Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are: $\bullet$ Carolyn always has the first turn. $\bullet$ Carolyn and Paul alternate turns. $\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list. $\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed. $\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers. For example, if $n=6,$ a possible sequence of moves is shown in this chart: \begin{tabular}{|c|c|c|} \hline Player & Removed \# & \# remaining \\ \hline Carolyn & 4 & 1, 2, 3, 5, 6 \\ \hline Paul & 1, 2 & 3, 5, 6 \\ \hline Carolyn & 6 & 3, 5 \\ \hline Paul & 3 & 5 \\ \hline Carolyn & None & 5 \\ \hline Paul & 5 & None \\ \hline \end{tabular} Note that Carolyn can't remove $3$ or $5$ on her second turn, and can't remove any number on her third turn. In this example, the sum of the numbers removed by Carolyn is $4+6=10$ and the sum of the numbers removed by Paul is $1+2+3+5=11.$ Suppose that $n=6$ and Carolyn removes the integer $2$ on her first turn. Determine the sum of the numbers that Carolyn removes.

The inverse function theorem says

[tex]\dfrac{\mathrm df^{-1}}{\mathrm dx}(16)=\dfrac1{\frac{\mathrm df}{\mathrm dx}(f^{-1}(16))}[/tex]

We have

[tex]f(x)=x^2+4x-5[/tex]

defined on [tex]x>-2[/tex], for which we get

[tex]f^{-1}(x)=-2+\sqrt{x+9}[/tex]

and

[tex]f^{-1}(16)=-2+\sqrt{16+9}=3[/tex]

The derivative of [tex]f(x)[/tex] is

[tex]f'(x)=2x+4[/tex]

So we end up with

[tex]\dfrac{\mathrm df^{-1}}{\mathrm dx}(16)=\dfrac1{\frac{\mathrm df}{\mathrm dx}(3)}=\dfrac1{10}[/tex]

Answer:

Step-by-step explanation: distribute -3 to the parenthesis (-2y-4) to eliminate the parenthesis. you’ll be left with 6y +12 -5y-2. From there you combine like terms. do 6y-5y= 1y or just y and 12-2 = 10. your answer would be 10

Answer:

Hi, there!!!

I hope you mean to evaluate cosA+ sonA /cosA - sinA.

so, i hope the answer in pictures will help you.

Answer: The missing number is 5.

Step-by-step explanation:

In the table we can only have numbers between 1 and 9,

The pattern that i see is:

We have sets of 3 numbers.

"the bottom number is equal to the difference between the two first numers, if the difference is negative, change the sign, if the difference is zero, there goes a 9 (the next number to zero)"

Goin from right to left we have:

9 - 6 = 3

6 - 2 = 4

4 - 9 = - 5 (is negative, so we actually use -(-5) = 5)

4 - 4 = 0 (we can not use zero, so we use the next number, 9)

3 - 3 = 0 (same as above)

? - 1 = 4

? = 4 + 1 =  5

The missing number is 5.

Answer:

x = 3, y = -2

Step-by-step explanation:

Since y=y

then, -3x +7 = 2x-8

7+8 = 3x+2x

15 = 5x

x=3

substitute

y = 2(3) - 8

y = -2

Hope that helped!!! k

Answer:

y = -2

x = 3

Step-by-step explanation:

Solve using elimination

1. Rearrange the equations to make it easier to solve

y = -3x + 7 → 3x + y = 7

y = 2x - 8 → 2x - y = 8

2. Multiply the equations to have a matching coefficient

2(3x + y = 7) = 6x + 2y = 14

3(2x - y = 8) = 6x - 3y = 24

3. Subtract

 6x + 2y = 14

- 6x - 3y = 24

  0 + 5y = -10

4. Solve for y

5y = -10

y = -2

5. Substitute y in any equation to solve for x

-2 = -3x + 7

-3x = -9

x = 3

Answer: 44

Step-by-step explanation:

Answer:

2.8

Step-by-step explanation:

11.1-8.3=2.8

HOPE I HELPED

PLS MARK BRAINLIEST

DESPERATELY TRYING TO LEVEL UP

✌ -ZYLYNN JADE ARDENNE

JUST A RANDOM GIRL WANTING TO HELP PEOPLE!

                            PEACE!

Answer:

Step-by-step explanation:

( See the attached picture )

Now,

Mean = [tex] \mathsf{\frac{Σfx}{n} }[/tex]

[tex] \mathsf{ = \frac{250}{19} }[/tex]

[tex] \mathsf{ = 13.15}[/tex]

------------------------------------------------------------------------

In the case of repeated data , follow the steps given below to calculate the mean :

Hope I helped!

Best regards!

Answer:

1. elements it contains =  (1,3)

2.  elements it contains = 35

3.  elements it contains = 8

4.  elements it contains = 17

Step-by-step explanation:

A. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 1; B: the numbers add to 4; C: at least one of the numbers is 1; and D: the numbers do not add to 10. Express the given event in symbols.

The red die shows 1 and the numbers add to 4.

How many elements does it contain?

B. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 3; B: the numbers add to 2; C: at least one of the numbers is 1; and D: the numbers do not add to 10. Express the given event in symbols. HINT [See Example 5.]

The numbers do not add to 2.

How many elements does it contain?

C. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 1; B: the numbers add to 2; C: at least one of the numbers is 3; and D: the numbers do not add to 11. Express the given event in symbols. HINT [See Example 5.]

Either the numbers add to 11 or the red die shows a 1.

How many elements does it contain?

D. Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows 4; B: the numbers add to 5; C: at least one of the numbers is 1; and D: the numbers do not add to 9. Express the given event in symbols. HINT [See Example 5.]

Either the numbers add to 5, or they add to 9, or at least one of them is 1.

How many elements does it contain?

NB. Attached is the solution to the problems stated above

Answer:

15x - y = - 126

or y = 15x + 126

Step-by-step explanation:

will make it simple and short

to find the equation... we need to find slope first.

                   y2 - y1             -9  -   6

slope = m  = ---------    =       -----------  =  15

                   x2 - x1             -9  -  (-8)

so we know that the equation of the line using point (-8,6) and slope 15             y - 6 = 15( x + 8)

y - 6 = 15x + 120

Writing the equation in the form   Ax + By = C

15x - y = -120-6

therefore.... 15x - y = - 126   or simplify it as or y = 15x + 126

Hope this helps

Answer:

Using Anova for a tri linear probability at ∝= 0.005

Step-by-step explanation:

Here simple probability cannot be used because  we want to enter your answer as a tri-linear inequality using decimals (not percents).

So we can use ANOVA

Null hypothesis

H0: µA = µB=µC

all the means are equal

Alternative hypothesis

H1: Not all means are equal.

The significance level is set at α-0.005

The test statistic to use is

F = sb²/ sw²

Which if H0 is true has an F distribution with v₁=k-1 and v₂= n-k degrees of freedom .

The computations are as follows

                     XA (XA)²            XB (XB)²         XC (XC)²       Total   ∑X²

Male                   19(361)          4(16)                 12(144)           35     521    

Female               3(9)                13 (169)           5  (25)            21      203  

TotalTj                  22                   17                      17              56     724

T²j                       (22)(22)

                             484                 289                  289        1062  

∑X²                     370               285                     169

Correction Factor = CF = Tj²/n = (56)²/6= 522.67

Total SS ∑∑X²- C. F = 724- 522.67= 201.33

Between SS ∑T²j/r - C.F = 1062/ 2 - 522.877 =8.33

Within SS = Total SS - Between SS

=201.33- 8.33= 193

The Analysis of Variance Table is

Source Of                Sum of          Mean         Computed

Variation        d.f    Squares      Squares                    F

Between

Samples           1        8.33               8.33           8.33/ 48.25= 0.1726

Within

Samples          4       193              48.25                                  

The critical region is F >F ₀.₀₀₅ (1,4) = 31.3328

Calculated value of F = 0.1726

Since it is smaller than 5 %  reject H0.

