How many integers between 10000 and 99999, inclusive, are divisible by 3 or
5 or 7?

Final answer:

P(D ∪ F') is the probability of drawing a black card that is not a 10 from a standard 52-card deck, which is 24 black non-10 cards out of 52 total cards, resulting in a probability of 12/13.

Explanation:

The student is asking about probability in relation to drawing cards from a standard 52-card deck. Specifically, they want to find the probability of the event D (drawing a black card) or the complement of event F (not drawing a 10 card), denoted as P(D ∪ F'). In a standard deck, there are 26 black cards and four 10 cards (two of which are black), so the complement of F (F') is drawing any card that is not a 10, which is 52 - 4 = 48 cards. To find P(D ∪ F'), we consider the number of black cards that are not 10s, which is 24, since there are 26 black cards and 2 are 10s. Therefore, P(D ∪ F') is the probability of drawing one of these 24 cards out of the 52-card deck.

Calculating this probability:

P(D ∪ F') = number of black cards that are not 10s / total number of cards = 24/52 = 12/13.

The key concept here is that we're looking for the union of a black card and a non-10 card, which includes black cards that are also not the number 10.

The probability that her pulse rate is between 69 beats per minute and 77 beats per minute is 25.1%

Z score is used to determine by how many standard deviations the raw score is above or below the mean.

It is given by:

z = (raw score - mean) / standard deviation

Mean = 73, standard deviation = 12.5

For x = 69:

z = (69 - 73) / 12.5 = -0.32

For x = 77:

z = (77 - 73) / 12.5 = 0.32

P(-0.32 <z < 0.32) = P(z < 0.32) - P(z < -0.32) = 0.6255 - 0.3745 = 0.251

The probability that her pulse rate is between 69 beats per minute and 77 beats per minute is 25.1%

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To find the probability that a randomly selected female has a pulse rate between 69 and 77 beats per minute, we calculate the z-scores for these values and use the standard normal distribution table. The probability is approximately 0.2481.

To find the probability that a randomly selected female has a pulse rate between 69 and 77 beats per minute, we need to calculate the z-scores for these values and use the standard normal distribution table.

First, we calculate the z-score for 69 using the formula: z = (x - mu) / sigma, where x is the value, mu is the mean, and sigma is the standard deviation. Plugging in the values, we get z = (69 - 73) / 12.5 = -0.32.

Next, we calculate the z-score for 77: z = (77 - 73) / 12.5 = 0.32.

From the standard normal distribution table, we find that the probability of a z-score between -0.32 and 0.32 is approximately 0.2481.

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The probability that a drum meets the guarantee is approximately 0.3625. The standard deviation needed for a 0.97 probability is -0.691 pounds.

To find the probability that a drum meets the guarantee, we need to calculate the z-score for the value of 100 pounds using the formula z = (x - mean) / standard deviation. Plugging in the values, we get z = (100 - 101.3) / 3.68 = -0.353. Using a z-score table or a calculator, we can find that the probability is approximately 0.3625.

To find the standard deviation that would give a probability of 0.97, we need to find the z-score that corresponds to that probability. Using a z-score table or a calculator, we find that the z-score is approximately 1.88. Plugging this value into the z-score formula and rearranging for the standard deviation, we get standard deviation = (100 - 101.3) / 1.88 = -0.691. Rounded to three decimal places, the standard deviation would need to be -0.691 pounds.

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(a) The probability that a drum meets the guarantee is approximately 0.6381.

(b) To achieve a 97% probability of meeting the guarantee, the standard deviation would need to be approximately 1.383 pounds.

(a) To determine the probability that a drum contains at least 100 lbs of solvent, we need to find the Z-score. The Z-score formula is:

Z = (X - μ) / σ

Where:

First, compute the Z-score:

Z = (100 - 101.3) / 3.68 = -1.3 / 3.68 ≈ -0.3533

Next, we look up the Z-score in the standard normal distribution table or use a calculator to find the probability:

P(Z > -0.3533) ≈ 0.6381

So, the probability that a drum meets the guarantee is approximately 0.6381.

(b) To find the standard deviation such that the probability of the drum meeting the guarantee is 0.97, we need to solve for σ when P(Z > Z₀) = 0.97.

We know P(Z > Z₀) = 0.97 implies P(Z < Z₀) = 0.03 (since it is the complementary probability).

