Show that the sum of any two odd numbers is even.

Answer: About 16 years

Step-by-step explanation:

The formula to find the compound amount if compounded continuously is given by :-

[tex]A=Pe^{rt}[/tex], where P is Principal amount, r is the rate of interest ( in decimal) and t is time ( in years).

Given : P= $1000   ;    r= 4.6%=0.046

let t be the time it will take to double the amount, the  we have

[tex]2(1000)=(1000)e^{0.046\times t}[/tex]

Dividing 1000 both sides, we get

[tex]2=e^{0.046 t}[/tex]

Taking natural log on each side, we get

[tex]\ln2=\ln(0.046\times t)\\\\\Rightarrow\ 0.6931=0.046t\\\\\Rightarrow\ t=\dfrac{0.6931}{0.046}=15.0673913043\approx16\text{ years}[/tex]

Hence, it will take about 16 years to double the amount.

Answer:

It is a chemical change ⇒ 1st answer

Step-by-step explanation:

* Lets explain the statements to solve the problem

- A chemical change occurs when a new substance is formed through

 a chemical reaction

- Ex: cooking an egg

- Change of reaction is the rate of reaction it can be decreases or

  increasing

- A phase change is a change from one state to another without a

 change in chemical composition

- Ex: Condensation: the substance changes from a gas to a liquid

- A physical change, such as a state change or dissolving, but does

 not create a new substance

- Ex: Breaking a glass

* Lets solve the problem

- A piece of toast came out of the toaster very overcooked.

∵ It is like the cooking an egg

∴ It is a chemical change

Answer:

Chemical

Step-by-step explanation:

Answer:

1) There were 33 $4,000 investors and 27 $8,000 investors.

2) The solution in x = 4, y = 9.

3) There were 24 nickels and 56 dimes.

Step-by-step explanation:

1) A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?

I am going to say that:

x is the number of investors that contributed 4,000.

y is the number of investors that contributed 8,000.

Building the system:

There are 60 investors. So:

[tex]x + y = 60[/tex]

In all, the partnership raised $348,000. So:

[tex]4000x + 8000y = 348000[/tex]

I am going to simplify by 4000. So:

[tex]x + 2y = 87[/tex]

Solving the system:

The elimination method is a method in which we can transform the system such that one variable can be canceled by addition. So:

[tex]1)x + y = 60[/tex]

[tex]2)x + 2y = 87[/tex]

I am going to multiply 1) by -1. So we have

[tex]1)-x - y = -60[/tex]

[tex]2)x + 2y = 87[/tex]

By addition, the x are going to cancel each other

[tex]-x + x - y + 2y = -60 + 87[/tex]

[tex]y = 27[/tex]

For x:

[tex]x + y = 60[/tex]

[tex]x = 60-y = 60-27 = 33[/tex]

There were 33 $4,000 investors and 27 $8,000 investors.

2) Solve the system by row-reducing the corresponding augmented matrix.

[tex]2x + y = 17[/tex]

[tex]x + y = 13[/tex]

This system has the following augmented matrix:

[tex]\left[\begin{array}{ccc}2&1&17\\1&1&13\end{array}\right][/tex]

To help the row reducing, i am going to swap the first with the second line:

[tex]L1 <-> L2[/tex]

So we have:

[tex]\left[\begin{array}{ccc}1&1&13\\2&1&17\end{array}\right][/tex]

Now, reducing the first column.

[tex]L2 = L2 - 2L1[/tex]

So we have:

[tex]\left[\begin{array}{ccc}1&1&13\\0&-1&-9\end{array}\right][/tex]

Now we do:

[tex]L2 = -L2[/tex]

And the matrix is:

[tex]\left[\begin{array}{ccc}1&1&13\\0&1&9\end{array}\right][/tex]

Now to reduce the second column, we do:

[tex]L1 = L1 - L2[/tex]

[tex]\left[\begin{array}{ccc}1&0&4\\0&1&9\end{array}\right][/tex].

So the solution is:

x = 4, y = 9.

3) A jar contains 80 nickels and dimes worth $6.80. How many of each kind of coin are in the jar?

I am going to say that x is the number of nickels and y is the number of dimes.

Each nickel is worth 5 cents and each dime is worth 10 cents.

