Find the general solution of the following nonhomogeneous second order differential equation: y" - 4y = e^2x

Answer:

Definition of mid-point

Step-by-step explanation:

Midpoint is the center point of a line segment

At midpoints the line segment is divided into two equal lengths

Point K is the midpoint of segment MJ and length of segment MK =  length of segment KJ

Point K is the midpoint of segment OL and length of segment OK = length of segment KL

Answer:

The 95% confidence interval for the true mean speed of thunderstorms is [10.712, 13.688].

Step-by-step explanation:

Given information:

Sample size = 10

Sample mean = 12.2 mph

Standard deviation = 2.4

Confidence interval = 95%

At confidence interval 95% then z-score is 1.96.

The 95% confidence interval for the true mean speed of thunderstorms is

[tex]CI=\overline{x}\pm z*\frac{s}{\sqrt{n}}[/tex]

Where, [tex]\overline{x}[/tex] is sample mean, z* is z score at 95% confidence interval, s is standard deviation of sample and n is sample size.

[tex]CI=12.2\pm 1.96\frac{2.4}{\sqrt{10}}[/tex]

[tex]CI=12.2\pm 1.487535[/tex]

[tex]CI=12.2\pm 1.488[/tex]

[tex]CI=[12.2-1.488, 12.2+1.488][/tex]

[tex]CI=[10.712, 13.688][/tex]

Therefore the 95% confidence interval for the true mean speed of thunderstorms is [10.712, 13.688].

Answer:

The number of ways are 1,048,576.

Step-by-step explanation:

Consider the provided information.

Product rule: If one event occurs in n contexts and the second event occurs in m contexts, then the number of ways in which the two events happen is n×m.

There are 10 questions and each question has 4 choices.

Therefore, for first question we have 4 choices, for second question we have 4 choices similarly for 10th question we have 4 choices which can be represented as:

4×4×4×4×4×4×4×4×4×4 = [tex]4^{10}[/tex]

4×4×4×4×4×4×4×4×4×4 = 1048576

Thus, the number of ways are 1,048,576.

Answer:  0.0013

Step-by-step explanation:

Given : The test scores are normally distributed with

Mean : [tex]\mu=\ 60,000[/tex]

Standard deviation :[tex]\sigma= 4,000[/tex]

Sample size : [tex]n=4[/tex]

The formula to calculate the z-score :-

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

For x = 66,000

[tex]z=\dfrac{66000-60000}{\dfrac{4000}{\sqrt{4}}}=3[/tex]

The p-value = [tex]P(z>3)\=1-P(z<3)=1- 0.9986501\approx0.0013[/tex]

Hence, the likelihood the mean tire life of these four tires is more than 66,000 miles = 0.0013

Answer:

i = 5.7%

n = 12

Step-by-step explanation:

Compounded annually means once per year.  So the rate per period is 5.7%, and the number of periods is 12.

Answer:

i = 5.7%  

n = 12

Step-by-step explanation:

i​ (the rate per​ period) and n​ (the number of​ periods) for the following loan at the given annual rate.

Annual payments of ​$3,600 are made for 12 years to repay a loan at 5.7​% compounded annually.

Therefore,

i = 5.7%  

n = 12

Answer:

hence rate interest r = 41.176%

Step-by-step explanation:

The amount borrowed= $680

amount payed back = $750

therefore, interest incurred = 750-680= $70

time, t= 3 months = 3/12= 0.25 years

rate%, r

we know that SI= [tex]\frac{PRT}{100}[/tex]

70= [tex]\frac{680\timesr\0.25}{100}[/tex]

r=[tex]\frac{7000}{0.25\times680} = 41.176[/tex]

hence rate interest r = 41.176%

Answer:

The percent of Increase is of 11.08% (0.1108)

Step-by-step explanation:

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

Since we have two different values for two different years, we can use the following Algebraic Expression to calculate the percent difference of sales between both years. The Expression would be the following,

[tex]370 * (x+1) = 411[/tex]

Where x is the percent difference. Now we solve for x,

[tex]370 * (x+1) = 411[/tex]

[tex]x+1 = 1.1108[/tex]

[tex]x = 0.1108[/tex]

so now we see that the percent of Increase is of 11.08% (0.1108)

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

The number of units sold increased by about 11.08%.