Using the Z-table or a calculator, we find the Z-score for the 3rd percentile, which is approximately:

Z₀ ≈ -1.88

Now, use the Z-score formula in reverse to solve for σ:

Z₀ = (X - μ) / σ

Plugging in the values:

-1.88 = (100 - 101.3) / σ

Solving for σ, we get:

σ = (101.3 - 100) / 1.88 ≈ 1.383

Thus, the standard deviation would need to be approximately 1.383 lbs to achieve a 97% probability that each drum meets the guarantee.

Answer:

61 in Egyptian numeral is ∩∩∩∩∩∩l

61 is written as - т т

Step-by-step explanation:

Egyptian numeral:

 ∩ mean 10

l = 1

for 60 we can use  6 number of   ∩∩∩∩∩

i.e.

therefore 61 in Egyptian numeral is ∩∩∩∩∩∩l

Babylonian numeral : Basically Babylonian number system is 60 based instead of 10.

there are total number of 59 numerals made up by two symbol only.

61 is written as - т т with a space between two symbol

Answer:

Step-by-step explanation:

Given that there are 10 people 4 men and 6 women

i) No of ways to select 6 members with no restrictions

= 10C6 = 210

ii) If more women than men should be there then we can have any one of the above possibilities

Women, men = (6,0) , (5,1) (4,2)

So No of ways will be sum of these three possibilities

= 6C6(4C0)+(6C5)(4C1)+(6C4)(4C2)

= 1+24+90

=115

Answer with explanation:

The expansion  of

  [tex](1+x)^n=1 + nx +\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+......[/tex]

where,n is a positive or negative , rational number.

Where, -1< x < 1

Expansion of

 [tex](1-x^2)^{-5}=1-5 x^2+\frac{(5)\times (6)}{2!}x^4-\frac{5\times 6\times 7}{3!}x^6+\frac{5\times 6\times 7\times 8}{4!}x^8-\frac{5\times 6\times 7\times 8\times 9}{5!}x^{10}+\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}x^{12}+....[/tex]

Coefficient of [tex]x^{12}[/tex] in the expansion of [tex](1-x^2)^{-5}[/tex] is

        [tex]=\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}\\\\=\frac{15120}{6\times 5 \times 4\times 3 \times 2 \times 1}\\\\=\frac{151200}{720}\\\\=210[/tex]

⇒As the expansion [tex](1-x^2)^{-5}[/tex] contains even power of x , so there will be no term containing [tex]x^{17}[/tex].

Answer: 789 calories burned while combining diet with 2 hours of exercise

Step-by-step explanation:

we have that

[tex]f(x)=2x+510[/tex]

[tex]g(x)=200x-125[/tex]

we know that

[tex](f+g)(x)=f(x)+g(x)[/tex]

substitute

[tex](f+g)(x)=2x+510+200x-125[/tex]

[tex](f+g)(x)=202x+385[/tex]

Find [tex](f+g)(2)[/tex]

For x=2 hours

substitute

[tex](f+g)(2)=202(2)+385[/tex]

[tex](f+g)(2)=789\ calories[/tex]

therefore

The answer is

789 calories burned while combining diet with 2 hours of exercise

Answer:

option d)378

Step-by-step explanation:

Given that a researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure.

Margin of error should be at most 6% = 0.06

Let us assume p =0.5 as when p =0.5 we get maximum std deviation so this method will give the minimum value for n the sample size easily.

We have std error = [tex]\sqrt{\frac{pq}{n} } =\frac{0.5}{\sqrt{n} }[/tex]

For 98%confident interval Z critical score = 2.33

Hence we have margin of error = [tex]2.33(\frac{0.5}{\sqrt{n} } <0.06\\n>377[/tex]

Hence answer is option d)378

The size of the sample needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 6% is; 376

We are told that Margin of error should be at most 6% = 0.06

Formula for margin of error is;

M = z√(p(1 - p)/n)

we are given the confidence level to be 98% and the z-score at this confidence level is 2.326

Since no standard deviation then we assume it is maximum and as such  assume p =0.5 which will give us the minimum sample required.

Thus;

0.06 = 2.326√(0.5(1 - 0.5)/n)

(0.06/2.326)² = (0.5²/n)

solving for n gives approximately n = 376

Thus, the size of the sample required is 376

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Answer:

The next number in the sequence is 36.

Step-by-step explanation:

Consider the provided sequence.

4, 9, 16, 25

The number 4 can be written as 2².

