Determine whether the data shows a linear relationship. If so, write an equation of a line of it.

Answer:

e) none

Step-by-step explanation:

The first thing that we need to realise is that this question is set to a practical example of a ball being tossed into the air for a period of time. Thus, we know that:

a) time cannot be negative

b) a ball cannot travel a negative distance in height from where it was tossed it into the air (given that this starting position is also the ground)

In the context of what the problem asks us to achieve, which is to find the domain, we now know that:

a) t ≥ 0

b) the function h(t) is only practical when h(t) ≥ 0

In other words, we know that the domain starts at t = 0, however we need to find when the ball drops back onto the ground. We can do this by solving h(t) = 0:

h(t) = -16t^(2) + 96t

0 = -16t^(2) + 96t

0 = -16t(t - 6) (Factorise -16t^(2) + 96t)

So, now we get:

-16t = 0, therefor t = 0

or

t - 6 = 0, therefor t = 6

Thus, the ball is on the ground at t = 0 seconds and t = 6 seconds; it is between these two values of t that the domain exists and that the problem is practical.

Now, looking at the multiple choice options, it seems as though none of them are correct, therefor the answer would be e) none.

The other way to work through this question (particularly if it is just multiple choice) is that, after realising that the ball is tossed into the air at t = 0 seconds and then drops at some point later, we could already discount a), b) and c) as answers since they include infinity in the domain, which is not practical for this problem.

Then, we could substitute t = 5 into the equation to get:

h(t) = -16(5)^2 + 96*5

h(t) = -16*5*5 + 96*5

h(t) = -80*5 + 96*5

Since we need h(t) to be 0 at the end value of the domain, this is not the correct answer. Thus, the answer is e) none.

Answer:

16.

Step-by-step explanation:

The ratio is 4:1 so 4 / (4 + 1) = 4/5 of the total is green peppers.

So it is 20 * 4/5 = 16 .

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!

Final answer:

To find (g°f)(0), evaluate f(0) to get 6, then apply g to this result to also get 6. Hence, (g°f)(0) equals 6.

Explanation:

The question is asking to evaluate the composition of two functions, (g°f)(x), at the value x = 0. Composition of functions involves applying one function to the result of another. In this case, since f(x) = 2x + 6 and g(x) = x, we first find f(0), which is the result of plugging 0 into f(x).

Let's calculate:
f(0) = 2(0) + 6 = 6.
Next, we apply g to this result:
g(f(0)) = g(6) = 6.
Therefore, (g°f)(0) = 6.

Answer:

The amount of principal Jamie borrowed was $3600.

Step-by-step explanation:

Let the principal of loan is = Po

[tex]P=Po\times0.12\times0.5[/tex]

Here r = 12% or 0.12

t = 6 months or 0.5 years

Now as given : Po+Interest=3816

[tex]Po+0.06Po=3816[/tex]

[tex]=>1.06Po=3816[/tex]

So, Po=3600

Hence, the amount of principal Jamie borrowed was $3600.

Answer and Explanation:

Given : A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample.

The target accuracy is a variance in measurements of 1.2 or less. If the lab technician takes 16 measurements and the variance of the measurements in the sample is 2.2.

To find :

1) Does this provide enough evidence to reject the claim that the lab technician’s accuracy is within the target accuracy?

2)Compute the value of the appropriate test statistic ?

Solution :

1) n=16 number of sample

The target accuracy is a variance in measurements of 1.2 or less i.e. [tex]\sigma_1^2 =1.2[/tex]

The variance of the measurements in the sample is 2.2 i.e. [tex]\sigma_2^2=2.2[/tex]

According to question,  

We state the null and alternative hypotheses,

Null hypothesis [tex]H_o : \text{var}^2 \geq 1.2[/tex]

Alternative hypothesis [tex]H_a : \text{var}^2<1.2[/tex]

We claim the alternative hypothesis.

2) Compute the value of the appropriate test statistic.

Using Chi-square,

[tex]\chi =\frac{(n-1)\sigma_2^2}{\sigma_1^2}[/tex]

[tex]\chi =\frac{(16-1)(2.2)}{(1.2)}[/tex]

[tex]\chi =\frac{(15)(2.2)}{1.2}[/tex]

[tex]\chi =\frac{33}{1.2}[/tex]

[tex]\chi =27.5[/tex]

Therefore, The value of the appropriate test statistic is 27.5.