Building the system:

There are 80 coins in all:

[tex]x + y = 80[/tex]

They are worth $6.80. So:

[tex]0.05x + 0.10y = 6.80[/tex]

Solving the system:

[tex]1)x + y = 80[/tex]

[tex]2)0.05x + 0.10y = 6.80[/tex]

I am going to divide 1) by -10, so we can cancel y. So:

[tex]1)-0.10x - 0.10y = -8[/tex]

[tex]2)0.05x + 0.10y = 6.80[/tex]

Adding:

[tex]-0.10x + 0.05x - 0.10y + 0.10y = -8 + 6.80[/tex]

[tex]-0.05x = -1.2[/tex] *(-100)

[tex]5x = 120[/tex]

[tex]x = \frac{120}{5}[/tex]

[tex]x = 24[/tex]

Also

[tex]x + y = 80[/tex]

[tex]y = 80-x = 80-24 = 56[/tex]

There were 24 nickels and 56 dimes.

Answer:

a) 30.726m/s and b) 5.5549s

Step-by-step explanation:

a.) What was Chris Huber’s speed in meters per second(m/s)?

Given the distance and time, the formula to obtain the speed is

[tex]v=\frac{d}{t}[/tex].

Applying this to our problem we have that

[tex]v=\frac{200m}{6.509s}= 30.726m/s[/tex].

So, Chris Huber’s speed in meters per second(m/s) was 30.726m/s.

b) What was Whittingham’s time through the 200 m?

In a) we stated that [tex]v=\frac{d}{t}[/tex]. This formula implies that

First, observer that [tex]19\frac{km}{h} =19,000\frac{m}{h}=\frac{19,000}{3,600}m/s= 5.2777m/s[/tex].

Then, Sam Whittingham speed was equal to Chris Huber’s speed plus 5.2777 m/s. So, [tex]v=30.726\frac{m}{s} +5.2777\frac{m}{s}= 36.003 m/s.[/tex]

Then, applying 1) we have that

[tex]t=\frac{200m}{36.003m/s}=5.5549s.[/tex]

So, Sam Whittingham’s time through the 200 m was 5.5549s.

Answer:

The number of tablets that can be prepared is 3076.

Step-by-step explanation:

The total amount of active ingredients in the tablet is the sum of the amounts provided in the formula:

[tex]325 mg + 2mg+15 mg=342 mg[/tex]

The percentages of each component in the formula are:

Acetaminophen:[tex]\frac{325mg*100}{342mg}=95.03[/tex]%

Chlorpheniramine maleate:[tex]\frac{2mg*100}{342mg} =0.58[/tex]%

Dextromethorphan hydrobromide:[tex]\frac{15mg*100}{342mg}=4.39[/tex]%

If 1 Kg=[tex]10^{6}[/tex] mg of acetaminophen is used, the needed amount of chlorpheniramine maleate would be:

[tex]\frac{10^{6} mg *0.58}{95.03}=6153.85 mg[/tex]

Since there are 125 g = 125000 mg of chlorpheniramine maleate, there is enough of these ingredient to run the available acetaminophen out. Thus, the total amount of active ingredients that can be prepared with 1 kg of acetaminophen is:

[tex]\frac{10^{6}mg*100}{95.03}=1052307.7mg[/tex]

Since each tablet weighs 342 mg, the number of tablets that can be prepared is:

[tex]\frac{1052307.7mg}{342mg}=3076.923[/tex]

Which means that 3076 tablets can be prepared and a there will be a remanent of 0.923*342 mg = 315.69 mg of active ingredients.

Ingredients are used in ratio to prepare a specific product, generally. The number of tablets that can be manufactured for given context is 3076

You can represent the unknown amounts by the use of variables. Follow whatever the description is and convert it one by one mathematically. For example if it is asked to increase some item by 4 , then you can add 4 in that item to increase it by 4. If something is for example, doubled, then you can multiply that thing by 2 and so on methods can be used to convert description to mathematical expressions.

We are given that the considered cold tablet consists:

acetaminophen to chlorpheniramine maleate to dextromethorphan hydrobromide  as 325 : 2 : 15 in weight.

Since we're given that there is unlimited quantities of dextromethorphan hydrobromid, the first two ingredients will be the one capping(limiting) the number of tablets that can be manufactured.

Suppose that 'n' tablets can be manufactured by the given amount of ingredients, then:

Amount of acetaminophen  needed = 325 mg × n ≤ 1 kg (which is available amount)

Amount of chlorpheniramine maleate = 2 mg × n ≤ 125 g

Converting all scales of weight to grams, we get two inequalities:

(since 1 g = 0.001 mg , and 1 kg = 1000 g)

Remember that 'n' is amount of tablets which is going to be a whole number.