The number 9 can be written as 3².

The number 16 can be written as 4².

The number 25 can be written as 5².

The general term of the sequence is: [tex]a_n=(n+1)^2[/tex]

Thus, the next term will be:

[tex]a_5=(5+1)^2[/tex]

[tex]a_5=(6)^2[/tex]

[tex]a_5=36[/tex]

Therefore, the next number in the sequence is 36.

The next number in the sequence 4, 9, 16, 25 is 36.

The given sequence is 4, 9, 16, 25. To find the next number, we need to look for a pattern. Notice that these numbers are perfect squares:

⇒ 4 = 2²

⇒ 9 = 3²

⇒ 16 = 4²

⇒ 25 = 5²

The pattern shows that the numbers are the squares of consecutive integers (2, 3, 4, 5). The next integer in this sequence is 6, and its square is:

⇒ 6² = 36

Thus, the next number in the sequence is 36.

Answer:  The required answers are

AB is of order  6 × 6.

BA is of order  7  × 7.

Step-by-step explanation:  Given that the sizes of the matrices A and B are as follows :

A is of size 6 × 7   and   B is of size 7 × 6.

We are to find the sizes of AB and BA whenever they are defined.

We know that

if a matrix P has m rows and n columns, then its size is written as m × n.

Also, two matrices P and Q of sizes m × n and r × s respectively can be multiplies if the number of columns in P is equal to the number of rows in Q.

That is, if n = r. And the size of the matrix P × Q is m × s.

Now, since the number of columns in A is equal to the number of rows in B, the product A × B is possible and is of order 6 × 6.

Similarly, the number of columns in B is equal to the number of rows in A, the product B × A is possible and is of order 7 × 7.

Thus, the required answers are

AB is of order  6 × 6.

BA is of order  7  × 7.

Answer:

D. 40 < t < 60

Step-by-step explanation:

Given function,

[tex]N(t) = -3t^3 + 450t^2 - 21,600t + 1,100[/tex]

Differentiating with respect to x,

[tex]N(t) = -9t^2+ 900t - 21,600[/tex]

For increasing or decreasing,

f'(x) = 0,

[tex]-9t^2+ 900t - 21,600=0[/tex]

By the quadratic formula,

[tex]t=\frac{-900\pm \sqrt{900^2-4\times -9\times -21600}}{-18}[/tex]

[tex]t=\frac{-900\pm \sqrt{32400}}{-18}[/tex]

[tex]t=\frac{-900\pm 180}{-18}[/tex]

[tex]\implies t=\frac{-900+180}{-18}\text{ or }t=\frac{-900-180}{-18}[/tex]

[tex]\implies t=40\text{ or }t=60[/tex]

Since, in the interval -∞ < t < 40, f'(x) = negative,

In the interval 40 < t < 60, f'(t) = Positive,

While in the interval 60 < t < ∞, f'(t) = negative,

Hence, the values of t for which N'(t) increasing are,

40 < t < 60,

Option 'D' is correct.

Answer:

D. 1200 mg

Step-by-step explanation:

In order to find the solution we need to understand that a dosage of 20 mg/kg means that 20 mg are administered to the patient for each kg of his/her weight.

So, if the patient weight is 60 kg then:

Total drug X = (20mg/Kg)*(60Kg)=1200mg.

In conclusion, 1200 mg will be administered to the patient, so the answer is D.

Final answer:

The student's numbers have been converted to scientific notation with the correct number of significant figures: 2.7 × 10^-8 for 0.000000027 × 10, 3.56 × 10^2 for 356 × 10, 4.7764 × 10^5 for 47,764 × 10, and 9.6 × 10^-2 for 0.096 × 10.

Explanation:

To express numbers in scientific notation, you need to write them in the form of a single digit from 1 up to 9 (but not 10), followed by a decimal point and the rest of the significant figures, and then multiplied by 10 raised to the power of the number of places the decimal point has moved.

Here are the conversions for the numbers provided:

(a) 0.000000027 × 10 is written in scientific notation as 2.7 × 10-8.

(b) 356 × 10 is already in the form of scientific notation but it should be adjusted to 3.56 × 102.

(c) 47,764 × 10 can be written as 4.7764 × 105 using significant figures.

(d) 0.096 × 10 should be written as 9.6 × 10-2.