To determine if the lab technician's accuracy is within the target accuracy, a one-sample variance test can be performed using the chi-square statistic.

To determine if the lab technician's accuracy is within the target accuracy, we can perform a hypothesis test. We will use a one-sample variance test to compare the sample variance to the target variance.

The appropriate test statistic for a one-sample variance test is the chi-square statistic. The chi-square statistic is calculated by taking the sample variance and dividing it by the target variance, then multiplying by the degrees of freedom.

In this case, we have 16 measurements and a target variance of 1.2. The sample variance is 2.2. We can calculate the test statistic using the formula chi-square = (n-1) * (sample variance / target variance). Plugging in the values, we get chi-square = (16-1) * (2.2 / 1.2) = 29.67.

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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:

a. If I Catch the 8:05 bus, then I will arrive on time for work.

b. If an integer is divisible by 9, then that integer is divisible by 3.

C. If the Cubs win tomorrow's game, then they win the pennant and if the Cubs have won the pennant it is because they will have won the game tomorrow.

Step-by-step explanation:

Final answer:

To rewrite the statements in 'If...Then' form: 1. If you catch the 8:05 bus, then you will be on time for work. 2. If a number is divisible by 9, then it is divisible by 3. 3. If the Cubs win the pennant, then they have won tomorrow's game.

Explanation:

Rewriting Statements in 'If...Then' Form

To rephrase the given statements in 'If...Then' form while identifying the sufficient and necessary conditions, we'll look at each statement individually:

Catching the 8:05 bus is a sufficient condition for being on time for work. This can be rewritten as: If you catch the 8:05 bus, then you will be on time for work.

Being divisible by 3 is a necessary condition for a number to be divisible by 9. The 'If...Then' form is: If a number is divisible by 9, then it is divisible by 3.

The Cubs will win the pennant only if they win tomorrow's game. In the 'If...Then' form: If the Cubs win the pennant, then they have won tomorrow's game.

Each of these rephrased statements now clearly shows the reliance of the consequent (the outcome) on the antecedent (the condition).

Answer:

There is a 62.7% probability that a transaction can be completed in less than 100 milliseconds.

Step-by-step explanation:

This a probability problem.

We have the following probabilities:

-70% probability that a transaction is an addition of an item.

-30% probability that a transaction is a change to an item

-81% probability that an addition can be completed in less than 100 milliseconds

-20% probability that change can be completed in less than 100 milliseconds

The probability P that a transaction can be completed in less than 100 milliseconds is:

[tex]P = P_{1} + P_{2}[/tex]

In which [tex]P_{1}[/tex] is the probability that the transaction is an addition and it takes less than 100 milliseconds. So

[tex]P_{1} = 0.7*0.81 = 0.567[/tex]

[tex]P_{2}[/tex] is the probability that the transaction is a change and it takes less than 100 milliseconds. So

[tex]P_{2} = 0.2*0.3 = 0.06[/tex]

So

[tex]P = P_{1} + P_{2} = 0.567 + 0.06 = 0.627[/tex]

There is a 62.7% probability that a transaction can be completed in less than 100 milliseconds.

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:

The fist four terms are 0, -1, -4, -6.

Step-by-step explanation:

Since [tex] k\geq 0 [/tex], the fist four terms of the sequence correspond to those in wich [tex] k [/tex] takes the values 0, 1, 2 and 3, that is

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:

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

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The required simplified expression for 11 x 10³ is 11,000. Option B is correct.

Given that,
A simplified expression form of the expression 11 x 10³, is to be determined.

The process in mathematics to operate and interpret the function or expression to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
= 11 x 10³
Since 10³ = 10 x 10 x 10 = 1000
= 11 x 1000
= 11,000

Thus, the required simplified expression for 11 x 10³ is 11,000. Option B is correct.

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Answer:

4.00512 cents

Step-by-step explanation:

Given:

Power output = 48 Watts

Time for which owner paddles = 8 hours

Selling price of the electricity = 10.43 cents/kWh

Now,

Power = Energy × Time

or

Power generated = 48 × 8 = 384 Wh = 0.384 kWh

now,

Money earned will be = Power generated × selling price per kWh

or

Money earned = 0.384 kWh ×  10.43 cents/ kWh = 4.00512 cents

Answer:  15

Step-by-step explanation:

Given : Elena has 60 colored pencils, and Lucy has 26 colored pencils.

Total colored pencils = [tex]60+26=86[/tex]

Let 'x' denotes the number of pencils Lucy has when she get pencils from Elena , and Elena left with 'y' pencils.

Then, by considering the given information we have the following system :-

[tex]\text{Total pencils : }x+y=86-----(1)\\\\\text{Colored pencils left with Elena : }y=4+x-----(2)[/tex]

We can rewritten the equation (2) as

[tex]y-x=4[/tex]

Now, add equation (1) and (2), we get

[tex]2y=90\\\\\Rightarrow\ y=\dfrac{90}{2}=45[/tex]

Put the value of y in (2), we get

[tex]45=4+x\\\\\Rightarrow\ x=45-4=41[/tex]

It means, the number of pencils Lucy has now =41

The number of pencils Elena give to Lucy= [tex]41-26=15[/tex]

Hence, Elena must give 15 pencils to lucy so that elena will have 4 more colored pencils than Lucy.

Answer:

a)The linear function for [tex]P(t)[/tex] is:

[tex]P(t) = 2000 - 100*t[/tex]

b)The exponential function for [tex]P(t)[/tex] is:

[tex]P(t) = 2000e^{-0.055t}[/tex]

Step-by-step explanation:

(a) Suppose that P(t) is a linear function. Find a formula for P(t):

[tex]P(t)[/tex] can be modeled by a linear function in the following format.

[tex]P(t) = P_{0} - r*t[/tex], in which [tex]P_{0}[/tex] is the initial number of bacteria cells in the dish, t is the time and r is the rate that the number decreases.

Since the dish initially contains 2000 bacteria cells, [tex]P_{0} = 2000[/tex]

We have

[tex]P(t) = 2000 - r*t[/tex]

An antibiotic is introduced and after 4 hour, there are now 1600 bacteria cells present. So [tex]P(4) = 1600[/tex]. With this information, we can find the value of r.

[tex]P(t) = 2000 - r*t[/tex]

[tex]1600 = 2000 - r*(4)[/tex]

[tex]4r = 400[/tex]

[tex]r = \frac{400}{4}[/tex]

[tex]r = 100[/tex]

So, the linear function for [tex]P(t)[/tex] is:

[tex]P(t) = 2000 - 100*t[/tex]

b) Suppose that P(t) is an exponential function. Find a formula for P(t)

[tex]P(t)[/tex] can also be modeled by an exponential function in the following format:

[tex]P(t) = P_{0}e^{rt}[/tex]

The values mean the same as in a). We use the fact that [tex]P(4) = 1600[/tex] to find r.

[tex]P(t) = 2000e^{rt}[/tex]

[tex]1600 = 2000e^{4r}[/tex]

[tex]e^{4r} = \frac{1600}{2000}[/tex]

[tex]e^{4r} = 0.8[/tex]

[tex]ln e^{4r} = ln 0.8[/tex]

[tex]4r = -0.22[/tex]

[tex]r = \frac{-0.22}{4}[/tex]

[tex]r = -0.055[/tex]

So, the exponential function for [tex]P(t)[/tex] is:

[tex]P(t) = 2000e^{-0.055t}[/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:

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:

The energy of each photon is [tex]6.468 \times 10^{-26}[/tex] Joule.

Step-by-step explanation:

Consider the provided information.

According to the plank equation:

[tex]E=h\nu[/tex]

Where E is the energy of photon, h is the plank constant and [tex]\nu[/tex] is the frequency.

It is given that [tex]h= 6.6 \times10^{-34}[/tex] and [tex]\nu=98MHz[/tex]

98Mhz = [tex]98\times 10^6Hz[/tex]

Substitute the respective value in plank equation.

[tex]E=6.6\times 10^{-34}\times 98\times 10^6[/tex]

[tex]E=6.6\times 98\times 10^{-34+6}[/tex]

[tex]E=646.8 \times10^{-28}[/tex]

[tex]E=6.468 \times 10^{-26}[/tex]

Hence, the energy of each photon is [tex]6.468 \times 10^{-26}[/tex] Joule.

Answer: 92378

Step-by-step explanation:

Given : At the grocery store, Hosea has narrowed down his selections to 7 vegetables, 6 fruits, 5 cheeses, and 6 whole grain breads.