Solving the inequalities, we get:

[tex]0.325 n \leq 1000\\\\n \leq \dfrac{1000}{0.325} = 3076.9\\\\n \leq 3076[/tex]

and

[tex]0.002n \leq 125\\\\n \leq \dfrac{125}{0.002} = 62500[/tex]

So, we see that by the given amount of acetaminophen , we can only make 3076 tablets, but we can make 62500 tablets by the second ingredient chlorpheniramine maleate,

since both ingredients are necessary, so after 3076 tablets, first ingredient will exhaust.

Thus, The number of tablets that can be manufactured for given context is 3076

Learn more about inequalities here:

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Answer:

Given differential equation,

[tex]\frac{dy}{dx}+5y=7[/tex]

[tex]\frac{dy}{dx}=7-5y[/tex]

[tex]\implies \frac{dy}{7-5y}=dx[/tex]

Taking integration both sides,

[tex]\int \frac{dy}{7-5y}=\int dx[/tex]

Put 7 - 5y = u ⇒ -5 dy = du ⇒ dy = -du/5,

[tex]-\frac{1}{5} \int \frac{du}{u} = \log x + C[/tex]

[tex]-\frac{1}{5} \log u = \log x + C[/tex]

[tex]-\frac{1}{5}\log(7-5y) = \log x + C---(1)[/tex]

Here, x = 0, y = 0

[tex]\implies -\frac{1}{5} \log 7= C[/tex]

Hence, from equation (1),

[tex]-\frac{1}{5}\log(7-5y)=\log x -\frac{1}{5}log 7[/tex]

[tex]\log(7-5y)=\log (\frac{x}{7^\frac{1}{5}})[/tex]

[tex]7-5y=\frac{x}{7^\frac{1}{5}}[/tex]

[tex]7-\frac{x}{7^\frac{1}{5}}=5y[/tex]

[tex]\implies y=\frac{1}{5}(7-\frac{x}{7^\frac{1}{5}})[/tex]

Answer: 0.516

Step-by-step explanation:

Binomial probability distribution formula to find the probability of getting success in x trial:-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where n is the number of trials , p is the probability of getting success in each trial.

Given :  People with type O-negative blood are  universal donors.

The proportion of  the American population  have O-negative blood =0.07

For n=10, the probability that at least 1 of them is a  O-negative blood :-

[tex]P(x\geq1)=1-P(x=0)\\\\=1-[^{10}C_0(0.07)^0(1-0.07)^{10}]\\\\=1-[(1)(1)(0.93)^{10}]\ \ \ [\text{ Since}^nC_0=1]\\\\=1-0.483982307179\approx1-0.4840=0.516[/tex]

Hence, the probability that at least 1 of them is a  universal donor = 0.516

Answer:

We are given that the rate of change is proportional to its size S

So, [tex]\frac{dS}{dt} \propto S[/tex]

[tex]\frac{dS}{dt} = kS[/tex]

[tex]\frac{dS}{S} = kdt[/tex]

Integrating both sides

[tex]\log(S)= kt + log c[/tex]

[tex]\frac{S}{S_0}=e^{kt}[/tex]

[tex]S=S_0 e^{kt}[/tex]

S is the population after t hours

[tex]S_0[/tex] is the initial population

Now we are given that After 6 hours there will be 1200 bacteria

[tex]1200=200 e^{6k}[/tex]

[tex]6=e^{6k}[/tex]

[tex]6^{\frac{1}{6}=e^{k}[/tex]

So, [tex]S=200 \times 6^{\frac{t}{6}[/tex]

a)Now the population after t hours as a function of t; [tex]S=200 \times 6^{\frac{t}{6}[/tex]

b)  What will be the population after 7 hours?

Substitute t = 7 hours

A bacteria culture starts with 200 bacteria

[tex]S=200 \times 6^{\frac{7}{6}}[/tex]

[tex]S=1617.607[/tex]

c) How long will it take for the population to reach 1750 ?

[tex]1750=200 \times 6^{\frac{t}{6}[/tex]

[tex]\frac{1750}{200} =6^{\frac{t}{6}[/tex]

[tex]8.75 =6^{\frac{t}{6}[/tex]

[tex]t=7.26[/tex]

So,  it will take 7.2 hours for the population to reach 1750

Final answer:

To determine the population growth function in terms of time and find the population after a specific duration, use the exponential growth formula N(t) = N0 x 2^t. Calculate the growth rate using given data points like the initial and final population. Finally, substitute the desired time into the population function to find the population at that specific time.