Step-by-step explanation:

at time = 0min,

height, h0 = 10.5cm

area, a0 = 95cmsq

base, b0 = a0 x2/h0

=> b0 = 95 x2 / 10.5 = 18.1cm

at time = 1 min,

increase of height, rh = 1.5cm/min

height at 1 min, h1 = h0 x rh

=> h1= 10.5 × 1.5 = 15.75cm

increase of area, ra = 4.5cmsq/min

area after 1 min, a1 = a0 x ra

=> a1= 95 x 4.5 = 427.5cm/sq

base at 1 min, b1 = a1x2/h1

=> b1 = 427.5 x 2 /15.75 = 54.3 cm

rate of increase for base, rb = b1/b2

=> rb = 54.3/18.1 = 3cm/min

Answer:

The water level changing by the rate of -0.0088 feet per hour ( approx )

Step-by-step explanation:

Given,

The volume of a segment of height h in a hemisphere of radius r is,

[tex]V=\pi h^2(r-\frac{h}{3})[/tex]

Where, r is the radius of the hemispherical tank,

h is the water level, ( in feet )

Here, r = 9 feet,

[tex]\implies V=\pi h^2(9-\frac{h}{3})[/tex]

[tex]V=9\pi h^2-\frac{\pi h^3}{3}[/tex]

Differentiating with respect to t ( time ),

[tex]\frac{dV}{dt}=18\pi h\frac{dh}{dt}-\frac{3\pi h^2}{3}\frac{dh}{dt}[/tex]

[tex]\frac{dV}{dt}=\pi h(18-h)\frac{dh}{dT}[/tex]

Here, [tex]\frac{dV}{dt}=-2\text{ cubic feet per hour}[/tex]

And, h = 6 feet,

Thus,

[tex]-2=\pi 6(18-6)\frac{dh}{dt}[/tex]

[tex]\implies \frac{dh}{dt}=\frac{-2}{72\pi}=-0.00884194128288\approx -0.0088[/tex]

Answer:

3 (x-2) (x+2) (x^2+4)

Step-by-step explanation:

3x^4 − 48

Factor out a 3

3(x^4 -16)

Inside the parentheses is the difference of squares (a^2 - b^2) = (a-b) (a+b)

where a = x^2  and b = 4

3 (x^2-4) (x^2+4)

Inside the first parentheses is the difference of squares where a = x and b=2

3 (x-2) (x+2) (x^2+4)

Answer:

3 (x-2) (x+2) (x^2+4)

Step-by-step explanation:

Answer: [tex]H_0:p\leq0.53[/tex]

[tex]H_a:p>0.53[/tex]

Step-by-step explanation:

Claim :  A a political strategist wants to test the claim that the percentage of residents who favor construction is more than 53%.

Let 'p' be the percentage of residents who favor construction .

Claim : [tex]p> 0.53[/tex]

We know that the null hypothesis has equal sign.

Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-

[tex]H_0:p\leq0.53[/tex]

And the alternative hypothesis must be :-

[tex]H_a:p>0.53[/tex]

Thus, the null hypothesis and the alternative hypothesis for this test :

[tex]H_0:p\leq0.53[/tex]

[tex]H_a:p>0.53[/tex]

Answer: 12.10

Step-by-step explanation:

Given : Mean : [tex]\mu = 15.45[/tex]

Standard deviation : [tex]\sigma = 13.70[/tex]

The formula to calculate the z-score :-

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

For x= 5 degrees

[tex]z=\dfrac{5-15.45}{13.70}=-0.7627737226\approx-0.76[/tex]

For x= 10 degrees

[tex]z=\dfrac{10-15.45}{13.70}=-0.397810218\approx-0.40[/tex]

The P-value : [tex]P(-0.76<z<-0.40)=P(z<-0.40)-P(z<-0.76)[/tex]

[tex]=0.3445783-0.2236273=0.120951\approx0.1210[/tex]

In percent , [tex]0.1210\times100=12.10\%[/tex]

Hence, the percentage of days had a low temperature between 5 degrees and 10 degrees = 12.10%

Answer:

-$ 0.79

Step-by-step explanation:

Since, the player has made 282 out of 393 free throws,

So, the probability of a free throw = [tex]\frac{282}{393}[/tex],

Thus, the probability of 2 free throws = [tex]\frac{282}{393}\times \frac{282}{393}=\frac{8836}{17161}[/tex]

And, the probability of not getting 2 free throws = [tex]1-\frac{8836}{17161}=\frac{8325}{17161}[/tex]