To find the percent increase or decrease in the number of units sold, we need to calculate the difference between the number of units sold in 2012 and 2011, and then divide that difference by the number of units sold in 2011.

The amount of increase or decrease is calculated as: (Number of units sold in 2012 - Number of units sold in 2011)/Number of units sold in 2011 x 100

In this case, the calculation is: (411 - 370)/370 x 100 = (41/370) x 100 = 11.08%

Therefore, the number of units sold increased by about 11.08%.

We have

[tex]\dfrac{\partial(y\cos(xy)+3x^2)}{\partial y}=\cos(xy)-xy\sin(xy)[/tex]

[tex]\dfrac{\partial(x\cos(xy)+2y)}{\partial x}=\cos(xy)-xy\sin(xy)[/tex]

so the ODE is indeed exact. Then there's a solution of the form [tex]f(x,y)=C[/tex] such that

[tex]\dfrac{\partial f}{\partial x}=y\cos(xy)+3x^2[/tex]

[tex]\implies f(x,y)=\sin(xy)+x^3+g(y)[/tex]

Differentiating wrt [tex]y[/tex] gives

[tex]\dfrac{\partial f}{\partial y}=x\cos(xy)+2y=x\cos(xy)+g'(y)[/tex]

[tex]\implies g'(y)=2y\implies g(y)=y^2+C[/tex]

Then the solution to the ODE is

[tex]f(x,y)=\boxed{\sin(xy)+x^3+y^2=C}[/tex]

We are asked to prove the statement,

For all sets A, B, and C, if A ⊆ B then A∪C ⊆ B∪C

Let us consider set A as:

         A={1,3,4,5}

and B={1,2,3,4,5,6,7}

Clearly we may observe that A is a subset of B.

( Since, all the elements of set A are contained in set B .

Hence, A is a subset of B)

Now let us consider set C as:

C={1,2}

Hence,

A∪C={1,2,3,4,5}

and

B∪C={1,2,3,4,5,6,7}

Still we observe that:

          A∪C ⊆ B∪C

Since all the elements of the set A∪C are contained in the set B∪C.

Answer:

a.The probability that two randomly selected tulip bulbs both are red=[tex]\frac{45}{406}[/tex].

b.The probability that the first bulb selected is red and second yellow=[tex]\frac{50}{406}[/tex].

c.The probability that the first bulb selected is yellow and the second red=[tex]\frac{50}{406}[/tex].

d.The probability that one bulb is red and other yellow=[tex]\frac{50}{203}[/tex].

Step-by-step explanation:

Given

Total number of bulbs= 29

Number of bulbs of red=10

Number of yellow bulbs=10

Number of purple bulbs=9

Formula of probability, P(E)=[tex]\frac{favourable \; cases}{total\;number\; of \; cases}[/tex]

a.The probability that two randomly selected tulip bulbs are both red=[tex]\frac{10}{29}\times\frac{9}{28}=\frac{45}{406}[/tex].

b.The probability of getting first bulb is red=[tex]\frac{10}{29}[/tex].