Explanation:

The population of bacteria after t hours can be represented by the formula N(t) = N0 x 2t.

(a) To express the population after t hours as a function of t, you can use the given data points to find the growth rate. For the provided data, the growth rate is calculated as r = log2(1200/200) / 6 = 0.1333 per hour.

(b) To find the population after 7 hours, substitute t=7 into the function: N(7) = 200 x 27x0.1333 = 3481.49.

c) To find out how long it will take for the population to reach 1750, we get

p(t)=1750 and solving it

t ≈7.846

Answer: Hi!

First, UxV = sin(a)*IUI*IVI

where a is the angle between U and V, in this case 45°.

First, the cross product of UxV points:

Here you can use the right hand method,

Put your hand flat, so the point of your fingers point in the same direction that the first vector, in this case U, so your fingers will point to the north.

Now roll your fingers in the direction of the second vector, so here you will roll your fingers in the northeast direction. Now you will see that your thumb is pointing down, so the cross product of UxV points down.

And the magnitude is 6*5*sin(45) = 21.213

The cross-product u×v points: Option D: Down, and its magnitude  |u×v| evaluates to  [tex]|u \times v| = 15\sqrt{2}[/tex]

Suppose that two vectors in consideration are u and v, then their cross product is evaluated as:

[tex]u \times v = |u|.|v|.sin(\theta)\hat{n}[/tex]

where [tex]\hat{n}[/tex] is the normal unit vector whose direction is decided by right hand thumb rule, and theta is the angle between u and v vector.

The two bars around a vector represents the magnitude of that vector.

Cross product returns the result as a vector itself.

For this case, we have:

Thus, as north and northeast have 45 degrees in between them, therefore, we get:

[tex]u \times v = 6\times 5 \times sin(45^\circ) \hat{n} = 15\sqrt{2} \: \hat{n}[/tex]

Directing index finger of right hand to north direction, and middle to northeast makes the thumb go down, therefore, the direction of normal vector (and therefore direction of the resultant cross product vector too) is downside of this whole north south east west plane.

The magnitude of cross product is [tex]|u \times v| = 15\sqrt{2}[/tex]

Thus, the cross-product u×v points: Option D: Down, and its magnitude  |u×v| evaluates to  [tex]|u \times v| = 15\sqrt{2}[/tex]

Learn more about cross product here:

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Answer:

 The standard divisor is 22.48.

Step-by-step explanation:

There are a total of 3259 students at the 5 schools. Then dividing that number by the number of teachers (145) we get the "standard divisor" of ...

  3259/145 ≈ 22.48

__

By Hamilton's method, that divisor is used to divide the number of students at each school, and the result is rounded down. This is the initial allocation of teachers to schools. The remainders from the division are examined. Starting with the largest and working down, one additional teacher is assigned until all the unassigned teachers have been assigned.

For this problem, the initial assignment results in 142 teachers being assigned, so there are 3 more that can be allocated. In order, the highest three remainders are associated with the number of students at East, Central, and South. Each of those schools gets one more teacher than the number initially assigned. The final allocation of teachers is highlighted in the attachment.

Answer:

a)Reflexive, not symmetric, transitive

b)Reflexive, not symmetric, transitive

c)Not reflexive, symmetric, not transitive

d)Reflexive, not symmetric, transitive

Step-by-step explanation:

a)

[tex]R=\left \{ (a,b)\epsilon  \mathbb{R} \times \mathbb{R} \mid a \leq b\right \}[/tex]

The relation R is reflexive for

[tex]a\leq a[/tex] for every real number a

it is not symmetric because 0 is less than 1, but 1 is not less than 0

it is transitive

[tex]a\leq[/tex] and [tex] b\leq c\Rightarrow a\leq c[/tex]

So if aRb and bRc, then aRc

b)  

[tex]R=\left \{ (m,n)\epsilon  \mathbb{Z} \times \mathbb{Z} \mid \exists k\in \mathbb{Z} \ni m=kn \right \}[/tex]

R is reflexive because m=1.m for every integer m

R is not symmetric: 2 is a factor of 4, but 4 is not a factor of 2

R is transitive:  if mRn and nRp if m=kn and n=qp, so m=(kq)p and kq is an integer , so mRp

c)

[tex]R=\left \{ (m,n)\epsilon  \mathbb{Z} \times \mathbb{Z} \mid m\neq n\right \}[/tex]

R is obviously not reflexive because all numbers equals themselves

R is symmetric: if a different to b, then b different to a

R is not transitive: 1R2 and 2R1 (because 1 different to 2), but 1 = 1

d)

[tex]R=\left \{ A,B\mid A\subseteq B \right \}[/tex]

R is reflexive for every set A is a subset of itself

R is not symmetric {1,2} is a subset of {1,2,3} but {1,2,3} is not a subset of {1,2}

R is transitive: if A is subset of B and B is subset of C, then A is subset of C

There are no natural numbers that satisfy the equation x + 2 = 1. Therefore, using the roster method, the set is empty.