Given, the price of winning ( getting 2 free throws ) is $6 while the price of losing ( not getting 2 free throws ) is - $ 8 ( ∵ there is a loss of $ 8 ),

Hence, the expected value of the proposition = probability of winning × winning value + probability of losing × losing value

[tex]= \frac{8836}{17161}\times 6 + \frac{8325}{17161}\times -8[/tex]

[tex]=\frac{53016}{17161}-\frac{66600}{17161}[/tex]

[tex]=-\frac{13584}{17161}[/tex]

[tex]=-\$ 0.79156226327[/tex]

[tex]\approx -\$ 0.79[/tex]

Answer: 16

Step-by-step explanation:

Given : Mean : [tex]\mu=100[/tex]

Standard deviation : [tex]\sigma =15[/tex]

The value of z-score is given by :-

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

For x= 115

[tex]z=\dfrac{115-100}{15}=1[/tex]

The p-value : [tex]P(z>1)=1-P(z<1)=1- 0.8413447=0.1586553[/tex]

Now, the number of people smarter than Many is given by :_

[tex]100\times0.1586553=15.86553\approx16[/tex]

Hence, there are 16 members smarter than Mandy.

Answer: $4.61 ⇒ this much amount expect to earn

Step-by-step explanation:

Given that,

Total number of tickets that do not have prize = 880

Tickets that win a free adoption (valued at $20) = 11

Ticket that wins $123 worth of pet supplies and toys = 1

So, total ticket sold = 880 + 11 + 1

                                = 892

Probability of tickets that not getting any prize = [tex]\frac{880}{892}[/tex]

Probability of tickets that win a free adoption = [tex]\frac{11}{892}[/tex]

Probability of tickets that wins $123 worth of pet supplies and toys = [tex]\frac{1}{892}[/tex]

Ticket value for no prize = $5

Ticket value that win a free adoption = -$20 + $5 = -$15

Ticket value that wins $123 worth of pet supplies and toys = -$123 + $5 = -$118

Expected return for each ticket = Σ(probability)(value of ticket)

 =  [tex]\dfrac{880}{892} \times5 + \dfrac{11}{892} \times (-$15) +\dfrac{1}{892}\times(-118)[/tex]

= [tex]\frac{4117}{892}[/tex]

= $4.61 ⇒ this much amount expect to earn.

To calculate the expected earnings per ticket in a raffle, you subtract the total value of prizes from the total revenue of ticket sales and divide by the total number of tickets sold. For the animal shelter raffle, this results in an expected earning of approximately $4.62 per ticket sold.

The subject of your question falls under the category of Mathematics, specifically dealing with the concept of expected value in probability. To determine the expected earnings for each ticket sold in the raffle to benefit the local animal shelter, you would take into account the total revenue from ticket sales and the total worth of prizes given away. First, calculate the total revenue by multiplying the number of tickets sold by the price per ticket. Then, add up the total value of all the prizes. Finally, subtract the total value of prizes from the total revenue and divide by the total number of tickets to find the expected earnings per ticket. Remember to round to the nearest cent.

Here is an example calculation based on the figures provided:

Therefore, the shelter should expect to earn approximately $4.62 for each ticket sold, after rounding to the nearest cent.

Answer:

-1/4 meter per minute

Step-by-step explanation:

Since, the volume of a cube,

[tex]V=r^3[/tex]

Where, r is the edge of the cube,

Differentiating with respect to t ( time )

[tex]\frac{dV}{dt}=3r^2\frac{dr}{dt}[/tex]

Given, [tex]\frac{dV}{dt}=-3\text{ cubic meters per minute}[/tex]

Also, V = 8 ⇒ r = ∛8 = 2,

By substituting the values,

[tex]-3=3(2)^2 \frac{dr}{dt}[/tex]

[tex]-3=12\frac{dr}{dt}[/tex]

[tex]\implies \frac{dr}{dt}=-\frac{3}{12}=-\frac{1}{4}[/tex]

Hence, the rate of change of an edge is -1/4 meter per minute.

The rate of change of an edge of the cube when the volume is exactly 8 cubic meters is -0.25 meters per minute, calculated using the formula for the volume of a cube and the chain rule for differentiation.

The student seeks to find the rate of change of an edge of a cube when the volume is decreasing at a specific rate. Given that the volume decreases at a rate of 3 cubic meters per minute, we can find the rate at which the edge length changes using the formula for the volume of a cube, which is V = s^3, where V is volume and s is the edge length.

To determine the rate of change of the edge length, we can use the chain rule in calculus to differentiate the volume with respect to time: dV/dt = 3( s^2 )(ds/dt). We know that dV/dt = -3 m^3/min and that when the volume V = 8 m^3, the edge length s can be found by taking the cube root of the volume, which is 2 meters. By substituting these values, we solve for ds/dt, which is the rate of change of the edge length. The resulting calculation is ds/dt = (dV/dt) / (3s^2) = (-3 m^3/min) / (3(2m)^2) = -0.25 m/min.

Answer:

∠AMX=72°

Step-by-step explanation:

we know that

An isosceles triangles has two equal sides and two equal interior angles

In the isosceles triangle MAX

we have that

XA=MA

and ∠AXM= ∠AMX -----> angles base    

we have that

∠AXM=72°

therefore

∠AMX=72°

Answer: C. 2,272 mg

Step-by-step explanation:

Given : The recommended dose of a particular drug is 0.1 g/kg.

We know that 1 kilogram is equals to approximately 2.20 pounds.

Then ,[tex]\text{1 pound}=\dfrac{1}{2.20}\text{ kilogram}[/tex]

[tex]\Rightarrow\text{50 pounds}=\dfrac{1}{2.20}\times50\approx22.72text{ kilogram}[/tex]

Now, the dose of drug should be given to a 22.72 kilogram patient is given by :-

[tex]22.72\times0.1=2.272g[/tex]

Since 1 grams = 1000 milligrams

[tex]2.272\text{ g}=2,272\text{ mg}[/tex]

Hence , 2,272 mg of the drug should be given to a 50 lb. patient.

Answer:

A transitive property

Step-by-step explanation:

There isn't much to this.

This is the the transitive property.

I guess I can go through each choice and tell you what the property looks like or postulate.

A)  If x=y and y=z, then x=z.

This is the exact form of your conditional.

x is AB here

y is BC here

z is DE here

B) Segment Addition Postulate

If A,B, and C are collinear with A and B as endpoints, then AB=AC+CB.

Your conditional said nothing about segment addition (no plus sign).

C) Distributive property is a(b+c)=ab+ac.

This can't be applied to any part of this.  There is not even any parenthesis.

D) The symmetric property says if a=b then b=a.

There is two parts to our hypothesis where this is only part to the symmetric property for the hypothesis .  

The statement 'If AB = BC  and BC = DE, then AB = DE' is justified by the Transitive Property of Equality, stating that, if two quantities both equal a third, they are equal to each other.

The justification for the statement 'If AB = BC  and BC = DE, then AB = DE' is the Transitive Property of Equality. This property states that if two quantities are both equal to a third quantity, then they are equal to each other. In this case, AB and DE are both equal to BC, therefore, according to the transitive property, AB must be equal to DE.

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When we evaluate the line integral by the two following methods the answer is: [tex]\frac{1}{3}[/tex].

(a) Directly:

We will evaluate the line integral directly by breaking it up into three parts, one for each side of the triangle.

1. Along the line from (0, 0) to (1, 0),  y = 0 , so  dy = 0 . The integral simplifies to:

[tex]\[ \int_{(0,0)}^{(1,0)} xy \, dx + x^2y^3 \, dy = \int_{0}^{1} 0 \, dx + 0 \, dy = 0 \][/tex]

 2. Along the line from (1, 0) to (1, 2),  x = 1 , so [tex]\( dx = 0 \)[/tex]. The integral simplifies to:

[tex]\[ \int_{(1,0)}^{(1,2)} 1 \cdot y \, dx + 1^2 \cdot y^3 \, dy = \int_{0}^{2} y \, dy = \left[ \frac{1}{2}y^2 \right]_{0}^{2} = 2 \][/tex]

3. Along the line from (1, 2) to (0, 0),  x  varies from 1 to 0, and  y  varies from 2 to 0. We can express y  as [tex]\( y = 2 - 2x \)[/tex] and [tex]\( dx = -dx \)[/tex] (since x  is decreasing). The integral becomes:

[tex]\[ \int_{(1,2)}^{(0,0)} x(2 - 2x) \, dx + x^2(2 - 2x)^3(-dx) \] \[ = \int_{1}^{0} 2x - 2x^2 \, dx - \int_{1}^{0} 8x^2(1 - x)^3 \, dx \] \[ = \left[ x^2 - \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{8}{3}x^3(1 - x)^3 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{1}{3} \right) - 0 = \frac{1}{3} \][/tex]

