Find the general solution for the following homogeneous equations. PLEASE SHOW SOLUTIONS

1) (xy+y2)dx−x2dy=0 (hint: let u=y/x, so y=ux, dy=xdu+udx))

2) 2x2ydx=(3x3+y3)dy. (hint: let v=x/y, so x=vy, dx=vdy+ydv)

Answer:

The current yield is 3.11%.

Step-by-step explanation:

Given - A $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450.

Now the face value of the bond is = $500

The rate of interest is = 2.8%

Then interest on $500 becomes:

[tex]0.028\times500=14[/tex] dollars

The current market value is $450

So, current yield is = [tex]\frac{14}{450}\times100= 3.11[/tex]%

The current yield is 3.11%.

Final answer:

The current yield on a $500 Treasury bond with a 2.8% coupon rate and a market value of $450 is 3.11%, calculated by dividing the annual coupon payment by the market value of the bond and then converting to a percentage.

Explanation:

To calculate the current yield on the described $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450, follow these steps:

Therefore, the current yield on the bond is 3.11% when rounded to two decimal places.

Answer:

B. 3√2/2

Step-by-step explanation:

The value of x can be found with the tan rule.

Step 1: Identify the sides as opposite, adjacent and hypotenuse to apply the formula

Opposite = (opposite from angle 45 degrees)

Adjacent = x (between angle 45 degrees and 90 degrees)

Hypotenuse = 3 (opposite from 90 degrees)

Step 2: Apply the tan formula

Cos (angle) = adjacent/hypotenuse

Cos (45) = x/3

√2/2 = x/3

x = √2/2 x 3

x = 3√2/2

Therefore, the correct answer is B; x = 3√2/2

!!

Answer:

Rita paid 47,500 dollars for the purchase price.

Step-by-step explanation:

We are given the sales tax on a new car is directly proportional to the purchase price of the car which means there is is something k such that

when you multiply it to the sales tax you get the purchase price.

Let's set this equation:

y=kx

Let y represent the purchase price and x the sales tax.

The second sentence tells us that (x,y)=(1500,30000).

We can plug this into y=kx to find the constant k.  (Constant means it stays the same no matter what the input and output is).

So we have:

30000=k(1500)

300    =k(15)      I went ahead and divided previous equation by 100.

Now divide both sides by 15:

300/15=k

Simplify:

20=k

So the equation to use the answer the question is

y=20x

where y is purchase price and x is sales tax.

So we want to know the purchase price on a car if the sales tax is 2375.

So replace x with 2375:

y=20(2375)

y=47500

Answer:

$47500

Step-by-step explanation:

If the amount of sales tax on a new car is directly proportional to the purchase price of the car and Victor bought a new car for $30,000 and paid $1,500 in sales tax and Rita bought a new car from the same dealer and paid $2,375 sales tax, Rita payed $47,500 for her car.

y=20(2375)

y=47500

Answer:

The correct answer is 206550 pigs.

Step-by-step explanation:

First of all you need to calculate the constant growth rate, which is given by the formula [tex]p=((\frac{f}{s} )^{\frac{1}{y} } -1)*100[/tex] where f is the value at the end of the year, s is the start value of that year, and y is the number of years.

From the excercise facts, we know that for 1 year (y=1), the final value is 421 (f=421), and the start value is 16 (s=16). Replacing them in the formula we get : [tex]((\frac{421}{16}) ^{\frac{1}{1} } -1)*100 = 2531.25[/tex] So, the constant growth rate equals 2531.25.

Next, we have to multiply the starting amount of pigs times the constant growth rate times the amount of time that passed to get the final quantity of pigs. This would be [tex]16*2531.25*5.1[/tex] and this gives us a total amount of 206550 pigs.

Have a nice day.

Answer:

Since living expenses are 11% higher in the new city. Marla would need an 11% increase in her current income to maintain her current buying power when moving. We can solve this by simply multiplying her current income by 11%.

Step-by-step explanation:

[tex]95,000*1.11 = 105,450[/tex]

If we multiply her current income of $95,000 by 11% we see that Marla would need an income of $105,450 in order to maintain her buying power in the new city. Since her new job offer is paying $118,000 she will have more buying power in the new city.

Answer: Marla must earn an Income of $105,450 to maintain her buying power.

