A total of 363 were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was two times the number of adult tickets sold. How many adult tickets were sold?

Answer:[tex]18\sqrt{3}[/tex]

Step-by-step explanation:

Given data

we haven given a parabola and a straight line

Parabola is [tex]{y^2}={-\left ( x-10\right )[/tex]

line is [tex]x=7[/tex]

Find the point of intersection of parabola and line

[tex]y=\pm \sqrt{3}[/tex] when[tex]x=7[/tex]

Area enclosed is the shaded area which is given by

[tex]Area=\int_{0}^{\sqrt{3}}\left ( 10-y^2 \right )dy[/tex]

[tex]Area=_{0}^{\sqrt{3}}10y-_{0}^{\sqrt{3}}\frac{y^3}{3}[/tex]

[tex]Area=10\sqrt{3}-\sqrt{3}[/tex]

[tex]Area=9\sqrt{3}units[/tex]

Required area will be double of calculated because it is symmetrical about x axis=[tex]18\sqrt{3}units[/tex]

To find the area of the region enclosed by the graphs of[tex]x=10-y^2[/tex]and x=7, we need to find the points of intersection between the two equations and then integrate the curve between those points.

To find the area of the region enclosed by the graphs of  [tex]x=10-y^2[/tex] and x=7, we need to find the points of intersection between the two equations. Setting x equal to each other, we have  [tex]10-y^2=7.[/tex]Solving for y, we get y=±√3.

Now we can integrate the curve between the two values of y, as y goes from -√3 to √3. So the area is given by  [tex]\int (10 - y^2 - 7) \, dy[/tex] from -√3 to √3.

Evaluating the integral, we get A=√3*10-2√3/3 ≈ 30.78.

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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:

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.

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:   [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:    

1.  6 fives.

2.  1 ten and 4 fives.

3.  2 tens and 2 fives.

4.  3 tens.

5.  1 twenty and 2 fives.

6.  1 twenty and 1 ten.

Step-by-step explanation:

Given : A customer is owed $30.00.

To find : How many different combinations of bills,using only five, ten, and twenty dollars bills are possible to give his or her change?

Solution :

We have to split $30 in terms of only five, ten, and twenty dollars.

1) In terms of only five we required 6 fives as

[tex]6\times 5=30[/tex]

So, 6 fives.

2) In terms of only ten and five,

a) We required 1 ten and 4 fives as

[tex]1\times 10+4\times 5=10+20=30[/tex]

So, 1 ten and 4 fives.

b) We required 2 tens and 2 fives as

[tex]2\times 10+2\times 5=20+10=30[/tex]

So, 2 tens and 2 fives

3) In terms of only tens we require 3 tens as

[tex]3\times 10=30[/tex]

So, 3 tens.

4)  In terms of only twenty and five, we required 1 twenty and 2 fives as

[tex]1\times 20+2\times 5=20+10=30[/tex]

So, 1 twenty and 2 fives.

5)  In terms of only twenty and ten, we required 1 twenty and 1 ten as

[tex]1\times 20+1\times 10=20+10=30[/tex]

So, 1 twenty and 1 ten.

Therefore, There are 6 different combinations.

(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]    

With a 95% confidence level, the population mean is estimated to be between approximately 96.85 and 111.15 based on a sample size of 10, a mean of 104, and a standard deviation of 10.

With a sample size (n) of 10, a mean \bar{x}104, and a standard deviation (s) of 10, we can find the 95% confidence interval for the population mean (μ).

First, we calculate the standard error of the mean (SE). The standard error of the mean can be calculated by dividing the standard deviation by the square root of the sample size.

SE = s/√n.  
By substituting s = 10 and n = 10 into the equation, we get SE = 3.162277660168379.

Next, we need to find the critical value (t) for a 95% confidence interval based on a t-distribution. Since we're using a confidence level of 95% and the sample size is 10, which means degree of freedom is n-1=9, the critical value (t) is 2.2621571627409915 based on the t-distribution table.

To calculate the lower bound and the upper bound of the 95% confidence interval, you should subtract and add to the mean the product of the critical value and the standard error respectively.

So,
Lower Bound = \bar{x} - t * SE
Upper Bound = \bar{x} + t * SE

Substituting from our known values, we get:
Lower Bound = 104 - 2.2621571627409915 * 3.162277660168379 = 96.84643094047428
Upper Bound = 104 + 2.2621571627409915 * 3.162277660168379 = 111.15356905952572

So, with a 95% confidence level, the confidence interval estimate of the population mean is (96.84643094047428, 111.15356905952572). This means we are 95% confident that the true population mean lies somewhere between approximately 96.85 and 111.15.