However the decimal probability will be

Male 19 4 12 35

Female 3 13 5 21

Total 22 17 17 56

There are total 22 people who get an A but only 19 males who get an A

So the probability that a male gets an A is = 19/22= 0.8636

Answer:

First use the protractor then round the number to the nearest 10

Answer:

Round to the nearest tenth

Step-by-step explanation:

This sequence converges to 0.

Proof: Recall that

[tex]\displaystyle\lim_{n\to\infty}\frac1{\sqrt n}=0[/tex]

is to say that for any given [tex]\varepsilon>0[/tex], there is some [tex]N[/tex] for which [tex]\left|\frac1{\sqrt n}-0\right|=\frac1{\sqrt n}<\varepsilon[/tex] for all [tex]n>N[/tex].

Let [tex]N=\left\lceil\frac1{\varepsilon^2}\right\rceil[/tex]. Then

[tex]n>\left\lceil\dfrac1{\varepsilon^2}\right\rceil\ge\dfrac1{\varepsilon^2}[/tex]

[tex]\implies\dfrac1n<\varepsilon^2[/tex]

[tex]\implies\dfrac1{\sqrt n}<\varepsilon[/tex]

as required.

Answer:

h=1/2fg

Step-by-step explanation:

Solve for x, h=1/2fg

It is true for all x; h=1/2fg

h=1/2fg

Both sides are equal

It is true for all x; h=1/2fg

Answer:

The answer is option B

Step-by-step explanation:

To find the coefficient of the third term in

[tex](x + 5)^{7} [/tex]

Rewrite the expansion in the form

[tex](a + x)^{n} [/tex]

where n is the index

So we have

[tex] ({5 + x})^{7} [/tex]

After that we use the formula

[tex]nCr( {a}^{n - r} ) {x}^{r} [/tex]

where r is the term we are looking for

For the third term we are looking for the term containing x²

that's

r + 1 = 3

r = 2

So to find the coefficient of the third term

We have

[tex]7C2[/tex]

Hope this helps you

first of all, the notation is wrong it should be [tex] {}^nC_r \text{ and more usual notation is } {n \choose k} [/tex]

second, the

[tex](r+1)^{\text{th}} \text{ term } T_{r+1} \text{ in the expansion of } (x+a)^n \text{ is } {n \choose r}x^{(n-r)}a^r[/tex]

here [tex] a=5 \text{ and } n=7 \text{ and for } 3^{\text{rd}} \text{ term } T_3, \quad r+1=3 \implies r=2 [/tex]

so the coefficient of third term is, [tex]{7 \choose 2}={7\choose 5}[/tex]

an important property of binomial coefficient you should know:

[tex] {n \choose k}={n \choose {n-k}}[/tex]

and if you interchange [tex] x \text{ and } a[/tex]

only the "order" will get reversed. i.e. the series will start from back.

another thing, the [tex] k^{\text{th}} \text{ term from beginning, is the } (n-k+2)^{\text{th}} \text{ term from behind}[/tex]

Answer:

The probability that he answered neither of the problems correctly ​is 0.0625.

Step-by-step explanation:

We are given that a student ran out of time on a multiple-choice exam and randomly guess the answers for two problems each problem have four answer choices ABCD and only one correct answer.

Let X = Number of problems correctly ​answered by a student.

The above situation can be represented through binomial distribution;

[tex]P(X=r)=\binom{n}{r}\times p^{r}\times (1-p)^{n-r};x=0,1,2,3,....[/tex]    

where, n = number of trials (samples) taken = 2 problems

           r = number of success = neither of the problems are correct

           p = probability of success which in our question is probability that

                 a student answer correctly, i.e; p = [tex]\frac{1}{4}[/tex] = 0.75.

So, X ~ Binom(n = 2, p = 0.75)

Now, the probability that he answered neither of the problems correctly ​is given by = P(X = 0)

             P(X = 0) = [tex]\binom{2}{0}\times 0.75^{0}\times (1-0.75)^{2-0}[/tex]

                            = [tex]1 \times 1\times 0.25^{2}[/tex]

                            = 0.0625

Answer:

3.5

Step-by-step explanation:

Well you have to find first 2/3 of 5.25. This means multiplication, which is 3.5. so to find how much to add to this to get 7, we have to subtract 3.5 from 7. 7-3.5=3.5. so we must add 3.5 to get 7. Hope this helps :D

Answer:

Total amount in dollars= $64614.00

Step-by-step explanation:

Initial starting salary is $40000.

Rate of increase is 3%

Number of years is 16 years

The salary is compounded yearly.

Amount A after 16 years is given as

A= p (1+r/n)^ (nt)

A=40000(1+0.03/16)^(16*16)

A= 40000(1.001875)^(256)

A=40000(1.61534824)

A= 64613.92959

Total amount in dollars= $64614.00

Answer: the answer is $806275

Step-by-step explanation:

A p e x

Answer:

m∠AOD = 140°

Step-by-step explanation:

In the diagram attached,

OA⊥OC and OB⊥OD

m∠AOD = 3.5(m∠BOC)

Since, m∠BOD = 90°  [Given: OA⊥OC]

m∠BOC + m∠COD = 90° ---------(1)

Similarly, m∠AOC = 90° [Given : OA⊥OC]

m∠AOB + m∠BOC = 90° --------(2)

Equation (1) - Equation(2)

(m∠BOC + m∠COD) - (m∠AOB + m∠BOC) = 90°- 90°

m∠COD = m∠AOB

m∠AOB + m∠BOC + m∠COD = m∠AOD --------(3)

m∠AOB + m∠BOC + m∠AOB = 3.5(m∠BOC) [Since m∠COD = m∠AOB]

2m∠AOB = 3.5(m∠BOC) - m∠BOC

2m∠AOB = 2.5(m∠BOC)

m∠AOB = 1.25(m∠BOC)

From equation (2),

m∠AOB + m∠BOC = 90°

1.25(m∠BOC) + m∠BOC = 90°

2.25(m∠BOC) = 90°

m∠BOC = 40°

From equation (1),

m∠BOC + m∠COD = 90°

m∠COD + 40° = 90°

m∠COD = 50°

Now by putting these values in equation (3)

m∠AOB + m∠BOC + m∠COD = m∠AOD

m∠COD + m∠BOC + m∠COD = m∠AOD

50° + 40° + 50°= m∠AOD

m∠AOD = 140°

Answer:

[tex]\boxed{\frac{1}{2} }[/tex]

Step-by-step explanation:

Hey there!

Well given,

[tex]\frac{2}{5} * 1 \frac{3}{12}[/tex]

We need to make 1 3/12 improper,

1*12 = 12

12 + 3 = 15

[tex]\frac{2}{5} * \frac{15}{12}[/tex]

2*15 = 30

5*12 = 60

[tex]\frac{30}{60}[/tex]

Simplified

[tex]\frac{1}{2}[/tex]

Hope this helps :)

Answer:

255.8

Step-by-step explanation:

first

1/6*1535

=255.8

Answer:

i. Has a liver problems?

= 0.08

ii. Is a heavy drinker ?

= 0.066

iii. If a person is found to be a heavy drinker, what is the probability that this person has liver problem?

= 0.303

iv. If a person is found to have liver problems, what is the probability that this person is a heavy drinker?

= 0.25

v. If a person is found to be a non –drinker, what is the probability that this person has liver problems?

= 0.104

Step-by-step explanation:

We have 2 Events in this question

Event A: People with liver problems

Event B : People without liver problems

Event A: People with liver problems

Let us represent people with liver problems as = (L)

a)8% have liver problems. = P(L)

Under liver problems we have:

b) 25% are heavy drinkers = P( L & H)

c) 35% are social drinkers = P( L & S)

d) 40% are non-drinkers. = P( L & N)

Event B( no liver problem)

Let us represent no liver problem as NL

We are not given in the question but Probability of having no liver problem = 100 - Probability of having liver problem

= 100 - 8% = 92 %

P(NL ) = 92%

From the question, For people without liver problems, we have:

a) 5% are heavy drinkers = P(NL & H)

b) 65% are social drinkers = P( NL & S)

c) 30% do not drink at all = P( NL & N)

An adult is chosen at random, what is the probability that this person

i. Has a liver problems?

P(L) = 8% or 0.08

ii. Is a heavy drinker ?

From the question, we have:

Probability of people that have liver problems and are heavy drinkers P(L & H) = 25% = 0.25

Probability of people that have do not have liver problems and are heavy drinkers P(NL & H) = 5% = 0.05

Probability ( Heavy drinker) =

P(L) × P(L & H) + P(NL) × P(NL & H)

= 0.25 × 0.08 + 0.05 × 0.92

= 0.066

iii. If a person is found to be a heavy drinker, what is the probability that this person has liver problem?

Probability (Heavy drinker and has liver problem) = [P(L) × P(L & H)] ÷ [P(L) × P(L & H)] + [P(NL) × P(NL & H) ]

= [0.25 × 0.08] ÷ [0.25 × 0.08] + [0.05 × 0.92]

= 0.303030303

Approximately = 0.303

iv. If a person is found to have liver problems, what is the probability that this person is a heavy drinker?

P(L & H) = 25% = 0.25

v. If a person is found to be a non –drinker, what is the probability that this person has liver problems.?

People with liver problems are non-drinkers. = P( L & N) = 40% = 0.4

People without liver problems and do not drink at all = P( NL & N) = 30% = 0.3

Probability (non drinker and has liver problem) = [P( L & N) × P(L & H)] ÷ [P( L & N) × P(L & H)] + [ P( NL & N) × P(NL & H) ]

= [0.4× 0.08] ÷ [0.4 × 0.08] + [0.3 × 0.92]

= 0.1038961039

Approximately ≈ 0.104

Answer:

Step-by-step explanation:

Note that the perimeter of a rectangle P = 2(Length + Breadth)

The distance between the upper-left coordinates on a rectangle and the upper-right coordinates is the breadth of the rectangle. To get the breadth of the rectangle, we will use tgw formula for calculating the distance between two points as shown.

D = √(y2-y1)²+(x2-x1)²

Given the coordinates (-1,4) and (3,4), the distance between the coordinates where x1 = -1, y1 = 4, x2 = 3 and y2 = 4 will be expressed as.

B = √(4-4)²+(3-(-1))²

B = √0+4²

B = √16

B = 4

Hence the breadth of the rectangle is 4 units.

Substituting the breadth into the formula for calculating the perimeter will give;

P = 2(L+B)

24 = 2(L+4)

L+4 = 24/2

L+4 =12

L = 12-4

L = 8

Hence the length of the rectangle is 8 units.

The diagram of the rectangle on a coordinate is as given in the attachment below.

Answer:

0.4 cm

Step-by-step explanation:

The magnifying glass basically zooms into smaller objects. If the insect appears to be 2cm, then it is actually smaller than this. It cannot be 10 cm.

If the scale factor is 5, then this means that the insect is zoomed in 5 times through the magnifying glass. Use the following ratio:

[tex]\frac{2}{5}[/tex]

This fraction can also be seen as division, so:

[tex]2[/tex]÷[tex]5=0.4[/tex]

The insect is actually 0.4 cm long.

(or 4 millimeters)

:Done

Answer:

(1) B

(2) A

(3) C

Step-by-step explanation:

A random variable is a variable that denotes a set of all the possible outcomes of a random experiment. It is denotes by a single capital letter such as X or Y.

There are two types of random variables.

(1)

Exact weight of quarters now in circulation in the United States.

The variable weight is a continuous variable.

Thus, the exact weight of quarters now in circulation in the United States is a continuous random variable.

(2)

Shoe sizes of humans.

The shoe size of a person are discrete and finite values.

Thus, the shoe sizes of humans are discrete random variables.

(3)

Political party affiliations of adults in the United States.

This variable is not a quantitative variable.

It is a qualitative variable.

Thus, the political party affiliations of adults in the United States is no random variable.

Answer:

4110

Step-by-step explanation:

One ton is equal to 2000 pounds and one ounce is equal to 0.0625 pounds.

2 tons*2000 lbs per ton  = 4000 lbs

1760 ounces*0.0625 lebs per ounce = 110 lbs

4000+110=4110 lbs

Answer:

The events are independent.

The probability of showing heads on both toss is equal to 1/2

Step-by-step explanation:

The sample space for this experiment consists of 2⁴= 16 sample points, as each toss can result in two outcomes we assume that the events are equally likely.

Two events are independent in the sample space if the probability of one event occurs, is not affected by whether the other event has or has not occurred.

In general the k events are defined to be mutually independent if and only if the probability of the intersection of  any 2,3,--------, k  equals the product of their respective probabilities.

P (A∩B) = P(A). P(B)

P (A∩B)   = 1/2. 1/2= 1/4

                                                                  Head          Tail

 P(E1)= 1/2  ----------          Coin 1               H,H              T,H

                                                                1/4                  1/4

  P(E2)= 1/2  ---------------  Coin 2             H, H               H,T

                                                                      1/4           1/4

So the events are independent.

The probability of showing heads on both toss is equal to 1/2

The sample space for this experiment consists of 2⁴= 16 sample points, out of which eight will have heads on both toss.

Or in other words ( 1/4* 1/4) = 2/4 = 1/2

Answer: There will 486 bacteria in 31 hours.

Step-by-step explanation:

The population decay in bacteria is exponential.

Exponential function : [tex]y=Ab^x[/tex], where A = initial population, b multiplication decay factor, t= time:

As per given:

Initial population: [tex]A=120,000[/tex]

After 36 hours, population = [tex]120000(b^{36})=200[/tex]

Divide both sides by 120,000 we get

[tex]b^{36}= 0.00167[/tex]

Taking natural log on both sides , we get

[tex]36\ln b=\ln 0.00167\\\\\Rightarrow\ b=e^{\left(\frac{\ln0.00167}{36}\right)}=0.83724629\approx0.8372[/tex]

At x= 31,

[tex]y=120000(0.8372)^{31}=120000\times0.00405234\approx486[/tex]

Hence, there will 486 bacteria in 31 hours.

Answer:

About $1.75 per tennis balls

Step-by-step explanation:

A tin of 4 tennis balls costs $6.99. We are asked to find the price of one tennis ball.

We need to find the unit price, or price per ball.

Divide the cost by the number of tennis balls.

cost / tennis balls

cost = $6.99

tennis balls = 4 tennis balls

$6.99 / 4 tennis balls

Divide 6.99 by 4.

$1.7475 / 1 tennis ball

Round to the nearest cent or hundredth. The 7 in the thousandth place tells us to round the 4 to a 5 in the hundredth place.

$1.75 / 1 tennis ball

It would cost approximately $1.75 for one tennis ball.

Answer:

Option A. 4√5

Step-by-step explanation:

To obtain the value of x, we must first obtain the value of y as shown in the attached photo.

The value of y can be obtained by using the pythagoras theory as illustrated below:

In this case y is the longest side i.e the Hypothenus.

y² = 4² + [4√3]²

y² = 4² + [4² × (√3)²]

y² = 4² + [4² × 3]

y² = 16 + [16 × 3]

y² = 16 + 48

y² = 64

Take the square root of both side

y = √64

y = 8

Finally, we shall determine the value of x by using the pythagoras theory as illustrated below.

Note: x is the longest side i.e the Hypothenus in this case.

x² = 4² + 8²

x² = 16 + 64

x² = 80

Take the square root of both side

x = √80

x = √(16 × 5)

x = √16 × √5

x = 4√5

Therefore, the value of x is 4√5.