Marla would need to earn $105,450 in the new city to maintain her current buying power after taking into account an 11 percent increase in living expenses.

The student has asked how much Marla Opper must earn at the new job in another city to maintain her current buying power, considering that the new city has living expenses that are 11 percent higher. To calculate this, we first need to figure out what amount of income in the new city would be equivalent to Marla's current income of $95,000, given the increase in living expenses.

Firstly, we find 11 percent of Marla's current income:

This means Marla needs an additional $10,450 on top of her current income to maintain the same buying power in the new city. So, the amount Marla needs to earn in the new city is:

Thus, Marla would need to earn $105,450 in the new city to keep her current buying power.

Answer:

A (9, 3)

Step-by-step explanation:

First the point is rotated 90° counterclockwise about the origin.  To do that transformation: (x, y) → (-y, x).

So S(-3, -5) becomes S'(5, -3).

Next, the point is translated +4 units in the x direction and +6 units in the y direction.

So S'(5, -3) becomes S"(9, 3).

Answer: 0.8023

Step-by-step explanation:

Given : [tex]\text{Mean}=\mu=481 \text{ millimeters}[/tex]

[tex]\text{Standard deviation}=41 \text{ millimeters}[/tex]

Assuming these lengths are normally distributed.

The formula to calculate the z-score is given by :-

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

For x= [tex]516 \text{ millimeters}[/tex]

[tex]z=\dfrac{516-481}{41}=0.853658536585\approx0.85[/tex]

The p-value = [tex]P(z\leq0.85)=0.8023374\approx0.8023[/tex]

Hence, the required probability : 0.8023

Answer: 0.4772

Step-by-step explanation:

Given : The distribution is bell shaped , then the distribution must be normal distribution.

Mean : [tex]\mu=\$280[/tex]

Standard deviation :[tex]\sigma= \$40[/tex]

The formula to calculate the z-score :-

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

For x = $200

[tex]z=\dfrac{200-280}{40}=-2[/tex]

For x = $280

[tex]z=\dfrac{280-280}{40}=0[/tex]

The p-value = [tex]P(-2<z<0)=P(z<0)-P(z<-2)[/tex]

[tex]0.5-0.0227501=0.4772499\approx0.4772[/tex]

Hence, the proportion of students will spend between $200 and $280 this semester = 0.4772

Approximately 34% of the college students will spend between $200 and $280 on book purchases for a semester according to the empirical rule for a bell-shaped and symmetric distribution.

The distribution of the amount of money spent on book purchases for a semester by college students is described as bell-shaped and symmetric with a mean of $280 and a standard deviation of $40. To find the proportion of students that will spend between $200 and $280, we can use the empirical rule (68-95-99.7 rule) which indicates that approximately 68% of the data fall within one standard deviation from the mean. Since we are concerned with the range from $200 (which is 2 standard deviations below the mean) to $280 (the mean), we are effectively looking at half of this 68% range. Therefore, approximately 34% of the students are expected to spend between $200 and $280 on books for a semester.

To calculate the proportion, we take 68% of the range (which covers -1 to +1 standard deviation from the mean) and divide it by 2:

Thus, the proportion of students spending between $200 and $280 is 0.34, which when rounded to four decimal places, gives us 0.3400.

Answer:

The cumulative percentage of fathers who completed only elementary school is nearle 33%.

Step-by-step explanation:

Among 30 fathers:

You can fill these numbers into the table:

[tex]\begin{array}{cccc}&\text{Frequency}&\text{Cumulative frequency}&\text{Cumulative percentage}\\\text{Elementary school}&10&10&\approx 33\%\\\text{El. and high school}&10&20&\approx 67\%\\\text{El., high schools and college}&7&27&90\%\\\text{El., high, college and grad. sch.}&3&30&100\%\end{array}[/tex]

The cumulative percentage of fathers who completed only elementary school is nearle 33%.

Answer: 33%

Step-by-step explanation:

Answer:  [tex]\dfrac{1}{100,000}[/tex]

Step-by-step explanation:

Given : The total number of digits in number system (0 to 9) = 10

The number of digits in a social security number = 9

After last four digits are printed, the number of digits remaining to print = 9-4=5

Since , repetition of digits is allowed, then the total number of ways to print 5 remaining digits is given by :-

[tex]10\times10\times10\times10\times10=100,000[/tex]

Now, the probability of getting the correct Social Security number of the person who was given the​ receipt is given by:-

[tex]\dfrac{\text{Number of correct code}}{\text{Total number of codes}}\\\\=\dfrac{1}{100,000}[/tex]

Final answer:

The probability of correctly guessing an entire Social Security number with the last four digits known is 0.001%, calculated by multiplying the probability of guessing each of the five unknown digits correctly, which is 1/10, resulting in (1/10)^5 or 1/100,000.