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The 95% confidence interval for the population mean IQ score of professional athletes, based on a sample size of 10 with a mean of 104 and standard deviation of 10, is estimated to be between 96.83 and 111.17.

To find the 95% confidence interval estimate of the population mean [tex](\( \mu \))[/tex] given the sample data, we'll use the formula for the confidence interval for a population mean when the population standard deviation is unknown:

[tex]\[ \text{Confidence interval} = \bar{x} \pm t \left( \frac{s}{\sqrt{n}} \right) \][/tex]

Where:

-[tex]\( \bar{x} \)[/tex] is the sample mean,

-  s  is the sample standard deviation,

-  n  is the sample size, and

-  t  is the critical value from the t-distribution for the desired confidence level and degrees of freedom.

Given:

- Sample size  n  = 10

- Sample mean [tex](\( \bar{x} \))[/tex]= 104

- Sample standard deviation  s  = 10

First, we need to find the critical value t  for a 95% confidence level with 9 degrees of freedom (since n - 1 = 10 - 1 = 9 ).

Using a t-table or statistical software, [tex]\( t \approx 2.262 \)[/tex] for a 95% confidence level and 9 degrees of freedom.

Now, let's plug in the values into the formula:

[tex]\[ \text{Confidence interval} = 104 \pm 2.262 \left( \frac{10}{\sqrt{10}} \right) \][/tex]

Now, let's calculate the margin of error:

[tex]\[ \text{Margin of error} = 2.262 \left( \frac{10}{\sqrt{10}} \right) \]\[ \text{Margin of error} \approx 7.17 \][/tex]

Finally, let's calculate the confidence interval:

[tex]\[ \text{Lower bound} = 104 - 7.17 \]\[ \text{Upper bound} = 104 + 7.17 \]\[ \text{Lower bound} \approx 96.83 \]\[ \text{Upper bound} \approx 111.17 \][/tex]

So, the 95% confidence interval estimate of the population mean IQ score of professional athletes is approximately between 96.83 and 111.17.

[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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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: 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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Answer:

The value of P(A or B or C) is 0.382.

Step-by-step explanation:

Given,

P(A) = 0.169,

P(B) = 0.041,

P(C) = 0.172

Since, if events A, B and C are mutually events ( in which no  element is common ),

Then, P(A∪B∪C) = P(A) + P(B) + P(C)

Or  P(A or B or C) = P(A) + P(B) + P(C),

By substituting the values,

P(A or B or C) = 0.169 +  0.041 +  0.172 = 0.382

[tex]u(x,y,z)=x^2+yz[/tex]

[tex]\begin{cases}x(p,r,\theta)=pr\cos\theta\\y(p,r,\theta)=pr\sin\theta\\z(p,r,\theta)=p+r\end{cases}[/tex]

At the point [tex](p,r,\theta)=(2,2,0)[/tex], we have

[tex]\begin{cases}x(2,2,0)=4\\y(2,2,0)=0\\z(2,2,0)=4\end{cases}[/tex]

Denote by [tex]f_x:=\dfrac{\partial f}{\partial x}[/tex] the partial derivative of a function [tex]f[/tex] with respect to the variable [tex]x[/tex]. We have

[tex]\begin{cases}u_x=2x\\u_y=z\\u_z=y\end{cases}[/tex]

The Jacobian is

[tex]\begin{bmatrix}x_p&x_r&x_\theta\\y_p&y_r&y_\theta\\z_p&z_r&z_\theta\end{bmatrix}=\begin{bmatrix}r\cos\theta&p\cos\theta&-pr\sin\theta\\r\sin\theta&p\sin\theta&pr\cos\theta\\1&1&0\end{bmatrix}[/tex]

By the chain rule,

[tex]u_p=u_xx_p+u_yy_p+u_zz_p=2xr\cos\theta+zr\sin\theta+y[/tex]

[tex]u_p(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_p(2,2,0)=16}[/tex]

[tex]u_r=u_xx_r+u_yy_r+u_zz_r=2xp\cos\theta+zp\sin\theta+y[/tex]

[tex]u_r(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_r(2,2,0)=16}[/tex]

[tex]u_\theta=u_xx_\theta+u_yy_\theta+u_zz_\theta=-2xpr\sin\theta+zpr\cos\theta[/tex]

[tex]u_\theta(2,2,0)=-2\cdot4\cdot2\cdot2\sin0+4\cdot2\cdot2\cos0\implies\boxed{u_\theta(2,2,0)=16}[/tex]

This problem is about using the Chain Rule to compute the partial derivatives of a function with respect to different variables, followed by substitution of specific values into the obtained derivatives.