The probability of getting second bulb is yellow=[tex]\frac{10}{28}[/tex]

Hence,the probability that the firs bulb selected is red and the second bulb yellow=[tex]\frac{10}{29}\times\frac{9}{28}=\frac{45}{406}[/tex]

c. The probability of getting firs bulb is yellow =[tex]\frac{10}{29}[/tex]

The probability of getting second bulb is red=[tex]\frac{10}{28}[/tex]

Hence,the probability that the firs bulb selected is yellow and the second bulb red=[tex]\frac{10}{29}\times\frac{10}{28}=\frac{50}{406}[/tex].

d.The probability of getting first bulb is red and second is yellow=[tex]\frac{50}{406}[/tex]

The probability of getting first bulb is yellow and second is red=[tex]\frac{50}{406}[/tex]

The probability that one bulbe is red and other is yellow= probability of getting first bulb is red and other yellow+ probability of getting first bulb is yellow and other is red

Hence, the probability of getting one bulb is red and other is yellow=[tex]\frac{50}{406}+\frac{50}{406}=\frac{50}{203}[/tex]

Answer:

$1210

Step-by-step explanation:

Let x be total amount

First John spent $110 on a radio and 4/11 of what was left on presents for his friends so he was left with

[tex]\frac{7}{11}(x-110)=\frac{7}{11}x-70[/tex]

Then he put 2/5 of his remaining money into a checking account

[tex]\frac{3}{5}\left(\frac{7}{11}x-70\right)[/tex]

Rest he donated to charity

[tex]420=\frac{3}{5}\left(\frac{7}{11}x-70\right)\\\Rightarrow \left(\frac{7}{11}x-70\right)=\frac{2100}{3}\\\Rightarrow \frac{7}{11}x-70=700\\\Rightarrow \frac{7}{11}x=770\\\Rightarrow x=1210[/tex]

Hence total amount of money John originally had was $1210

Answer:

163

Step-by-step explanation:

So n=391.

This means p=23 and q=17 where p*q=n.

[tex] \lambda (391)=lcm(23-1,17-1)=lcm(22,16)=2*8*11=16*11=176. [/tex]

We want to choose e so that e is between 1 and 176 and the gcd(e,176)=1.

There is only one number in your list that is between 1 and 176... Hopefully the gcd(163,176)=1.

It does. See notes below for checking it:

176=2(88)=2(4*22)=2(2)(2)(2)(11)

None of the prime factors of 176 divide 163 so we are good.

The answer is 163.

Answer with explanation:

Let m and n are integers

To prove that m+n and m-n are either both even or both odd.

1. Let m and n are both even

We know that sum of even number is even and difference of even number is even.

Suppose m=4 and n=2

m+n=4+2=6 =Even number

m-n=4-2=2=Even number

Hence, we can say m+n and m-n are both even .

2. Let m and n are odd numbers .

We know that sum of odd numbers is even and difference of odd numbers is even.

Suppose m=7 and n=5

m+n=7+5=12=Even number

m-n=7-5=2=Even number

Hence, m+n and m-n are both even .

3. Let m is odd and n is even.

We know that sum of an odd number and an even number is odd and difference of an odd and an even number is an odd number.

Suppose m=7 , n=4

m+n=7+4=11=Odd number

m-n=7-4=3=Odd number

Hence, m+n and m-n are both odd numbers.

4.Let m is even number and n is odd number .

Suppose m=6, n=3

m+n=6+3=9=Odd number

m-n=6-3=3=Odd number

Hence, m+n and m-n are both odd numbers.

Therefore, we can say for all inetegers m and n , m+n and m-n are either both even or both odd.Hence proved.

Final answer:

The problem is solved by expressing the conditions under which m and n are both even or odd, and their sum and difference in terms of 2k (for even) and 2k+1 (for odd), demonstrating that m+n and m-n are both even or both odd.

Explanation:

To prove that for all integers m and n, m + n and m - n are either both even or both odd, we start by recalling the definition of even and odd numbers. An even number can be expressed as 2k, where k is an integer, and an odd number can be expressed as 2k + 1, where k is an integer.

This argument shows that m + n and m - n must either be both even or both odd for any integers m and n.

Answer:

130 cars.

Step-by-step explanation:

The cost function is given by:

C(x) = 0.9x^2 -234x + 23,194; where x is the input and C is the total cost of production.

To find the minimum the unit cost, there must be a certain number of cars which have to be produced. To find that, take the first derivative of C(x) with respect to x:

C'(x) = 2(0.9x) - 234 = 1.8x - 234.