Explanation:

The question asks about the probability of correctly guessing an entire Social Security number given the last four digits. A Social Security number has nine digits and the digits can be repeated. If you know the last four digits, you need to guess the first five correctly.

Since each digit can be any number from 0 to 9, there are 10 possibilities for each digit. The probability of guessing one digit correctly is 1 out of 10 (1/10). To find the probability of guessing all five correctly, you need to multiply the probability for each digit, so the probability for all five is (1/10) x (1/10) x (1/10) x (1/10) x (1/10), which equals 1/100,000 or 0.00001. Therefore, the probability of getting the correct Social Security number is 0.00001 or 0.001%.

Each consecutive term in the sum is separated by a difference of 6, so the [tex]n[/tex]-th term is [tex]4+6(n-1)=6n-2[/tex] for [tex]n\ge1[/tex]. The last term is 70, so there are [tex]6n-2=70\implies n=12[/tex] terms in the sum.

Now,

[tex]S=4+10+\cdots+64+70[/tex]

but also

[tex]S=70+64+\cdots+10+4[/tex]

Doubling the sum and grouping terms in the same position gives

[tex]2S=(4+70)+(10+64)+\cdots+(64+10)+(70+4)=12\cdot74[/tex]

[tex]\implies\boxed{S=444}[/tex]

Step-by-step explanation:

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

Kyd's expected value for this game is -$0.38.

North's expected value for this game is  $0.38.Kyd is expected to lose money, and North is expected to gain money.  

North has the advantage.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Given

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

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Answer:

Step-by-step explanation:Find the slope of the line that passes through the points shown in the table.

The slope of the line that passes through the points in the table is

.

Answer:

The quotient is: x-7

The remainder is: -2

Step-by-step explanation:

We need to divide (x^2 - 13x +40) divided by (x- 6)

The quotient is: x-7

The remainder is: -2

The division is shown in the figure attached

Answer:

The probability that a randomly selected person is not color blind is 0.958

Step-by-step explanation:

Given,

C represents red-green color blindness,

Also, the probability that a randomly selected person is color blind,

P(C) = 0.042,

Thus, probability that a randomly selected person is not color blind,

P(C') = 1 - P(C) = 1 - 0.042 = 0.958

Answer:

D. x = 9√3 and y = 18

Step-by-step explanation:

This is an isosceles triangle divided into two equal parts.

Step 1: 18 can be divided into 2 parts which makes the base of both triangles 9.

Step 2: Find the value of x

The value of x can be found through the tan rule.

tan (angle) = opposite/adjacent

tan (60) = x/9

√3 = x/9

x = √3 x 9

x = 9√3

Step 3: Find the value of y

The value of y can be found through the cos rule.

Cos (angle) = adjacent/hypotenuse

Cos (60) = 9/y

1/2 = 9/y

y = 18

Therefore, the correct answer is D; x = 9√3 and y = 18

!!

Answer:

(2, 1)

Step-by-step explanation:

The best way to do this to avoid tedious fractions is to use the addition method (sometimes called the elimination method).  We will work to eliminate one of the variables.  Since the y values are smaller, let's work to get rid of those.  That means we have to have a positive and a negative of the same number so they cancel each other out.  We have a 2y and a 3y.  The LCM of those numbers is 6, so we will multiply the first equation by a 3 and the second one by a 2.  BUT they have to cancel out, so one of those multipliers will have to be negative.  I made the 2 negative.  Multiplying in the 3 and the -2:

3(-9x + 2y = -16)--> -27x + 6y = -48

-2(19x + 3y = 41)--> -38x - 6y = -82

Now you can see that the 6y and the -6y cancel each other out, leaving us to do the addition of what's left:

-65x = -130 so

x = 2

Now we will go back to either one of the original equations and sub in a 2 for x to solve for y:

19(2) + 3y = 41 so

38 + 3y = 41 and

3y = 3.  Therefore,

y = 1

The solution set then is (2, 1)

x=2 and y=1

proof:

-9x+2y=-16

-9(2)+2(1)=-16

which is a true statement

Notice that

[tex](2x^3-xy^2-2y+3)_y=-2xy-2[/tex]

[tex](-x^2y-2x)_x=-2xy-2[/tex]

so the ODE is exact, and we can find a solution [tex]F(x,y)=C[/tex] such that

[tex]F_x=2x^3-xy^2-2y+3[/tex]

[tex]F_y=-x^2y-2x[/tex]

Integrating both sides of the first equation wrt [tex]x[/tex] gives

[tex]F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x+g(y)[/tex]

Differentiating wrt [tex]y[/tex] gives

[tex]F_y=-x^2y-2x=-x^2y-2x+g'(y)\implies g'(y)=0\implies g(y)=C[/tex]

So we have

[tex]\boxed{F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x=C}[/tex]

The solution to this complex differential equation requires knowledge in calculus and differential equations. Without additional context, it's impossible to provide a specific solution. However, exploring technique utilization such as exact differential equations, integrating factors and substitution would be beneficial.

To solve the given differential equation, which is (2x^3 - xy^2 - 2y + 3)dx - (x^2y + 2x)dy = 0, we will use an approach of factorization or grouping like terms to simplify the equation. In some cases, you might need to rearrange terms and identify if it's a special type of differential equation, like exact, separable, or homogeneous, and then apply the relevant techniques accordingly.

This difficult task requires excellent knowledge in calculus and differential equations. Unfortunately, due to the complexity of this particular equation, without additional context or information, it is impossible to provide a specific solution. I would recommend you to go through topics such as exact differential equations, as well as methods involving integrating factors and substitution. These may help you to analyze and solve this complex equation.

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Answer:

C. 62 degrees

Step-by-step explanation:

Alright I see a circle and a half a rotation where the diameter is at. A half of 360 degrees is 180 degrees.

So the arc measure in degrees for EG is 180 degrees (both the left piece and right piece have this measure).

Since EG is 180 then FG=EG-EF=180-56=124.

To find x we have to half 124 since it is the arc measure where x is but x is the inscribed angle.

x=124/2=62

C.

62 degrees ! All the rest wouldn’t make sense (:

Answer:

The given statement is FALSE.

Step-by-step explanation:

If p1, p2, . . . , pn are prime, then A = p1p2 . . . pn1 + 1 is also prime.

No, this statement is false.

Let's take an example:

We take two prime numbers.

p1 = 3

p2 = 5

Now p1p2+1 becomes :

[tex](3\times5)+1=16[/tex]

And we know that 16 is not a prime number.

Note : A prime number is a number that is only divisible by 1 and itself like 3,5,7,11 etc.

Answer:

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Step-by-step explanation:

We are given that  a general solution of 6th order homogeneous linear equation

[tex] y=C_1+C_2e^{-t}+C_3 Cos(3t)+C_4 Sin(3t)+C_5 tCos(3t) +C_6 sin t (3t)[/tex]

We have to find the 6th order homogeneous linear differential equation whose general solution is given above.

We know that imaginary roots are in pair

There two values of imaginary roots and the values of imaginary roots are repeat.

From first term of general solution we get D=0

From second term of general solution we get D=-1

Last four terms are the values of imaginary roots and roots are repeated.

Therefore, D=[tex]\pm 3i[/tex] and D=[tex]\pm 3i[/tex]

Substitute all values then we get

[tex]D(D+1)(D^2+9)^2=0[/tex]

[tex]D(D+1)(D^4+18D^2+81)=0[/tex]

[tex]D^6+D^5+18D^4+18D^3+81 D^2+81 D=0[/tex]

[tex](D^6+D^5+18D^4+18D^3+81 D^2+81 D)y=0[/tex]

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Therefore, the 6th order homogeneous linear differential equation is

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Answer with explanation:

To test the Significance of the population which is Normally Distributed we will use the following Formula Called Z test

   [tex]z=\frac{\Bar X - \mu}{\frac{\sigma}{n}}[/tex]

[tex]\Bar X =31.1\\\\ \sigma=6.3\\\\ \mu=25\\\\n=15\\\\z=\frac{31.1-25}{\frac{6.3}{\sqrt{15}}}\\\\z=\frac{6.1\times\sqrt{15}}{6.3}\\\\z=\frac{6.1 \times 3.88}{6.3}\\\\z=3.756[/tex]

→p(Probability) Value when ,z=3.756 is equal to= 0.99992=0.9999

⇒Significance Level (α)=0.01

We will do Hypothesis testing to check whether population mean is different from 25 at the alpha equals 0.01 level of significance.

→0.9999 > 0.01

→p value > α

With a z value of 3.75, it is only 3.75% chance that ,mean will be different from 25.

So,we conclude that results are not significant.So,at 0.01 level of significance population mean will not be different from 25.

Answer:

Step-by-step explanation:

In March, Your Co. will collect 20% of January's sales, 30% of February's sales, and 50% of March's sales:

  .20×50 +.30×40 +.50×60 = 10 +12 +30 = 52

Similarly, in April, collections will be ...

  .20×40 + .30×60 + .50×30 = 8 +18 +15 = 41

Answer:Am nevoie de Puncte

Step-by-step explanation:

Answer: There are 4 people who only go to the game on Saturday.

Step-by-step explanation:

Let the number of people go on Saturday be n(A).

Let the number of people go on Sunday be n(B).

Since we have given that

n(A) = 8

n(B) = 12

n(A∪B)  = 16

According to rules, we get that

[tex]n(A)+n(B)-n(A\cap B)=n(A\cup B)\\\\8+12-n(A\cap B)=16\\\\20-n(A\cap B)=16\\\\n(A\cap B)=20-16=4[/tex]

So, n(only go on Saturday) = n(only A) = n(A) - n(A∩B) = 8-4 = 4

Hence, there are 4 people who only go to the game on Saturday.

Answer: [tex]\dfrac{2}{1001}[/tex]

Step-by-step explanation:

Given : The number of female applicants = 9

The number of male applicants = 6

Total applicants = 15

The number of ways to select 5 applicants from 15 applicants :-

[tex]^{15}C_5=\dfrac{15!}{5!(15-5)!}=3003[/tex]

The number to select 5 applicants from 15 applicants such that no female applicant is selected:-

[tex]^{9}C_0\times^6C_5=1\times\dfrac{6!}{5!(6-5)!}=6[/tex]

Now, the required probability :-

[tex]\dfrac{6}{3003}=\dfrac{2}{1001}[/tex]

Answer:

5377

Step-by-step explanation:

C(x) = x^2 - 400x + 45,377

To find the location of the minimum, we take the derivative of the function

We know that is a minimum since the parabola opens upward

dC/dx = 2x - 400

We set that equal to zero

2x-400 =0

Solving for x

2x-400+400=400

2x=400

Dividing by2

2x/2=400/2

x=200

The location of the minimum is at x=200

The value is found by substituting x back into the equation

C(200) = (200)^2 - 400(200) + 45,377

           =40000 - 80000+45377

            =5377

Answer:

The minimum unit cost is 5377

Step-by-step explanation:

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

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

For a quadratic function of the form

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

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

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

In this case the function is:

[tex]C(x) = x^2 - 400x + 45,377[/tex]

So:

[tex]a=1\\b=-400\\c=45,377[/tex]

Then the x coordinate of the vertex is:

[tex]x=-\frac{-400}{2(1)}[/tex]

[tex]x=200\ cars[/tex]

So the minimum unit cost is:

[tex]C(200) = (200)^2 - 400(200) + 45,377[/tex]

[tex]C(200) = 5377[/tex]

Answer:

R is not reflexive,not irreflexive ,symmetric,not asymmetric,not antisymmetric and transitive.

Step-by-step explanation:

1.Reflexive : Relation R is not reflexive because it does not contain identity relation on A.

Identity relation on A:{(1,1),(2,2),(3,3),(4,4)}

If we take a relation R on A :{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}

The relation R is reflexive on set A because it contain identity relation.

2.Irreflexive: Given relation R is not reflexive because it contain some elements of identity relation .If a relation is irreflexive then it does not contain any element of identity relation. We can say the intersection of R with identity relation is empty.

[tex]R\cap I=\left\{(1,1),(2,2),(4,4)\right\} \neq \phi[/tex]

If we take a relation on A R:{(1,2),(2,1)}

Then relation is irreflexive because it does not contain any element of identity relation

[tex]R\cap I=\phi[/tex]

3.Symmetric : The given relation R is symmetric because it satisfied the property of symmetric relation.