The question requires us to use the roster method to write the set of natural numbers x that satisfy the equation x + 2 = 1. Natural numbers, by definition, are counting numbers starting from 1. They are non-negative and do not include zero. So, if we try to find a natural number x that satisfies the equation x + 2 = 1, we see that x would need to be -1 (since -1 + 2 equals 1). However, -1 is not a natural number. Therefore, there are no natural numbers that satisfy the equation, so the set is empty.

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The set of natural numbers x that satisfy x + 2 = 1. The correct answer is EMPTY.

To solve the equation  x + 2 = 1  for natural numbers x  we would first try to isolate x subtracting 2 from both sides of the equation:

x + 2 - 2 = 1 - 2

x = -1

 However, natural numbers are defined as the set of positive integers, starting from 1 and increasing indefinitely.

Since the solution to the equation x = -1 is not a positive integer, it does not belong to the set of natural numbers. Therefore, there is no natural number x that satisfies the equation x + 2 = 1  

Since there are no elements that satisfy the condition, the set is empty. Hence, the correct representation of the set using the roster method is EMPTY.

Answer:

Step-by-step explanation:

We know that the total monthly payment is 3200, so if we call Pa (family A`s payment) and Pb (Family B's payment) the payments:

So if we replace Pa in the first ecuation:

then Pa+Pb=3200 => Pa= 3200-1350= $1.850

Good Luck!

Answer:

It is more convenient to fill up in the United States.

Step-by-step explanation:

We convert a US gallon to liters:

1 Gallon equals 3.78541 liters.

Therefore, 1 US Gallon costs (3.78541) x (1.29) = 4.8831789 Canadian dollars.

Now we convert the price of a US gallon in Canadian territory to US dollars:

4.8831789 * 0.79 = 3.85771133 US dollars.

Conclusion: A gallon purchased in the United States costs 2.58 US dollars, while a gallon in Canada is equivalent to 3.85771133 US dollars. This way it is more convenient to fill up in the United States.

It would be more economical to fill up the gas tank in the United States before crossing into Canada.

To determine whether it is more cost-effective to fill up the gas tank in the United States or Canada, we need to convert the Canadian gas price into U.S. dollars per gallon. The price of gas in Canada is $1.29 per liter. Since 1 gallon is equal to 3.78541 liters, the price per gallon in Canada would be $1.29 × 3.78541 = $4.88238 CAD. Now, we need to convert this price into USD, knowing the Canadian dollar is worth 79¢ in U.S. currency. Therefore, $4.88238 CAD × 0.79 = $3.85728 USD per gallon.

The price of gas in the United States is $2.58 per gallon. When comparing the two prices, it is clear that $2.58 per gallon in the United States is cheaper than the converted price of gas in Canada ($3.85728 per gallon in USD). Hence, it would be more economical to fill up the gas tank in the United States before crossing into Canada.

Answer:

A set S is closed under an operation * (we're denoting the operation as asterisk) IF for any two elements a,b in S, the result a*b is also in S.

Step-by-step explanation:

A set being closed under an operation means that whenever you operate elements from the set, the result you get out of it is ALWAYS inside the set. For example, think of the set Z of integer numbers and the operation + (usual addition). If we add ANY two integers, we're going to get another integer. Or said in terms of sets, for any two numbers a,b in Z, a+b is also in Z.

On the other side, not being closed under an operation means you do NOT ALWAYS get results inside the same set. Think of the set of natural numbers N, and the operation - (usual difference). If we do the operation 5-12, we get -7 which is NOT in the set of natural numbers. So N is not closed under subtraction.

Answer:

y = x'

z = y'

z' = x + t

Step-by-step explanation:

Hi!

You need to define two new variables y and z:

y = x'

z = y'

Then:

z = y' = x''

z' = x''' = x + t

Now you have a system of 3 equations with only first derivatives

Answer:

The answers are the same you stated.