To minimize the cost, put C'(x) = 0. Therefore:

1.8x - 234 = 0.

Solving for x gives:

1.8x = 234.

x = 234/1.8.

x = 130 units of cars.

To check whether the number of cars are minimum, the second derivative of C(x) with respect to x:

C''(x) = 1.8. Since 1.8 > 0, this shows that x = 130 is the minimum value.

Therefore, the cars to be made to minimize the unit cost = 130 cars!!!

[tex]2^{2013}=2^{4\cdot503+1}\\\\2^4=16\equiv 1\pmod{15}\\2^{4\cdot 503}\equiv 1\pmod{15}\\2^{4\cdot 503+1}\equiv 2\pmod{15}\\\\2^{2013}\equiv 2\pmod{15}[/tex]

Answer:

D. $393.00

Step-by-step explanation:

We have to sum the three distances to get the perimeter.

P= 34 ft + 67 ft+ 30 ft

P = 131 ft

And each foot of material cost $3.00

we get the cost with the distance and de cost of the material:

Cost = 131 ft * 3.00 $ per ft = $393.00

Sarah needs $393.00 to fence her backyard garden

a. The marginal densities

[tex]f_X(x)=\displaystyle\int_0^1(x+y)\,\mathrm dy=x+\frac12[/tex]

and

[tex]f_Y(y)=\displaystyle\int_0^1(x+y)\,\mathrm dx=y+\frac12[/tex]

b. This can be obtained by integrating the joint density over [0.25, 1] x [0.5, 1]:

[tex]P(X>0.25,Y>0.5)=\displaystyle\int_{1/4}^1\int_{1/2}^1(x+y)\,\mathrm dx\,\mathrm dy=\frac{33}{64}[/tex]

Final answer:

To find the marginal distributions of X and Y, we integrate the joint density function over the range of the other variable. The marginal distribution of X is f(x) = x+1/2, for 0≤x≤1. The marginal distribution of Y is f(y) = y+1/2, for 0≤y≤1.

Explanation:

To find the marginal distributions of X and Y, we need to integrate the joint density function over the range of the other variable. For the marginal distribution of X, we integrate f(x,y) with respect to y from 0 to 1:

(∫⁰₁(x+y) dy) = (x+y/2)∣⁰₁ = x+1/2

So, the marginal distribution of X is given by f(x) = x+1/2, for 0≤x≤1.

Similarly, for the marginal distribution of Y, we integrate f(x,y) with respect to x from 0 to 1:

(∫⁰₁(x+y) dx) = (x2/2+xy)∣⁰₁ = y+1/2

Therefore, the marginal distribution of Y is given by f(y) = y+1/2, for 0≤y≤1.

Answer:

total balance due at the end of 1 year is  $769.75

Step-by-step explanation:

Given data

loan amount = $1000

time period = 1 year

return = $300

rate = 9%

to find out

balance due at the end of one year

solution

we know in question $300 return after 3 month so we first calculate interest of $1000 for 3 month and than we after 3 month remaining 9 month we calculate interest for $700

interest for first 3 month = ( principal × rate × time ) / 100    .............1

here time is 3 month so = 3/12  will take and rate 9 % and principal $1000

put all these value in equation 1 we get interest for first 3 month

interest for first 3 month = ( principal × rate × time ) / 100

interest for first 3 month = ( 1000 × 9 × 3/12 ) / 100

interest for first 3 month = $22.5

now we calculate interest for remaining 9 months i.e.

interest for next 9 months = ( principal × rate × time ) / 100

here principal will be $700 because we pay $300 already

interest for next 9 months = ( 700 × 9 × 9/12 ) / 100

interest for next 9 months = $47.25

now we combine both interest that will be

interest for first 3 months +interest for next 9 months = interest of 1 year

interest of 1 year  =   $22.5  + $47.25

interest of 1 year  =   $69.75

so amount will be paid after 1 year will be loan amount + interest

amount will be paid after 1 year = 1000 + 69.75

amount will be paid after 1 year is $1069.75

so total balance due at the end of 1 year = amount will be paid after 1 year - amount paid already

total balance due at the end of 1 year = $1069.75 - $300

total balance due at the end of 1 year is  $769.75

Answer:

  6.9%

Step-by-step explanation:

Interest rate is the one variable in an amortization formula that cannot be determined explicitly. An iterative solution is required, which means the computation must be done by a calculator, spreadsheet, or web site.

My TI-84 TVM Solver tells me that for the given loan amount and payment schedule, the APR is about 6.9%.

The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is (1/3.5) × e[tex]^{(-x/3.5)[/tex].

The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is given by:

f(x) = (1/3.5) × e[tex]^{(-x/3.5)[/tex]

Where x is the time it takes to ring up a customer.

In this case, the exponential distribution models the time between events, which in this context is the time between customer arrivals at the grocery store.

The exponential distribution is a continuous probability distribution that is often used to model random events that occur independently and exponentially over time.

Answer:

d) the disenfranchisement of individuals or groups based on the media construed images of members of those groups.

Step-by-step explanation:

Prejudice operates mainly through the use of stereotyping. A stereotype is/are the disenfranchisement of individuals or groups based on the media construed images of members of those groups.

Answer:

[tex]\text{16.57 dollars}[/tex]

Step-by-step explanation:

A proportion is needed to find the value of x

[tex]$\frac{22.50}{19} =\frac{x}{14}\Longrightarrow 19x = 315 \Longrightarrow x = 16.57$ \\ \\ \text{It would cost 16.57 dollars to purchase 14 gallons.}[/tex]

The cost of 14 gallons of gasoline is $16.58.

Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.

In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.

Given, The price of gasoline purchased varies directly with the number of gallons of gas purchased and 19 gallons are purchased for $22.50.

Let k be the proportionality constant.

∴ y = kx.

22.50 = 19k.

k = 22.50/19.

k = 1.184.

So, the cost of 14 gallons of gasoline is (14×1.184) = $16.58.

learn more about proportion here :

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The Lagrangian is

[tex]L(x,y,\lambda)=x^2-y^2+\lambda(x^2+y^2-81)[/tex]

with critical points where the partial derivatives are identically zero:

[tex]L_x=2x+2\lambda x=0\implies 2x(1+\lambda)=0\implies x=0\text{ or }\lambda=-1[/tex]

[tex]L_y=-2y+2\lambda y=0\implies-2y(1-\lambda)=0\implies y=0\text{ or }\lambda=1[/tex]

[tex]L_\lambda=x^2+y^2-81=0\implies x^2+y^2=81[/tex]

So we have four critical points to consider: (0, -9), (0, 9), (-9,0), and (9, 0). We have

[tex]f(0,-9)=-81[/tex]

[tex]f(0,9)=-81[/tex]

[tex]f(-9,0)=81[/tex]

[tex]f(9,0)=81[/tex]

So the maximum value is 81 and the minimum value is -81.

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are -81 and 81

The function is given as:

[tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]

[tex]\mathbf{x^2 + y^2 = 81}[/tex]

Subtract 81 from both sides of [tex]\mathbf{x^2 + y^2 = 81}[/tex]

[tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]

Using Lagrange multiplies, we have:

[tex]\mathbf{L(x,y,\lambda) = f(x,y) + \lambda(0)}[/tex]

Substitute [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex] and [tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]

[tex]\mathbf{L(x,y,\lambda) = x^2 - y^2 + \lambda(x^2 + y^2 - 81)}[/tex]

Differentiate

[tex]\mathbf{L_x = 2x + 2\lambda x}[/tex]

[tex]\mathbf{L_y = -2y + 2\lambda y}[/tex]

[tex]\mathbf{L_{\lambda} = x^2 + y^2 -81}[/tex]

Equate to 0

[tex]\mathbf{2x + 2\lambda x = 0}[/tex]

[tex]\mathbf{-2y + 2\lambda y = 0}[/tex]

[tex]\mathbf{x^2 + y^2 -81 = 0}[/tex]

So, we have:

[tex]\mathbf{2\lambda x = -2x}[/tex]

[tex]\mathbf{2\lambda y = 2y}[/tex]

Divide both sides of [tex]\mathbf{2\lambda x = -2x}[/tex] by -2x

[tex]\mathbf{\lambda = -1}[/tex]

Divide both sides of [tex]\mathbf{2\lambda y = 2y}[/tex] by 2y

[tex]\mathbf{\lambda = 1}[/tex]

The above means that:

[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex]

[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex]

Recall that: [tex]\mathbf{x^2 + y^2 = 81}[/tex]

When x = 0, we have:

[tex]\mathbf{0^2 + y^2 = 81}[/tex]

Take square roots of both sides

[tex]\mathbf{y = \±9}[/tex]

When y = 0, we have:

[tex]\mathbf{x^2 + 0^2 = 81}[/tex]

Take square roots of both sides

[tex]\mathbf{x = \±9}[/tex]

To determine the critical points, we consider:

[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex] or [tex]\mathbf{y = \±9}[/tex]

[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex] or [tex]\mathbf{x = \±9}[/tex]

So, the critical points are:

[tex]\mathbf{(x,y) = \{ (0, -9), (0, 9), (-9,0), (9, 0)\}}[/tex]

Substitute the above values in [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]

[tex]\mathbf{f(0,-9) = 0^2 - (-9)^2 = -81}[/tex]

[tex]\mathbf{f(0,-9) = 0^2 - (9)^2 = -81}[/tex]

[tex]\mathbf{f(-9,0) = (-9)^2 - 0^2 = 81}[/tex]

[tex]\mathbf{f(9,0) = (9)^2 - 0^2 = 81}[/tex]

Considering the above values, we have:

[tex]\mathbf{Minimum= -81}[/tex]

[tex]\mathbf{Maximum= 81}[/tex]

Hence, the extreme values are -81 and 81, respectively.

Read more about extreme values at;

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Final answer:

If 3 letters are followed by 2 numbers and repetitions are allowed, there are a total of 1757600 possible different account numbers. Each letter position has 26 choices and each number position has 10 choices.

Explanation:

The student has asked to determine the number of different customer account "numbers" a company can have if the accounts consist of 3 letters followed by 2 numbers, with repetitions allowed for both letters and numbers.

To calculate the total number of possible account numbers, we can use the multiplication principle of counting. The number of options for each position of the account number is multiplied together to get the total number of combinations.

For the 3 letters, each position can contain any letter from A-Z, which gives us 26 choices per position. Since repeats are allowed, each of the 3 positions has 26 possible choices.

For the 2 numbers, each position can contain any digit from 0-9, which gives us 10 choices per position.

Therefore, to find the total number of possible account numbers, we calculate:

26 × 26 × 26 × 10 × 10 = 1757600 possible account numbers.

Answer:

Step-by-step explanation:

In the case of each of the answers listed above, the second derivative is equal to -4 times the function, as required by the differential equation.

For y = 2/3t^3, the second derivative is 4t, not -4y.

For y = e^(2t), the second derivative is 4y, not -4y.

__

The graph shows the sum of the second derivative and 4y is zero for the answers indicated above, and not zero for the other two proposed answers.

If (2, -1, 2, 1) is a linear combination of the three given vectors, then there should exist [tex]c_1,c_2,c_3[/tex] such that

[tex](2,-1,2,1)=c_1(1,-2,0,3)+c_2(2,3,0,-1)+c_3(3,9,-4,-2)[/tex]

or equivalently, there should exist a solution to the system

[tex]\begin{cases}c_1+2c_2+3c_3=2\\-2c_1+3c_2+9c_3=-1\\-4c_3=2\\3c_1-c_2-2c_3=1\end{cases}[/tex]

Right away we get [tex]c_3=-\dfrac12[/tex], so the system reduces to

[tex]\begin{cases}c_1+2c_2=\dfrac72\\\\-2c_1+3c_2=\dfrac72\\\\3c_1-c_2=0\end{cases}[/tex]

Notice that the first equation is the sum of the latter two. The third equation gives us

[tex]3c_1-c_2=0\implies 3c_1=c_2[/tex]

so that in the second equation,

[tex]-2c_1+3c_2=\dfrac72\implies7c_1=\dfrac72\implies c_1=\dfrac12[/tex]

which in turn gives

[tex]3c_1=c_2\implies c_2=\dfrac32[/tex]

and hence the (2, -1, 2, 1) is a linear combination of the given vectors, with

[tex]\boxed{(2,-1,2,1)=\dfrac12(1,-2,0,3)+\dfrac32(2,3,0,-1)-\dfrac12(3,9,-4,-2)}[/tex]

Answer:

x^2+y^2 = 3^2

Step-by-step explanation:

We need to eliminate the parameter t

Given:

x = 3 cos t

y = 3 sin t

Squaring the above both equations

(x)^2=(3 cos t)^2

(y)^2 =(3 sin t)^2

x^2 = 3^2 cos^2t

y^2=3^2 sin^2t

Now adding both equations

x^2+y^2=3^2 cos^2t+3^2 sin^2t

Taking 3^2 common

x^2+y^2=3^2 (cos^2t+sin^2t)

We know that cos^2t+sin^2t = 1

so, putting the value

x^2+y^2=3^2(1)

x^2+y^2 = 3^2

Hence the parameter t is eliminated.

To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we can substitute cos(t) and sin(t) in terms of x and y to eliminate the parameter. The resulting equations represent the line y = x.

To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we need to express x and y in terms of each other without the parameter 't'. Using the identity cos^2(t) + sin^2(t) = 1, we can solve for cos(t) and sin(t), and substitute them into the equations to eliminate the parameter.

Using the fact that cos(t) = x/3 and sin(t) = y/3, we can rewrite the equations as x = 3 cos(t) = 3(x/3) = x and y = 3 sin(t) = 3(y/3) = y. Therefore, eliminating the parameter results in x = x and y = y, which simply means that the equations represent the line y = x.

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Answer:

8710 units

Step-by-step explanation:

Step 1: Write all the data

Fixed cost: $9000

Average variable cost: 9.3 per unit

Total cost: 90,000

Total units: x

Step 2: Find the total variable cost

Average variable cost is per unit so it has to be multiplied by the number of units to find the total variable cost.

Total variable cost = average variable cost per unit x number of units

Total variable cost = 9.3x

Step 3: Make the formula for finding x

Total cost = total fixed cost + total variable cost

90,000 = 9000 + 9.3x

81000 = 9.3x

x = 8709.67

Rounded off to 8710 units

!!

In matrix form, the system is

[tex]\dfrac{\mathrm d}{\mathrm dt}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1&1\\4&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}[/tex]

First find the eigenvalues of the coefficient matrix (call it [tex]\mathbf A[/tex]).

[tex]\det(\mathbf A-\lambda\mathbf I)=\begin{vmatrix}1-\lambda&1\\4&1-\lambda\end{vmatrix}=(1-\lambda)^2-4=0\implies\lambda^2-2\lambda-3=0[/tex]

[tex]\implies\lambda_1=-1,\lambda_=3[/tex]

Find the corresponding eigenvector for each eigenvalue:

[tex]\lambda_1=-1\implies(\mathbf A+\mathbf I)\vec\eta_1=\vec0\implies\begin{bmatrix}2&1\\4&2\end{bmatrix}\begin{bmatrix}\eta_{1,1}\\\eta_{1,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]

[tex]\lambda_2=3\implies(\mathbf A-3\mathbf I)\vec\eta_2=\vec0\implies\begin{bmatrix}-2&1\\4&-2\end{bmatrix}\begin{bmatrix}\eta_{2,1}\\\eta_{2,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]