(1,2)belongs to relation (2,1) also belongs to relation ,(1,4) belongs to relation and (4,1) also belongs to given relation Hence we can say it is symmetric relation.

4.Asymmetric: The given relation is not asymmetric relation.Because it does not satisfied the property of asymmetric relation

Asymmetric relation: If (a,b) belongs to relation then (b,a) does not belongs to given relation.

Here (1,2) belongs to given relation and (2,1) also belongs to given relation Therefore, it is not asymmetric.

If we take a relation R on A

R:{(1,2),(1,3)}

It is asymmetric relation because it does not contain (2,1) and (3,1).

5.Antisymmetric: The given relation is not antisymmetric because it does not satisfied the property of antisymmetric relation.

Antisymmetric relation: If (a,b) and (b,a) belongs to relation then a=b

If we take a relation R

R;{(1,2) (1,1)}

It is antisymmetric because it contain (1,1) where 1=1 .Hence , it is antisymmetric.

6.Transitive: The given relation is transitive because it satisfied the property of transitive relation

Transitive relation: If (a,b) and (b,c) both belongs to relation R then (a,c) belongs to R.

Here, we have (1,2) and (2,1)belongs to relation then (1,1)  and (2,2) are also belongs to relation.

(1,4) and (4,1)  then (1,1) and (4,4) are also belongs to the relation.

Answer:

D. Stanley Kubrick

Step-by-step explanation:

Answer:

[tex]\Large\textnormal{(D.) Stanley Kubrick}[/tex]

Step-by-step explanation:

Stanley Kubrick directed to Dr. Strangelove. I hope this helps, and have a wonderful day!

I assume the cone has equation [tex]z=\sqrt{x^2+y^2}[/tex] (i.e. the upper half of the infinite cone given by [tex]z^2=x^2+y^2[/tex]). Take

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

The volume of the described region (call it [tex]R[/tex]) is

[tex]\displaystyle\iiint_R\mathrm dx\,\mathrm dy\,\mathrm dz=\int_0^{2\pi}\int_0^{\sqrt2}\int_{\pi/4}^\pi\rho^2\sin\varphi\,\mathrm d\varphi\,\mathrm d\rho\,\mathrm d\theta[/tex]

The limits on [tex]\theta[/tex] and [tex]\rho[/tex] should be obvious. The lower limit on [tex]\varphi[/tex] is obtained by first determining the intersection of the cone and sphere lies in the cylinder [tex]x^2+y^2=1[/tex]. The distance between the central axis of the cone and this intersection is 1. The sphere has radius [tex]\sqrt2[/tex]. Then [tex]\varphi[/tex] satisfies

[tex]\sin\varphi=\dfrac1{\sqrt2}\implies\varphi=\dfrac\pi4[/tex]

(I've added a picture to better demonstrate this)

Computing the integral is trivial. We have

[tex]\displaystyle2\pi\left(\int_0^{\sqrt2}\rho^2\,\mathrm d\rho\right)\left(\int_{\pi/4}^\pi\sin\varphi\,\mathrm d\varphi\right)=\boxed{\frac43(1+\sqrt2)\pi}[/tex]

Answer:

The answer is [tex]\frac{11}{26}[/tex]

Step-by-step explanation:

Total number of cards in the deck = 52

Number of clubs = 13

Number of picture cards = 12

Number of picture cards that are clubs = 3

So, number of picture cards or clubs = [tex]13+12-3=22[/tex]

P(club or picture) = [tex]\frac{22}{52} =\frac{11}{26}[/tex]

The answer is [tex]\frac{11}{26}[/tex].

To determine the probability of selecting a club or a picture card from a standard deck of 52 cards, calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

To determine the probability that a card selected from a standard deck of 52 cards is a club or a picture card, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

There are 13 clubs and 12 picture cards (Jacks, Queens, and Kings) in a deck. However, we need to subtract the Queen of Clubs, as it has already been selected. So, the number of favorable outcomes is 13 + 12 - 1 = 24.

The total number of possible outcomes is 52, as there are 52 cards in a standard deck.

Therefore, the probability of selecting a club or a picture card is 24/52, which simplifies to 6/13 or approximately 0.4615.

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