The calculations are in the step-by-step explanation

Step-by-step explanation:

There are 3 prizes to be distributed among 50 tickets. The order these prizes are distributed does not matter. So the total number of prizes is a combination of 3 from 50.

The formula for a combination of n from m is:

[tex]C(m,n) = \frac{m!}{n!(m-n)!}[/tex]

So, the total number of prizes is:

[tex]T = C(50,3) = \frac{50!}{3!(50-3)!} = \frac{50*49*48*47!}{3!*47!} = 19600[/tex]

what is the probability that the four organizers

a) win all of the prizes?

The number of ways that the four organizers can will all of the prizes is a combination of 3 from 4.

[tex]C(4,3) = \frac{4!}{3!1!} = 4[/tex]

The probability that the win all of the prizes is the number of ways that they can win all the prizes divided by the total numbers of ways that the prizes can be distributed.

[tex]P = \frac{4}{19600}[/tex]

b) win exactly two of the prizes?

The total outcomes(total number of ways that the prizes can be distributed) is 19600.

For them to win exactly two of the prizes, we have a combination of 2 from 4(two organizers win prizes) multiplied by a combination of one from 46(one non-organizers wins a prize), so:

[tex]C(4,2)*C(46,1) = \frac{4!}{2!2!}*\frac{46!}{1! 45!} = 6*46 = 276[/tex]

The probability that they win exactly two of the prizes is

[tex]P = \frac{276}{19600}[/tex]

c) win exactly one of the prizes?

The total outcomes(total number of ways that the prizes can be distributed) is 19600.

For them to win exactly one of the prizes, we have a combination of 1 from 4(one organizer wins a prize) multiplied by a combination of two from 46(two non-organizers win prizes), so:

[tex]C(4,1)*C(46,2) = \frac{4!}{1!3!}*\frac{46!}{2! 44!} = 4*1035 = 4150[/tex]

The probability that they win exactly one prize is

[tex]P = \frac{4150}{19600}[/tex]

d) win none of the prizes?

The total outcomes(total number of ways that the prizes can be distributed) is 19600.

For them to win none of the prizes, we have a combination of 3 from 46(3 prizes distributed among 46 non-organizers). So:

[tex]C(46,3) = \frac{46!}{43!3!} = 15180[/tex]

The probability that they don't win any prize is:

[tex]P = \frac{15180}{19600}[/tex]

The probabilities of winning all of the prizes, winning exactly two of the prizes, winning exactly one of the prizes, and winning none of the prizes are [tex]\rm \dfrac{4}{19600},\dfrac{276}{19600},\dfrac{4140}{19600}, \ and \ \dfrac{15180}{19600}[/tex] respectively.

Probability means possibility. It deals with the occurrence of a random event. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.

Given

A local fraternity is conducting a raffle where 50 tickets are to be sold one per customer.

There are three prizes to be awarded.

If the four organizers of the raffle each buy one ticket.

The total outcomes will be

[tex]\rm ^{50}C_3 = \dfrac{50!}{47! *3!}\\\\\rm ^{50}C_3 = 19600[/tex]

a)  win all of the prizes.

The number of ways the four organizers can win all the prizes is given by

[tex]\rm ^4C_3 = \dfrac{4!}{3! *1!}\\\\\rm ^4C_3 = 4[/tex]

Then the probability of winning all the prizes will be

[tex]\rm Probability = \dfrac{4}{19600}[/tex]

b)  win exactly two of the prizes.

The number of ways is 2 from organizers and 1 from non-organizers given by

[tex]\rm ^4C_2 * ^{46}C_1 = \dfrac{4!}{2! *2!}*\dfrac{46!}{45! *1!}\\\\\rm ^4C_2 * ^{46}C_1 = 276[/tex]

Then the probability of 2 from organizers and 1 from non-organizers will be

[tex]\rm Probability = \dfrac{276}{19600}[/tex]

c)  win exactly one of the prizes.

The number of ways is 1 from organizers and 2 from non-organizers given by

[tex]\rm ^4C_1 * ^{46}C_2 = \dfrac{4!}{1! *3!}*\dfrac{46!}{44! *2!}\\\\\rm ^4C_1 * ^{46}C_2 = 4140[/tex]

Then the probability of 1 from organizers and 2 from non-organizers will be

[tex]\rm Probability = \dfrac{4140}{19600}[/tex]

d)  win none of the prizes.

The number of ways is 3 from non-organizers given by

[tex]\rm ^{46}C_3 = \dfrac{46!}{43! *3!}\\\\\rm ^{46}C_3 = 15180[/tex]

Then the probability of 3 from non-organizers will be

[tex]\rm Probability = \dfrac{15180}{19600}[/tex]

Thus, The probabilities of winning all of the prizes, winning exactly two of the prizes, winning exactly one of the prizes, and winning none of the prizes are [tex]\rm \dfrac{4}{19600},\dfrac{276}{19600},\dfrac{4140}{19600}, \ and \ \dfrac{15180}{19600}[/tex] respectively.

More about the probability link is given below.

https://brainly.com/question/795909

Answer:

$293,562.707

Step-by-step explanation:

As for the provided details we know,

Janet needs $51,500 from end of 7th year for upcoming 20 years.

The present value of 20 installments of $51,500 shall be @ 6% from year 7 to year 8.0858

Thus total value = $51,500 [tex]\times[/tex] 8.0858 = $416,418.7

Now the compound interest factor for 6 year @ 6 % = 1.4185

Thus, value to be invested today = $416,418.70/1.4185 = $293,562.707

As this when compounded annually will provide the balance as required at the end of 6 years.

Answer:

0.9503

Step-by-step explanation:

First of all, there are some wrong figures in the original text. Because there is a total of 362362 people, the figures should be 86086 (people in the 18-21 age bracket), 43043 (people in the 18-21 age bracket  who respond) and 43043 people in the 18-21 age bracket who refuse to respond. In the same way, because there are 276276 people in the 22-29 age bracket, it should be 18018 and not 1818 who refuse to respond in this subset of people. Now, let's define the following events:

R: a person respond

A: a person belongs to the 18-21 age bracket. So,

The number of people who respond is 43043 + 258258 = 301301, so

P(R) = 301301/362362 = 0.8315

P(A) = 86086/362362 = 0.2376

P(R | A) = 43043/86086 = 0.5

We are looking for P(A∪R) = P(A) + P(R) - P(A∩R),

P(A∩R) = P(R | A)P(A) = (0.5)(0.2376) = 0.1188, so,

P(A∪B) = 0.2376 + 0.8315 - 0.1188 = 0.9503

Answer:

[tex]S_{n} = \sum_{k=1}^{n} (2k-1) = n^2[/tex]

Step-by-step explanation:

Let's take a look at the first few odd numbers and their sum.

Lets define [tex]O_{k}[/tex] as the [tex]kth[/tex] odd number as:

[tex]O_{k} = 2k-1[/tex]

So we have:

[tex]O_{1} = 1\\O_{2} = 3\\O_{3} = 5\\O_{4} = 7\\[/tex]

And lets define the sum of all the odd numbers from [tex]k=1[/tex] to [tex]k=n[/tex] as:

[tex]S_{n} = \sum_{k=1}^n O_{k} = \sum_{k=1}^n (2k-1)[/tex]

Lets now check some values of said sum:

[tex]S_{1} = 1\\S_{2} = 1 + 3 = 4\\S_{3} = 1 + 3 + 5 = 9\\S_{4} = 1+3+5+7 = 16[/tex]

We can then observe than the sum up to [tex]n[/tex] equals [tex]n^2[/tex]

Let us then prove that this is the case by Induction.

First of all, we can prove this by an Induction Proof because we are taking all positive Integers. This is, we are working with the set of natural numbers [tex]\mathbb{N}[/tex].

We want to prove that

[tex]P(n) = S_{n} = \sum_{k=1}^n = n^2 \forall n\in \mathbb{N}[/tex]

This is, we want to prove that the sum of all odd numbers from [tex]1[/tex] to [tex]n[/tex] equals [tex]n^2[/tex] for all natural numbers.

Now, in order to prove something by Induction we need to check 2 things:

[tex]1) The\ base\ case . \ The\ statement\ holds\ for\ n=1\\2) The\ inductive\ step.\ Prove\ that\ if\ the\ statement\ holds\ for\ n\ then\ it\ must\ hold\ for\ n+1\\[/tex]

[tex]P(1)[/tex] is immediate:

[tex]P(1) = \sum_{k=1}^1 2k-1 = 1 = 1^2[/tex]

Now let's assume the statement holds for [tex]P(n)[/tex] and let's take a look at [tex]P(n+1)[/tex]

[tex]P(n+1) = \sum_{k=1}^{n+1} 2k-1[/tex]

And we can rewrite it by taking the last term out as:

[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) - 1[/tex]

And by inductive hypothesis we know that [tex]\sum_{k=1}^n 2k-1 = n^2[/tex]

and then:

[tex]P(n+1) = \sum_{k=1}^n 2k-1 \ + 2.(n+1) -1  = n^2 + 2n +2 -1 = n^2 +2n +1 = (n+1)^2[/tex]

And we have the proof we were looking for!