Adding up the three parts, we get the direct line integral:

[tex]\[ 0 + 2 + \frac{1}{3} = \frac{7}{3} \][/tex]

(b) Using Green's Theorem:

Green's Theorem states that for a vector field [tex]\( F(x, y) = P(x, y) \, \mathbf{i} + Q(x, y) \, \mathbf{j} \)[/tex]  and a simple closed curve C  oriented counter clockwise, the line integral around C  is equal to the double integral of the curl of F  over the region D  enclosed by C :

[tex]\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \][/tex]

For our vector field, [tex]\( P = xy \)[/tex] and [tex]\( Q = x^2y^3 \)[/tex], so:

[tex]\[ \frac{\partial Q}{\partial x} = 2xy^3 \] \[ \frac{\partial P}{\partial y} = x \][/tex]

The double integral over the triangle is:

[tex]\[ \int_{0}^{1} \int_{0}^{2x} (2xy^3 - x) \, dy \, dx \] \[ = \int_{0}^{1} \left[ \frac{1}{2}x \cdot y^4 - xy \right]_{0}^{2x} \, dx \] \[ = \int_{0}^{1} (4x^3 - 2x^2) \, dx \] \[ = \left[ x^4 - \frac{2}{3}x^3 \right]_{0}^{1} \] \[ = 1 - \frac{2}{3} = \frac{1}{3} \][/tex]

The result using Green's Theorem is: [tex]\[ \frac{1}{3} \][/tex]

For the line from (1, 2) to (0, 0), parameterizing  x  from 1 to 0 and  y = 2x , we have:

[tex]\[ \int_{1}^{0} x(2x) \, dx + x^2(2x)^3(-dx) \] \[ = \int_{1}^{0} 2x^2 \, dx - \int_{1}^{0} 8x^5 \, dx \] \[ = \left[ \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{4}{3}x^6 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{2}{3} \right) - 0 + \frac{4}{3} \] \[ = \frac{2}{3} + \frac{4}{3} = 2 \][/tex]

Now, adding up the corrected parts, we get:

[tex]\[ 0 + 2 + 2 = 4 \][/tex]

This corrected value matches the result obtained using Green's Theorem, which confirms that the correct answer is: [tex]\[ \boxed{\frac{1}{3}} \][/tex].

Hence, the answer is:

                           13860

Sally has 6 red​ flags, 4 green​ flags, and 2 white flags.

i.e. there are a total of 12 flags.

Now, we are asked to find the different number of arrangements that may be made with the help of these 12-flags.

We need to use the method of permutation in order to find the different number of arrangements.

The rule is used as follows:

If we need to arrange n items such that there are [tex]n_1[/tex] number of items of one type,[tex]n_2[/tex] items same of other type .

Then the number of ways of arranging them is:

[tex]=\dfrac{n!}{n_1!\cdot n_2!}[/tex]

Hence, here the number of ways of forming a flag signal is:

[tex]=\dfrac{12!}{6!\times 4!\times 2!}[/tex]

( since 6 flags are of same color i.e. red , 4 flags are of green color and 2 are of white colors )

[tex]=\dfrac{12\times 11\times 10\times 9\times 8\times 7\times 6!}{6!\times 4!\times 2!}\\\\\\=\dfrac{12\times 11\times 10\times 9\times 8\times 7}{4\times 3\times 2\times 2}\\\\=13860[/tex]

To determine how many different 12-flag signals Sally can run up a flag pole using 6 red flags, 4 green flags, and 2 white flags, we need to calculate the permutations of these flags, taking into account that flags of the same color are indistinguishable from each other.
Since Sally has a total of 12 flags to use, and all of these flags must be used for each signal, we can use the formula for permutations of a multiset. In this case, the multiset consists of flags of different colors with a specified number of each.
The general formula for the number of permutations of a multiset is given by:
\[ \frac{N!}{n_1! \cdot n_2! \cdot ... \cdot n_k!} \]
Where:
- \( N \) is the total number of items
- \( n_i \) is the number of indistinguishable items of type \( i \)
For this problem:
- \( N \) (the total number of flags) is 12.
- \( n_1 \) (the number of red flags) is 6.
- \( n_2 \) (the number of green flags) is 4.
- \( n_3 \) (the number of white flags) is 2.
Now we can plug these numbers into the formula:
\[ \frac{12!}{6! \cdot 4! \cdot 2!} \]
Calculating this, we have:
\[ 12! = 479,001,600 \]
\[ 6! = 720 \]
\[ 4! = 24 \]
\[ 2! = 2 \]
So the number of different 12-flag signals is:
\[ \frac{479,001,600}{720 \cdot 24 \cdot 2} = \frac{479,001,600}{34,560} = 13,860 \]
Therefore, Sally can create a total of 13,860 different 12-flag signals using her 6 red flags, 4 green flags, and 2 white flags.

Answer:

The solution of the given initial value problems in explicit form is [tex]y=x-x^2-2[/tex]  and the solutions are defined for all real numbers.

Step-by-step explanation:

The given differential equation is

[tex]y'=1-2x[/tex]

It can be written as

[tex]\frac{dy}{dx}=1-2x[/tex]

Use variable separable method to solve this differential equation.

[tex]dy=(1-2x)dx[/tex]

Integrate both the sides.

[tex]\int dy=\int (1-2x)dx[/tex]

[tex]y=x-2(\frac{x^2}{2})+C[/tex]                  [tex][\because \int x^n=\frac{x^{n+1}}{n+1}][/tex]

[tex]y=x-x^2+C[/tex]              ... (1)

It is given that y(1) = -2. Substitute x=1 and y=-2 to find the value of C.

[tex]-2=1-(1)^2+C[/tex]

[tex]-2=1-1+C[/tex]

[tex]-2=C[/tex]

The value of C is -2. Substitute C=-2 in equation (1).

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

Therefore the solution of the given initial value problems in explicit form is [tex]y=x-x^2-2[/tex] .

The solution is quadratic function, so it is defined for all real values.

Therefore the solutions are defined for all real numbers.

Answer:

6.a. Mean= 6

b. Median=6

c Midrange=6

d.Mode=4

7.Standard deviation=2.2079

Step-by-step explanation:

Given data

3,3,3,3,4,5,6,6,6,7,7,8,8,9,9,9

Total data items,n=16

Sum o data items=96

a. Mean=[tex]\frac{sum\;of\;data\;items}{total\;data\;items}[/tex]

Mean=[tex]\frac{96}{16}[/tex]

Mean=6

b.If total number of items are even then

Median=[tex]\frac{\frac{n}{2}^{th}\;observation+\left(\frac{n}{2}+1\right)^{th}}{2}[/tex]

Median=[tex]\frac{\frac{16}{2}^{th} observation+\left(\frac{16}{2}+1\right)^{th} observation}{2}[/tex]

Median=[tex]\frac{8^{th} observation+9^{th} observation}{2}[/tex]

Median= [tex]\frac{6+6}{2}[/tex]

Median= [tex]\frac{12}{2}[/tex]

Median=6

c. Midrange=[tex]\frac{lower\;value+highest\;value}{2}[/tex]

Lower data item=3

Highest data item=9

Midrange= [tex]\frac{3+9}{2}[/tex]

Midrange= 6

d.Mode : It is defines as  a number that appear most often in a set of numbers.

Mode=3

7. Mean[tex]\bar x=6[/tex]

[tex]\mid x-\bar x\mid[/tex]                       [tex]{\mid x-\bar x\mid}^2[/tex]

3                                           9    

3                                           9

3                                           9

3                                           9

2                                           4

1                                            1

0                                           0

0                                           0

0                                           0

1                                            1

1                                            1

2                                           4

2                                           4

3                                           9

3                                           9

3                                           9

[tex]\sum{\mid x-\bar x\mid}^2=78[/tex]

n=16

Standard deviation=[tex]\sqrt{\frac{\sum{\mid x-\bar x}^2}{n}}[/tex]

Standard devaition=[tex]\sqrt{\frac{78}{16}}[/tex]

Standard deviation=[tex]\sqrt{4.875}[/tex]

Standard deviation of data =2.2079

Answer:

The given set is infinite.

Step-by-step explanation:

If a set has finite number of elements, then it is known finite set.

If a set has infinite number of elements, then it is known infinite set. In other words a non finite set is called infinite set.

The given elements of a set are

124, 28, 32, 36,...

Let the given set is

A = { 124, 28, 32, 36,... }

The number of elements in set A is infinite. So, the set A is infinite.

Therefore the given set is infinite.