[tex]\implies\vec\eta_1=\begin{bmatrix}1\\-2\end{bmatrix},\vec\eta_2=\begin{bmatrix}1\\2\end{bmatrix}[/tex]

Then the system has general solution

[tex]\begin{bmatrix}x\\y\end{bmatrix}=C_1\vec\eta_1e^{\lambda_1t}+C_2\vec\eta_2e^{\lambda_2t}[/tex]

or

[tex]\begin{cases}x(t)=C_1e^{-t}+C_2e^{3t}\\y(t)=-2C_1e^{-t}+2C_2e^{3t}\end{cases}[/tex]

Given that [tex]x(0)=1[/tex] and [tex]y(0)=2[/tex], we have

[tex]\begin{cases}1=C_1+C_2\\2=-2C_1+2C_2\end{cases}\implies C_1=0,C_2=2[/tex]

so that the system has particular solution

[tex]\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}e^{3t}\\2e^{3t}\end{bmatrix}[/tex]

Final answer:

The linear system of differential equations can be written in matrix form as [dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]. By solving the system with the given initial conditions x(0) = 1 and y(0) = 2, the values of x and y at different time points can be determined.

Explanation:

To write the linear system of differential equations in matrix form, we can express the given equations as:

[dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]

Using the initial conditions x(0) = 1 and y(0) = 2, we can solve the system of equations to find the values of x and y at different time points.

Answer:

a) 0.7898

b) 0.0516

c) 0.1587

Step-by-step explanation:

Given : Mean : [tex]\mu=8.1\text{ minutes}[/tex]

Standard deviation : [tex]\sigma =1.9\text{ minutes}[/tex]

Since , the police response time has a normal distribution.

The formula to calculate the z-score :-

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

For x=5 minutes.

[tex]z=\dfrac{5-8.1}{1.9}=-1.63[/tex]

For x=10 minutes.

[tex]z=\dfrac{10-8.1}{1.9}=1[/tex]

a) The p-value =[tex]P(-1.63<z<1)=P(z<1)-P(z<-1.63)[/tex]

[tex]=0.8413447-0.0515507=0.789794\approx0.7898[/tex]

b) The p-value =[tex]P(z<-1.63)=0.0515507\approx0.0516[/tex]

c) The p-value =[tex]P(z>1)=1-P(z<1)[/tex]

[tex]=1-0.8413447=0.1586553\approx0.1587[/tex]

We calculated the probability of different police response times using the z-score method. The probability of a response time between 5 and 10 minutes is 0.7897, the probability for less than 5 minutes is 0.0516, and more than 10 minutes is 0.1587.

To answer this question, we need to first standardize the response times using the z-score formula: z = (X - μ) / σ, where X is the value we're interested in, μ is the mean, and σ is the standard deviation.

(a) To find the probability that the response time is between 5 and 10 minutes, we first calculate the z-scores for 5 and 10 minutes:

Z(5) = (5 - 8.1) / 1.9 = -1.632 Z(10) = (10 - 8.1) / 1.9 = 1

Next, we find these values in the z-table which yields: P(Z<1) = 0.8413, P(Z<-1.632) = 0.0516. The probability that the response time is between 5 and 10 minutes is the difference between these values, so P(5 < X < 10) = 0.8413 - 0.0516 = 0.7897.

(b) For the response time less than 5 minutes, we calculate the probability using the z-score for 5 minutes. Z(5) = -1.632, looking in the z-table, we find this value equals to 0.0516. Therefore, the response time is less than 5 minutes is 0.0516.

(c) Lastly, the probability for a response time more than 10 minutes is P(Z > 1) which is equal to 1 - P(Z < 1). From the z-table, we find P(Z<1) = 0.8413. Then, P(Z > 1) = 1 - 0.8413 = 0.1587. So, the probability that the response time is more than 10 minutes is 0.1587.

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