Step-by-step explanation:

The frequency of sine function is given by the number of periods in a given range. For example:

This means that, if we want another sine function with frequency 20% higher, we need that function to have a frequency of 1.2 in the interval [tex][0,2\pi][/tex].

To be easier to see we will consider interval [tex][0,10\pi][/tex] instead of [tex][0,2\pi][/tex]. In this interval [tex]sin(x)[/tex] has 5 periods, therefore our new sine function should have 6 periods.

Finally, as we can see in the graph, the function [tex]sin(\frac{6}{5}x )[/tex] (in blue) has a frequency 20% higher than [tex]sin(x)[/tex] (in red). This can be easily seen counting the number of periods between 0 and [tex]10\pi[/tex] for both functions. 5 for [tex]sin(x)[/tex] and 6 for [tex]sin(\frac{6}{5} x)[/tex].

Answer:

(D) Conclusion: Do not support the claim that the mean is greater than 9.4 minutes

Step-by-step explanation:

Let X be the mean duration of long distance telephone class originating in one town.

[tex]H_0: x bar = 9.4\\H_a: x bar >9.4[/tex]

(one tailed test)

The conclusion of the hypothesis test assuming that the results of the sampling don’t lead to rejection of the null hypothesis.

This means that there is no statistical evidence to support the alternate claim.

Hence option D is right.

(D) Conclusion: Do not support the claim that the mean is greater than 9.4 minutes

Answer:

1/2.

Step-by-step explanation:

There are 2 choices and he has to choose 1 , so the answer is 1/2.

Assuming Ciaran has no preference for white or brown bread, and each choice is equally likely, the probability of choosing brown bread is 1/2 or 50%.

The probability that Ciaran chooses brown bread depends on the assumption that he has no preference and that the choices are equally likely. If the only options available to Ciaran are white bread or brown bread, and each choice is equally likely, then the probability of choosing one over the other is 1 out of the total number of options.

In this case, there are 2 options (white or brown), so the probability that Ciaran will choose brown bread is 1/2 or 0.5, which can also be expressed as a 50% chance.

Answer:

We are given that  a manufacturer sells a product as $2 per unit.

Quantity = q units

So, Total revenue = [tex]\text{Cost per unit} \times quantity[/tex]

Total revenue = [tex]2q[/tex]

So, the total revenue function is  [tex]2q[/tex]

Marginal revenue is the derivative of the revenue functions

So, Marginal revenue = [tex]\frac{dR}{dq} =2[/tex]

The marginal revenue function is 2

The constant marginal revenue function mean that the revenue earned by the addition of the output is constant.

Answer:

There are two more sides on a hexagon than on a quadrilateral

Step-by-step explanation:

If the hexagon has 6 sides, and the quadrilateral has 4, then 6-4=2

Answer:

  corresponding angles

Step-by-step explanation:

Corresponding angles are congruent where a transversal crosses parallel lines. Such a geometry has 4 pairs of corresponding angles. The corresponding angles of each pair are congruent.

Answer:

corresponding angles

Step-by-step explanation:

Answer:

|-45|

Step-by-step explanation:

In mathematics, the absolute value of a real number is the numeric value of the  number, regardless the sign, either this is positive or negative.

The absolute value function can be definied as:

Using this definition, we have:

|-21| = -(-21) = 21

|14| = 14

|30| = 30

|-45| = -(-45) = 45

Therefore, the expression |-45| has the greatest value.

Answer:

[tex]h(t)=-\dfrac{11}{3}t[/tex]

Step-by-step explanation:

A snorkeler dives for a shell on a reef. After entering the water, the diver decends [tex]\frac{11}{3}[/tex] ft in one second.

Let t be the time passed after entering the water, in seconds, and h(t) be the position of the snorkeler under the water, in feet.  

The initial position of the snorkeler was 0 feet under the water.

An equation that models the divers position with respect to time is

[tex]h(t)=0-\dfrac{11}{3}t\\ \\h(t)=-\dfrac{11}{3}t[/tex]

Here the position is negative, because the diver decends (he deepens under the water)