The problem involves finding partial derivatives using the Chain Rule on the given equations with given parameters: p = 2, r = 2, θ = 0. By substituting the equations for x, y, z into u which gives us u = (prcosθ)² + prsinθ(p+r). The next step is to compute (partial u)/(partial p), (partial u)/(partial r), (partial u)/(partial theta) by using the Chain Rule to find each partial derivative. After computing, you just substitute the given values of p, r, θ into the obtained derivates to get the final answers.

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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:

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.

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: Hence, our required probability is 0.001077.

Step-by-step explanation:

Since we have given that

Number of red cars = 6

Number of silver cars = 9

Number of black cars = 3

Total number of cars = 6+9+3=18

We need to find the probability that 3 cars are red and 3 are black.

So, the required probability is given by

[tex]P(3R\ and\ 3B)=\dfrac{^6C_3\times ^3C_3}{^{18}C_6}\\\\P(3R\ and\ 3B)=0.001077[/tex]

Hence, our required probability is 0.001077.

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

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Answer with  explanation:

Let R be a communtative ring .

a and b elements in R.Let a and b are units

1.To prove that ab is also unit in R.

Proof: a and b  are units.Therefore,there exist elements u[tex]\neq0[/tex] and v [tex]\neq0[/tex] such that

au=1 and bv=1 ( by definition of unit )

Where u and v are inverse element  of a and b.

(ab)(uv)=(ba)(uv)=b(au)(v)=bv=1 ( because ring is commutative)

Because bv=1 and au=1

Hence, uv is an inverse element of ab.Therefore, ab is a unit .

Hence, proved.

2. Let a is a unit and b is a zero divisor .

a is a unit then there exist an element u [tex]\neq0[/tex]

such that au=1

By definition of unit

b is a zero divisor then there exist an element [tex]v\neq0[/tex]

such that bv=0 where [tex]b\neq0[/tex]

By definition of zero divisor

(ab)(uv)=b(au)v    ( because ring is commutative)

(ab)(uv)=b.1.v=bv=0

Hence, ab is a zero divisor.

If a is unit and b is a zero divisor then ab is a zero divisor.

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]

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:

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.

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:

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

Step-by-step explanation:

We are to compare the means of two samples. Since only sample std deviations are used, we have to use t test for this hypothesis

H0: Means are equal

Ha: Means are not equal

(Two tailed test at 5% )

Difference between means [tex]M1-M2 = -2587[/tex]

Std deviation combined = 3856

Std error for difference = 460.88

t statistic[tex]= -2587/460.88=-5.613[/tex]

p value =0

Since p <0.05 reject null hypothesis.

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

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

Answer:  0.1587

Step-by-step explanation:

Given : The amount dispensed is approximately normally distributed with Mean : [tex]\mu=\ 16[/tex]

Standard deviation : [tex]\sigma= 1[/tex]

The formula to calculate the z-score :-

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

For x= 17

[tex]z=\dfrac{17-16}{1}=1[/tex]

The p-value =[tex] P(17<x)=P(1<z)[/tex]

[tex]=1-P(z<1)=1-0.8413447\\\\=0.1586553\approx0.1587[/tex]

The proportion of bottles will have more than 17 ounces = 0.1587

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 value of det (3A) is -108.

Step-by-step explanation:

If M is square matrix of order n x n, then

[tex]|kA|=k^n|A|[/tex]

Let as consider a matrix A or order 3 x 3. Using the above mentioned property of determinant we get

[tex]|kA|=k^3|A|[/tex]

We need to find the value of det(3A).

[tex]|3A|=3^3|A|[/tex]

[tex]|3A|=27|A|[/tex]

It is given that the det(A)= -4. Substitute |A|=-4 in the above equation.

[tex]|3A|=27(-4)[/tex]

[tex]|3A|=-108[/tex]

Therefore the value of det (3A) is -108.

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.

To write the equation 5x + 3y = 15 in slope-intercept form, solve for y to get y = (-5/3)x + 5. The slope is -5/3 and the y-intercept is 5.

To convert the equation 5x + 3y = 15 into slope-intercept form, which is y = mx + b, we need to solve for y. Here are the steps: