarrow y=4-x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)[/tex]

Substituting the value of y from equation (iii) in equation (ii), we get

[tex]y=x^2-8x+16\\\\\Rightarrow 4-x=x^2-8x+16\\\\\Rightarrow x^2-8x+16-4+x=0\\\\\Rightarrow x^2-7x+12=0\\\\\Rightarrow x^2-4x-3x+12=0\\\\\Rightarrow x(x-4)-3(x-4)=0\\\\\Rightarrow (x-3)(x-4)=0\\\\\Rightarrow x-3=0,~~~~~~~x-4=0\\\\\Rightarrow x=3,~4.[/tex]

When, x = 3, then from (iii), we get

[tex]y=4-3=1.[/tex]

And, when x = 4, then from (iii), we get

[tex]y=4-4=0.[/tex]

Thus, the required solution of the given system is

(x, y) = (3, 1)  and  (4, 0).

Answer:

Linear function is specified by the equation ⇒ answer B

Step-by-step explanation:

* Look to the attached file

Answer:

B . Linear function.

Step-by-step explanation:

3x + 4y = 1

The degree of x and y is  1 and

if we drew a graph of this function we get a straight line.

Answer:

amount is $320243.25 need in your account at the beginning

Money pull in 25 years is $750000

money interest is $429756.75

Step-by-step explanation:

Given data

principal (P) = $30000

time (t) = 25 years

rate (r) = 8% = 0.08

to find out

amount need in beginning, money pull out , and interest money

solution

We know interest compounded annually so n = 1

we apply here compound annually formula i.e.

amount = principal ( 1 - [tex](1+r/n)^{-t}[/tex] / r/k

now put all these value principal, r , n and t in equation 1

amount = 30000 ( 1 - [tex](1+0.08/1)^{-25}[/tex] / 0.08/1

amount = 30000 × 0.853982  / 0.08

amount = $320243.25 need in your account at the beginning

Money pull in 25 years is $30000 × 25 i.e

Money pull in 25 years is $750000

money interest = total money pull out in 25 years - amount at beginning need

money interest = $750000 - $320243.25

money interest = $429756.75

The cash flow in the account are;

a. Amount in the account at the beginning is approximately $320,243.3

b. The total money pulled out is $750,000

c. Amount of in interest in money pulled out approximately $429,756.7

The reason the above values are correct are as follows;

The given parameter are;

The amount to be withdrawn each year, d = $30,000

The number of years of withdrawal, n = 25 years

The interest rate on the account = 8 %

a. The amount that should be in the account at the beginning is given by the payout annuity formula as follows;

[tex]P_0 = \dfrac{d \times \left(1 - \left(1 + \dfrac{r}{k} \right)^{-n\cdot k}\right) }{\left(\dfrac{r}{k} \right)}[/tex]

P₀ = The principal or initial balance in the account at the beginning

d = The amount to be withdrawn each year = $30,000

r =  The interest rate per annum = 8%

k = The number of periods the interest is applied in a year = 1

n = The number of years withdrawal is made = 25

We get;

[tex]P_0 = \dfrac{30,000 \times \left(1 - \left(1 + \dfrac{0.08}{1} \right)^{-25\times 1} \right) }{\left( \dfrac{0.08}{1} \right)} \approx 320,243.3[/tex]

The amount needed in the account at the beginning, P₀ ≈ $320,243.3

b. The amount of money pulled out, A = n × d

Therefore, A = 25 × $30,000 = $750,000

c. The amount of money received as interest, I = A - P₀

∴ I = $750,000 - $320,243.3 ≈ $429,756.7

Learn more about payout annuities here:

https://brainly.com/question/23553423

Answer:

1st year: $ 622,500

2nd year: $415,000

3rd year: $207,500

Step-by-step explanation:

Step 1: Write the beginning book value of the asset

$830,000

Step 2: Determine the asset's estimated useful life

8 years

Step 3: Determine the asset's salvage value

$75,000

Step 4: Subtract the salvage value from the beginning value to get the total depreciation amount for the asset's total life.

830,000 - 75,000 = $755,000

Step 5: Calculate the annual depreciation rate

Depreciation rate = 100%/8 years = 12.5%

Step 6: Multiply the beginning value by twice the annual depreciation rate to find the depreciation expense

Depreciation expense = 830,000 x 25% = $207,500

Step 7: Subtract the depreciation expense from the beginning value to find the ending period value

Ending period value for 1st year: Beginning value - depreciation expense

830,000 - 207,500 = $ 622,500

Ending period value for 2nd year: 622,500 - 207,500 = $ 415,000

Ending period value for 3rd year: 415,000 - 207,500 = $ 207,500

!!

Answer with Step-by-step explanation:

Let A= {a, b,c,d} and B= {1, 2, 3, 4, 5, 6}

Define f:A→B

where  f = {(a, 1), (b, 3), (c, 5), (d, 2)}

a. What is the co-domain of f  

The co-domain or target set of a function is the set into which all of the output of the function is constrained to fall.

Here, all the values are constrained to fall on the set B

Hence, co-domain={1, 2, 3, 4, 5, 6}

b. What is the range of f?

The set of all output values of a function is called range.

Here,  on putting the values of set A we get the output values as:1,2,3 and 5

Hence, Range={1,2,3,5}

c. Is f 1-1 function Explain.

A function for which every element of the range of the function corresponds to exactly one element of the domain.

Yes, f is 1-1

(Since, every element of range i.e. 1,2,3 and 5 corresponds to only one element of set A)

d. Can there exist a bijection between A and B? Explain.

No, there cannot exist bijection from A to B

because if a bijection exist between two sets then there cardinalities are same but A and B have different cardinalities

A has cardinality 4 and B has cardinality 6

Answer:

9 TV ads and 20 radio ads

Step-by-step explanation:

He has $37,000 to spend. He has to sum that amount of money between the TV and radio ads. Each TV ad costs $3000 while each radio ad costs $500, so the equation that represents that is 37000 = 3000x  + 500y  

The same happens with the amount of voters he needs to reach, the equation is 130000 = 10000x + 2000y

The system that represents this is

[tex]\left \{ {{3000x+500y=37000} \atop {10000x+2000y=130000}} \right.[/tex]

And the augmented matrix is

[tex]\left[\begin{array}{cc|c}3000&500&37000\\10000&2000&130000\end{array}\right][/tex]

First we divide the first row by 3000 and the second by 10000:

[tex]\left[\begin{array}{cc|c}1&1/6&37/3\\1&1/5&13\end{array}\right][/tex]

Then we multiply the second row by (-1) and we add the first row:

[tex]\left[\begin{array}{cc|c}1&1/6&37/3\\0&-1/30&-2/3\end{array}\right][/tex]

Now we multiply the second row by -30:

[tex]\left[\begin{array}{cc|c}1&1/6&37/3\\0&1&20\end{array}\right][/tex]

Finally, to the first row we add the second one multiply by (-1/6):

[tex]\left[\begin{array}{cc|c}1&0&9\\0&1&20\end{array}\right][/tex]

So, x = 9 and y = 20

That means 9 TV and 20 radio ads will contact 130,000 voters using the allocated funds

Answer:

[tex]6.1\frac{\text{ g}}{\text{ cm}^3}[/tex]

Step-by-step explanation:

We have been given that mass of an irregular object is 1220 g and it displaces 200 cubic cm of water when placed in a large overflow container. We are asked to find density of the object.

We will use density formula to solve our given problem.

[tex]\text{Density}=\frac{\text{Mass}}{\text{Volume}}[/tex]

Since the object displaces 200 cubic cm of water, so the volume of irregular object will be equal to 200 cubic cm.

Upon substituting our given values in density formula, we will get:

[tex]\text{Density}=\frac{1220\text{ g}}{200\text{ cm}^3}[/tex]

[tex]\text{Density}=\frac{61\times 20\text{ g}}{10\times 20\text{ cm}^3}[/tex]

[tex]\text{Density}=\frac{61\text{ g}}{10\text{ cm}^3}[/tex]

[tex]\text{Density}=6.1\frac{\text{ g}}{\text{ cm}^3}[/tex]

Therefore, the density of the irregular object will be 6.1 grams per cubic centimeters.

Hey there!:

Here , we choose the 10 marbles from the box of 40 marbles without replacement  

Therefore , probability is changes for every time  

Also , the trials are dependent  

Therefore ,the assumptions of binomial distributions are not satisfied

Therefore ,  Not binomial : the trials are not independent

Hope this helps!

The given procedure does not follow the characteristics of a binomial distribution.

The procedure of choosing marbles with replacement from a box with different colored marbles does not meet the criteria for a binomial distribution.

The given procedure does not result in a binomial distribution because in a binomial distribution, the trials must be independent, there must be a fixed number of trials, and there can only be two outcomes (success and failure).

In this case, choosing marbles from a box with replacement and tracking their colors does not meet the criteria for a binomial experiment, as the trials are not independent, the number of trials is not fixed, and there are more than two possible outcomes (purple, red, green).

Therefore, the given procedure does not follow the characteristics of a binomial distribution.

Answer: 95% confidence interval = 20,000 ± 2.12[tex]\times[/tex][tex]\frac{1500}{\sqrt{17} }[/tex]

( 19228.736 , 20771.263 ) OR ( 19229 , 20771 )

Step-by-step explanation:

Given :

Sample size(n) = 17

Sample mean = 20000

Sample standard deviation = 1,500

5% confidence

∴ [tex]\frac{\alpha}{2}[/tex] = 0.025

Degree of freedom ([tex]d_{f}[/tex]) = n-1 = 16

∵ Critical value at ( 0.025 , 16 ) = 2.12

∴ 95% confidence interval = mean ± [tex]Z_{c}[/tex][tex]\times[/tex][tex]\frac{\sigma}{\sqrt{n} }[/tex]

Critical value  at 95% confidence interval = 20,000 ± 2.12[tex]\times[/tex][tex]\frac{1500}{\sqrt{17} }[/tex]

( 19228.736 , 20771.263 ) OR ( 19229 , 20771 )

Answer:

[tex]f^{-1}(x)=\frac{x+15}{3}[/tex]

or

[tex]f^{-1}(x)=\frac{x}{3}+5[/tex]

Step-by-step explanation:

That means we want to find the inverse function of f(x)=3x-15.

The inverse function is just the swapping of x and y really.  We tend to remake the y the subject afterwards.  

So we are given:

y=3x-15

First step: Swap x and y

x=3y-15

Now it's time to solve for y.

Second step: Add 15 on both sides:

x+15=3y

Third step: Divide both sides by 3:

(x+15)/3=y

The inverse function is:

[tex]f^{-1}(x)=\frac{x+15}{3}[/tex]

You can also split up the fraction like so:

[tex]f^{-1}(x)=\frac{x}{3}+\frac{15}{3}[/tex]

The last fraction there can be reduced:

[tex]f^{-1}(x)=\frac{x}{3}+5[/tex]

For the corresponding homogeneous equation,

[tex]x^2y''+xy'+y=0[/tex]

we can look for a solution of the form [tex]y=x^m[/tex], with derivatives [tex]y'=mx^{m-1}[/tex] and [tex]y''=m(m-1)x^{m-2}[/tex]. Substituting these into the ODE gives

[tex]m(m-1)x^m+mx^m+x^m=0\implies m^2+1=0\implies m=\pm i[/tex]

which admits two solutions, [tex]y_1=x^i[/tex] and [tex]y_2=x^{-i}[/tex], which we can write as

[tex]x^i=e^{\ln x^i}=e^{i\ln x}=\cos(\ln x)+i\sin(\ln x)[/tex]

and by the same token,

[tex]x^{-i}=\cos(\ln x)-i\sin(\ln x)[/tex]

so we see two independent solutions that make up the characteristic solution,

[tex]y_c=C_1\cos(\ln x)+C_2\sin(\ln x)[/tex]

For the non-homogeneous ODE, we make the substitution

[tex]x=e^t\iff t=\ln x[/tex]

so that by the chain rule, the first derivative becomes

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac1x[/tex]

[tex]\dfrac{\mathrm dy}{\mathrm dx}=e^{-t}\dfrac{\mathrm dy}{\mathrm dt}[/tex]

Let [tex]f(t)=\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then the second derivative becomes

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm df}{\mathrm dx}=\dfrac{\mathrm df}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\left(-e^{-t}\dfrac{\mathrm dy}{\mathrm dt}+e^{-t}\dfrac{\mathrm d^2y}{\mathrm dt^2}\right)\dfrac1x[/tex]

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=e^{-2t}\left(\dfrac{\mathrm d^2y}{\mathrm dt^2}-\dfrac{\mathrm dy}{\mathrm dt}\right)[/tex]

Substituting these into the ODE gives

[tex]e^{2t}\left(e^{-2t}\left(\dfrac{\mathrm d^2y}{\mathrm dt^2}-\dfrac{\mathrm dy}{\mathrm dt}\right)\right)+e^t\left(e^{-t}\dfrac{\mathrm dy}{\mathrm dt}\right)+y=t^2+2e^t[/tex]

[tex]y''+y=t^2+2e^t[/tex]

Look for a particular solution [tex]y_p=a_0+a_1t+a_2t^2+be^t[/tex], which has second derivative [tex]{y_p}''=2a_2+be^t[/tex]. Substituting these into the ODE gives

[tex](2a_2+be^t)+(a_0+a_1t+a_2t^2+be^t)=t^2+2e^t[/tex]

[tex](2a_2+a_0)+a_1t+a_2t^2+2be^t=t^2+2e^t[/tex]

[tex]\implies a_0=-2,a_1=0,a_2=1,b=1[/tex]

so that the particular solution is

[tex]y_p=t^2-2+e^t[/tex]

Solving in terms of [tex]x[/tex] gives the solution

[tex]y_p=(\ln x)^2-2+x[/tex]

and the overall general solution is

[tex]y=y_c+y_p[/tex]

[tex]\boxed{y=C_1\cos(\ln x)+C_2\sin(\ln x)+(\ln x)^2-2+x}[/tex]

Answer:

42.25 feet

Step-by-step explanation:

The height function is a parabola.  The maximum value of a negative parabola is at the vertex, which can be found with:

x = -b/2a

where a and b are the coefficients in y = ax² + bx + c.

Here, we have y = -16t² + 52t.  So a = -16 and b = 52.  The vertex is at:

t = -52 / (2×-16)

t = 13/8

Evaluating the function:

h(13/8) = -16(13/8)² + 52(13/8)

h(13/8) = -169/4 + 169/2

h(13/8) = 169/4

h(13/8) = 42.25

Answer:

42.25 feet.

Step-by-step explanation:

The maximum height can be found by converting to vertex form:

h(t) = 52t - 16t^2

h(t) =   -16 (  t^2  - 3.25t)

h(t) = -16 [ (t - 1.625)^2 - 2.640625 ]

= -16(t - 1.625 ^2) + 42.25

Maximum height = 42.25 feet.

Another method of solving this is by using Calculus:

h(t) = 52t - 16t^2

Finding the derivative:

h'(t) = 52 - 32t

This = zero for  a maximum/minimum value.

52 - 32t = 0

t = 1.625 seconds at maximum height.

It is a maximum because  the  path is a parabola  which opens downwards. we know this because of the  negative coefficient of x^2.

Substituting in the original formula:h(t) = 52(1.625)- 16(1.625)^2

= 42.25 feet.

After working 6 hours, Manuel will have earned $72.

To find out how much money Manuel will have earned after doing 6 hours of yard work, we can use the given equation:

[tex]\[ d = 12h \][/tex]

Where ( d ) represents the amount of money earned and ( h ) represents the number of hours spent doing yard work.

Substitute the given value of [tex]\( h = 6 \)[/tex] into the equation:

[tex]\[ d = 12 \times 6 \][/tex]

Now, multiply 12 by 6:

[tex]\[ d = 72 \][/tex]

So, after working 6 hours, Manuel will have earned $72.

Answer:

[tex]\dfrac{2}{7}[/tex]

Step-by-step explanation:

3 different china dinner sets, each consisting of 5 plates consist of 15 plates.

A customer can select 2 plates in

[tex]C^{15}_2=\dfrac{15!}{2!(15-2)!}=\dfrac{15!}{13!\cdot 2!}=\dfrac{13!\cdot 14\cdot 15}{2\cdot 13!}=7\cdot 15=105[/tex]

different ways.

2 plates can be selected from the same dinner set in

[tex]3\cdot C^5_2=3\cdot \dfrac{5!}{2!(5-2)!}=3\cdot \dfrac{3!\cdot 4\cdot 5}{2\cdot 3!}=3\cdot 2\cdot 5=30[/tex]

different ways.

Thus, the probability that the 2 plates selected will be from the same dinner set is

[tex]Pr=\dfrac{30}{105}=\dfrac{6}{21}=\dfrac{2}{7}[/tex]

[tex]C[/tex], the boundary of [tex]S[/tex], is a circle in the [tex]x,y[/tex] plane centered at the origin and with radius 2, hence we can parameterize it by

[tex]\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath[/tex]

with [tex]0\le t\le2\pi[/tex]. Then the line integral is

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^{2\pi}(2\sin t\,\vec\imath+2\cos t\,\vec k)\cdot(-2\sin t\,\vec\imath+2\cos t\,\vec\jmath)\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^{2\pi}-4\sin^2t\,\mathrm dt[/tex]

[tex]=\displaystyle-2\int_0^{2\pi}(1-\cos2t)\,\mathrm dt=\boxed{-4\pi}[/tex]

By Stokes' theorem, the line integral of [tex]\vec F[/tex] along [tex]C[/tex] is equal to the surface integral of the curl of [tex]\vec F[/tex] across [tex]S[/tex]:

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]

Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(4-u^2)\,\vec k[/tex]

with [tex]0\le u\le2[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k[/tex]

The curl is

[tex]\nabla\times\vec F=-\vec\imath-\vec\jmath-\vec k[/tex]

Then the surface integral is

[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^2(-\vec\imath-\vec\jmath-\vec k)\cdot(2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle-\int_0^{2\pi}\int_0^2(2u^2\cos v+2u^2\sin v+u)\,\mathrm du\,\mathrm dv=\boxed{-4\pi}[/tex]

The circulation F · dr over the curve C is calculated both directly and using Stokes' Theorem. In both instances, the circulation equals zero, indicating there is no rotation of the vector field along the curve C.

To compute the circulation F · dr over the curve C, we can use either a direct calculation or Stokes' theorem. In the direct calculation, we parametrize C using polar coordinates (x = rcos(θ), y = rsin(θ), z = 0), resulting in dr = dx i + dy j + dz k where dx = -rsin(θ) dθ, dy = rcos(θ) dθ, and dz = 0. Then, F · dr = y dx + z dy + x dz = -r²cos(θ) sin(θ)dθ + 0 + 0 = 0, since the integrand is zero. So the circulation as calculated directly is zero.

For Stokes' theorem, we calculate the curl of F, ∇ x F = (i j k ∂/∂x ∂/∂y ∂/∂z) x (y z x) = (-1 -1 -1), and then integrate this over the surface S, yielding the same result of zero. Therefore, by both direct calculation and using Stokes' theorem, the circulation F · dr over the curve C is zero.

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Answer: 0.0062

Step-by-step explanation:

Given : Mean : [tex]\mu=\ 31[/tex]

Standard deviation :[tex]\sigma= 4[/tex]

Sample size : [tex]n=100[/tex]

Assume that age of people in the country is normally distributed.

The formula to calculate the z-score :-

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

For x = 32

[tex]z=\dfrac{32-31}{\dfrac{4}{\sqrt{100}}}=5[/tex]

The p-value = [tex]P(x\geq32)=P(z\geq5)[/tex]

[tex]=1-P(z<5)=1- 0.9937903\approx0.0062[/tex]

Hence, the the probability that the average age of our sample is at least =0.0062

The probability that the average age of the sample is at least 32 is approximately 0.62%.

To find the probability that the average age of our sample is at least 32, we can use the normal distribution. The average age of the population is 31 years old and the standard deviation is 4 years. Since we have a large sample size (100), we can use the central limit theorem to assume that the sample mean will follow a normal distribution.

To calculate the probability, we need to find the z-score for the value 32. The z-score formula is z = (x - μ) / (σ / √n), where x is the desired value, μ is the population mean, σ is the population standard deviation, and n is the sample size. Plugging in the values, we get z = (32 - 31) / (4 / √100) = 1 / (4 / 10) = 2.5.

Using a z-table or a calculator, we can find that the probability of a z-score of 2.5 or more is approximately 0.0062, or 0.62%.

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The equation of the plane that passes through the intersection of the planes x - z = 3 and y + 4z = 1, and is perpendicular to the plane x + y - 4z = 4, is s = 0.

To find the equation of a plane that passes through the intersection of two planes and is perpendicular to a third plane, we first need to find the intersection of the first two planes: x - z = 3 and y + 4z = 1. You can describe their line of intersection as x = z + 3 = s and y = 1 - 4z = 1 - 4(s - 3) = -4s + 13 by letting s be the parameter of the line.

Next, since our plane is perpendicular to the plane described by x + y - 4z = 4, we know the normal vector to our plane is (1,1,-4) which is the coefficients of x, y, and z in the equation of the perpendicular plane.

So, by using the point-normal form of the equation of a plane, which is (a(x-x0) + b(y-y0) + c(z-z0) = 0), where (a,b,c) is the normal vector and (x0,y0,z0) is a point on the plane. We use the point (z+3, -4z+13, z) that lies in the plane and put it all together, we get the equation of the plane as:  1(s - (s)) + 1((-4s + 13) - (-4s + 13)) - 4(s - (s)) = 0 , which simplifies to: s = 0.

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Answer:

(a) P(X=1)=0.46

(b) E[X]=1.3

Step-by-step explanation:

(a)

Let A be the event that first coin will land on heads and B be the event that second coin will land on heads.

According to the given information

[tex]P(A)=0.6[/tex]

[tex]P(B)=0.7[/tex]

[tex]P(A')=1-P(A)=1-0.6=0.4[/tex]

[tex]P(B')=1-P(B)=1-0.7=0.3[/tex]

P(X=1) is the probability of getting exactly one head.

P(X=1) = P(1st heads and 2nd tails ∪ 1st tails and 2nd heads)

          = P(1st heads and 2nd tails) + P(1st tails and 2nd heads)

Since the two events are disjoint, therefore we get

[tex]P(X=1)=P(A)P(B')+P(A')P(B)[/tex]

[tex]P(X=1)=(0.6)(0.3)+(0.4)(0.7)[/tex]

[tex]P(X=1)=0.18+0.28[/tex]

[tex]P(X=1)=0.46[/tex]

Therefore the value of P(X=1) is 0.46.

(b)

Thevalue of E[X] is

[tex]E[X]=\sum_{x}xP(X=x)[/tex]

[tex]E[X]=0P(X=0)+1P(X=1)+2P(X=2)[/tex]

[tex]E[X]=P(X=1)+2P(X=2)[/tex]                      ..... (1)

First we calculate  the value of P(X=2).

P{X = 2} = P(1st heads and 2nd heads)

             = P(1st heads)P(2nd heads)

[tex]P(X=2)=P(A)P(B)[/tex]

[tex]P(X=2)=(0.6)(0.7)[/tex]

[tex]P(X=2)=0.42[/tex]

Substitute P(X=1)=0.46 and P(X=2)=0.42 in equation (1).

[tex]E[X]=0.46+2(0.42)[/tex]

[tex]E[X]=1.3[/tex]

Therefore the value of E[X] is 1.3.

The probability of getting 1 head is 0.18. The expected value of X is 1.02.

To find P(X = 1), we need to find the probability of getting 1 head. Since the results of the flips are independent, we can multiply the probabilities of each flip. The probability of getting a head on the first coin is 0.6, and the probability of getting a tail on the second coin is 0.3. So, the probability of getting 1 head is 0.6 * 0.3 = 0.18.

To determine E[X], we can use the formula E[X] = Σ(x * P(X = x)), where x represents the possible values of X. In this case, the possible values of X are 0, 1, and 2. So, E[X] = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2). We already calculated P(X = 1) as 0.18. The probability of getting 0 heads is 0.4 * 0.3 = 0.12, and the probability of getting 2 heads is 0.6 * 0.7 = 0.42. So, E[X] = 0 * 0.12 + 1 * 0.18 + 2 * 0.42 = 1.02.

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Answer:   [tex]\dfrac{2}{9}[/tex]

Step-by-step explanation:

Let A be the event that the sum is 7 and and B be the event that the sum is 11 .

When two pair of dices rolled the total number of outcomes = [tex]n(S)=6\times6=36[/tex]

The sample space of event A ={(1,6), (6,1), (5,2), (2,5), (4,3), (3,4)}

Thus n(A)= 6

The sample space of event B = {(5,6), (6,5)}

n(B)=2

Since , both the events are independent , then the required probability is given by :-

[tex]P(A\cup B)=P(A)+P(B)\\\\=\dfrac{n(A)}{n(S)}+\dfrac{n(B)}{n(S)}=\dfrac{6}{36}+\dfrac{2}{36}=\dfrac{8}{36}=\dfrac{2}{9}[/tex]

Hence, the required probability = [tex]\dfrac{2}{9}[/tex]

Answer:

Probability that sum of numbers is either 7 or 11 is:

0.22

Step-by-step explanation:

A pair of dice is rolled.

Sample Space:

(1,1)       (1,2)        (1,3)       (1,4)          (1,5)        (1,6)

(2,1)      (2,2)       (2,3)      (2,4)         (2,5)       (2,6)

(3,1)      (3,2)       (3,3)      (3,4)        (3,5)       (3,6)

(4,1)      (4,2)       (4,3)      (4,4)         (4,5)       (4,6)

(5,1)     (5,2)      (5,3)      (5,4)         (5,5)       (5,6)

(6,1)      (6,2)       (6,3)      (6,4)         (6,5)       (6,6)

Total outcomes= 36

Outcomes with sum of numbers either 7 or 11 are in bold letters=8

i.e. number of favorable outcomes=8

So, P(sum of numbers is either 7 or 11 )=8/36

                                                                   =0.22

Answer:

720 ways to arrange

Step-by-step explanation:

Use the factorial of 6 to find this solution.  Namely, 6!

This means 6*5*4*3*2*1 which equals 720

It seems like a huge number, right?  But think of it like this:  For the first option, you have 6 collars.  After you fill the first spot with one of the 6, you have 5 left that will fill the second spot.  After the first 2 spots are filled and you used 2 of the 6 collars, there are 4 possibilities that can fill the next spot, etc.

Answer:

720 ways

Step-by-step explanation:

If you are arranging your dog's collars on a 6 hanger coat rack on the wall and if she has six collars, there are 720 ways to arrange them.

Factorial of 6 = 720

For example it could look something like,

Collar 1, Collar 2, Collar 3, Collar 4, Collar 3, Collar 2, Collar 1, and so on.

Follow the hints.

1. Let [tex]u=\dfrac yx[/tex], so that [tex]y=ux[/tex] and [tex]\mathrm dy=x,\mathrm du+u\,\mathrm dx[/tex]. Substituting into

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

gives

[tex](ux^2+u^2x^2)\,\mathrm dx-x^2(x\,\mathrm du+u\,\mathrm dx)=0[/tex]

[tex]u^2x^2\,\mathrm dx-x^3\,\mathrm du=0[/tex]

and the remaining ODE is separable:

[tex]x^3\,\mathrm du=u^2x^2\,\mathrm dx\implies\dfrac{\mathrm du}{u^2}=\dfrac{\mathrm dx}x[/tex]

Integrate both sides to get

[tex]-\dfrac1u=\ln|x|+C[/tex]

[tex]-\dfrac xy=\ln|x|+C[/tex]

[tex]\boxed{y=\dfrac x{Cx-\ln|x|}}[/tex]

2. [tex]Let [tex]v=\dfrac xy[/tex], so that [tex]x=vy[/tex] and [tex]\mathrm dx=v\,\mathrm dy+y\,\mathrm dv[/tex]. Then

[tex]2x^2y\,\mathrm dx=(3x^3+y^3)\,\mathrm dy[/tex]

becomes

[tex]2v^2y^3(v\,\mathrm dy+y\,\mathrm dv)=(3v^3y^3+y^3)\,\mathrm dy[/tex]

[tex]2v^3y^3\,\mathrm dy+2v^2y^4\,\mathrm dv=(3v^3y^3+y^3)\,\mathrm dy[/tex]

[tex]2v^2y^4\,\mathrm dv=(v^3y^3+y^3)\,\mathrm dy[/tex]

which is separable as

[tex]\dfrac{2v^2}{v^3+1}\,\mathrm dv=\dfrac{\mathrm dy}y[/tex]

Integrating both sides gives

[tex]\dfrac23\ln|v^3+1|=\ln|y|+C[/tex]

[tex]\ln|v^3+1|=\dfrac32\ln|y|+C[/tex]

[tex]v^3+1=Cy^{3/2}[/tex]

[tex]v=\sqrt[3]{Cy^{3/2}-1}[/tex]

[tex]\dfrac xy=\sqrt[3]{Cy^{3/2}-1}[/tex]

[tex]\boxed{x=y\sqrt[3]{Cy^{3/2}-1}}[/tex]

Answer:

Step-by-step explanation:

Did you perhaps mean what is the value of dx/dt at that instant?  You have a value for dy/dt to be 2dx/dt. I'm going with that, so if it is an incorrect assumption I have made, I apologize!

Here's what we have:

We have a right triangle with a reference angle (unknown as of right now), side y and side x; we also have values for y and x, and the fact that dθ/dt=-.01

So the game plan here is to use the inverse tangent formula to solve for the missing angle, and then take the derivative of it to solve for dx/dt.

Here's the inverse tangent formula:

[tex]tan\theta=\frac{y}{x}[/tex]

and its derivative:

[tex]sec^2\theta\frac{d\theta }{dt} =\frac{x\frac{dy}{dt}-y\frac{dx}{dt}  }{x^2}}[/tex]

We have values for y, x, dy/dt, and dθ/dt.  We only have to find the missing angle theta and solve for dx/dt.

Solving for the missing angle first:

[tex]tan\theta =\frac{32}{24}[/tex]

On your calculator you will find that the inverse tangent of that ratio gives you an angle of 53.1°.

Filling in the derivative formula with everything we have:

[tex]sec^2(53.1)(-.01)=\frac{24\frac{dx}{dt}-32\frac{dx}{dt}  }{24^2}[/tex]

We can simplify the left side down a bit by breaking up that secant squared like this:

[tex]sec(53.1)sec(53.1)(-.01)[/tex]

We know that the secant is the same as 1/cos, so we can make that substitution:

[tex]\frac{1}{cos53.1} *\frac{1}{cos53.1} *-.01[/tex] and

[tex]\frac{1}{cos53.1}=1.665500191[/tex]

We can square that and then multiply in the -.01 so that the left side looks like this now, along with some simplification to the right:

[tex]-.0277389=\frac{48\frac{dx}{dt} -32\frac{dx}{dt} }{576}[/tex]

We will muliply both sides by 576 to get:

[tex]-15.9776=48\frac{dx}{dt}-32\frac{dx}{dt}[/tex]

We can now factor out the dx/dt to get:

[tex]-15.9776=16\frac{dx}{dt}[/tex] (16 is the result of subtracting 32 from 48)

Now we divide both sides by 16 to get that

[tex]\frac{dx}{dt}=-.9986\frac{radians}{minute}[/tex]

The negative sign obviously means that x is decreasing

Step-by-step explanation:

-6(n-8)=4(12-5n)+14n

-6n+48=48-20n+14n

20n-6n-14n=48-48

-6n+6n=0

-6n=-6n

The statement is true for any value of n, because both sides are identical.

n€ R.

...

wait ... is this a graphical question ?!?!?!

The answer is:

The solution is all the values that belong to the real numbers,  and any value of "n" that we can substitute, will lead us to a satisfied equation.

We are given the following expression:

[tex]-6(n-8)=4(12-5n)+14n[/tex]

So, we need to perform the expressed operations using the distributive property, and then, add/subtract the like terms.

Now, solving we have:

[tex]-6(n-8)=4(12-5n)+14n[/tex]

[tex]-6*n+(-6*-8)=4*12+(4*-5n)+14n[/tex]

[tex]-6n+48=48-20n+14n[/tex]

[tex]-6n+48=48-6n[/tex]

[tex]0=0[/tex]

Hence, we can se that 0 is equal to 0, it means that the solution is all the values that belong to the real numbers,  and any value of "n" that we can substitute, will lead us to a satisfied equation.

Have a nice day!

Answer:

The percent of the parts are expected to fail before the 2100 hours is 0.15.

Step-by-step explanation:

Given :Life tests on a helicopter rotor bearing give a population mean value of 2500 hours and a population standard deviation of 135 hours.

To Find : If the specification requires that the bearing lasts at least 2100 hours, what percent of the parts are expected to fail before the 2100 hours?.

Solution:

We will use z score formula

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

Mean value = [tex]\mu = 2500[/tex]

Standard deviation = [tex]\sigma = 135[/tex]

We are supposed to find  If the specification requires that the bearing lasts at least 2100 hours, what percent of the parts are expected to fail before the 2100 hours?

So we are supposed to find P(z<2100)

so, x = 2100

Substitute the values in the formula

[tex]z=\frac{2100-2500}{135}[/tex]

[tex]z=−2.96[/tex]

Now to find P(z<2100) we will use z table

At z = −2.96 the value is 0.0015

So, In percent = [tex].0015 \times 100=0.15\%[/tex]

Hence The percent of the parts are expected to fail before the 2100 hours is 0.15.

Answer:

The weekly total revenue function is [tex]R(x,y)=200x-15x^2-220xy+160y-14y^2[/tex].

Step-by-step explanation:

Let the estimated quantities demanded each week of its roll top desks in

the finished and unfinished versions are x and y units respectively.

The unit price of finished furniture is

[tex]p=200-15x-110y[/tex]

The unit price of unfinished furniture is

[tex]q=160-110x-14y[/tex]

Total weekly revenue function is

[tex]R(x,y)=px+qy[/tex]

[tex]R(x,y)=(200-15x-110y)x+(160-110x-14y)y[/tex]

[tex]R(x,y)=200x-15x^2-110xy+160y-110xy-14y^2[/tex]

Combine like terms.

[tex]R(x,y)=200x-15x^2+(-110xy-110xy)+160y-14y^2[/tex]

[tex]R(x,y)=200x-15x^2-220xy+160y-14y^2[/tex]

Therefore the weekly total revenue function is [tex]R(x,y)=200x-15x^2-220xy+160y-14y^2[/tex].

Final answer:

The Weekly Total Revenue Function R(x, y) for Country Workshop's finished and unfinished roll top desks is found by multiplying their demand quantities by their respective unit prices, resulting in R(x, y) = -15x² - 220xy - 14y² + 200x + 160y.

Explanation:

The question asks us to find the weekly total revenue function R(x, y) for Country Workshop, which manufactures both finished and unfinished roll top desks with estimated weekly demands represented by x for finished and y for unfinished versions. The unit prices are given as p=200-15x-110y and q=160-110x-14y dollars, respectively. To calculate the total revenue, we multiply the price of each version by its quantity demanded and sum these values.

Total Revenue Calculation

To find the total revenue, R(x, y), we use the formula: R(x, y) = px + qy. By substituting the given price functions, we get:

This equation represents the weekly total revenue based on the quantities demanded of both the finished and unfinished roll top desks.