Total items : 7+6+5+6=24

If he wants all cheeses  , then the remaining items needed to be select = 15-5=10

Total items left from which he will select = 24-5=19

No. of combinations of r things out of n : [tex]C(n;r)=\dfrac{n!}{r!(n-r)!}[/tex]

The combination of 10 things selecting from 19 things given by :-

[tex]C(5;5)\timesC(19;10)=\dfrac{19!}{10!(19-10)!}\\\\=(1)\times\dfrac{19\times18\times17\times16\times15\times14\times13\times12\times11\times10!}{10!9!}\ \because ^nC_n=1\\\\=92378[/tex]

Hence, the number of ways he can choose 15 items to buy if he wants all 5 cheeses =92378

Final answer:

If Hosea wants to buy all 5 cheeses, there are 19 items he can choose from the remaining categories. Using the combination formula, the number of ways to choose 10 items from a set of 19 is calculated to be 92378.

Explanation:

If Hosea wants to buy all 5 cheeses, he needs to choose 10 more items from the remaining categories (vegetables, fruits, and whole grain breads). There are 7 vegetables, 6 fruits, and 6 whole grain breads, which means he can choose from a total of 7+6+6 = 19 items in those categories. Since Hosea can only choose 10 more items, he needs to calculate the number of ways to choose 10 items from a set of 19. This can be calculated using the combination formula.

The number of ways to choose 'r' items from a set of 'n' items is given by the combination formula: C(n, r) = n! / (r! * (n-r)!). In this case, n = 19 (the number of items to choose from) and r = 10 (the number of items to choose). Evaluating the formula, we get: C(19, 10) = 19! / (10! * (19-10)!).

By simplifying the factorial expressions and performing the calculations, we find that there are 92378 ways for Hosea to choose his 15 items.

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:

Part a) value of [tex]\lambda [/tex] such that all the solutions tend to zero equals 1.

Part b)

For a particular solution to tend to 0 will depend on the boundary conditions.

Step-by-step explanation:

The given differential equation is

[tex]y'+\lambda y=e^{-t}[/tex]

This is a linear differential equation of first order of form [tex]\frac{dy}{dt}+P(t)\cdot y=Q(t)[/tex] whose solution is given by

[tex]y\cdot e^{\int P(t)dt}=\int e^{\int P(t)dt}\cdot Q(t)dt[/tex]

Applying values we get

[tex]y\cdot e^{\int \lambda dt}=\int e^{\int \lambda dt}\cdot e^{-t}dt\\\\y\cdot e^{\lambda t}=\int (e^{(\lambda -1)t})dt\\\\y\cdot e^{\lambda t}=\frac{e^{(\lambda -1)t}}{(\lambda -1)}+c\\\\\therefore y(t)=\frac{c_{1}}{\lambda -1}(e^{-t}+c_{2}e^{-\lambda t})[/tex]

here [tex]c_{1},c_{2}[/tex] are arbitrary constants

part 1)

For all the function to approach 0 as t approaches infinity we have

[tex]y(t)=\lim_{t\to \infty }[\frac{c_{1}}{\lambda -1}(e^{-t}+c_{2}e^{-\lambda t})]\\\\y(\infty )=\frac{c_{1}}{\lambda -1}=0\\\\\therefore \lambda =1[/tex]

Part b)

For a particular solution to tend to 0 will depend on the boundary conditions as [tex]c_{1},c_{2}[/tex] are arbitrary constants

Answer:

[tex]I=\$47.25[/tex]

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Final Interest Value

P is the amount of money borrowed

r is the rate of interest  

t is Number of Time Periods

in this problem we have

[tex]t=9/12\ years\\ P=\$900\\r=0.07[/tex]

substitute in the formula above

[tex]I=900(0.07(9/12))[/tex]

[tex]I=\$47.25[/tex]

Final answer:

Margo will pay $47.25 in interest on a $900 loan with a 7% annual interest rate after 9 months, after converting the rate to a decimal and the time to years for the calculation.

Explanation:

The question asks how much interest Margo will pay on a $900 loan with an annual interest rate of 7% after 9 months. To calculate this, we need to know that Interest = Principal × Rate × Time, where the principal is the amount borrowed, the rate is the annual interest rate (as a decimal), and the time is the period of the loan in years

First, convert the annual interest rate from a percentage to a decimal by dividing by 100: 7% / 100 = 0.07. Next, convert the loan period from months to years since the interest rate is annual. There are 12 months in a year, so 9 months is equal to 9/12 or 0.75 years.

Now, calculate the interest: $900 (Principal) × 0.07 (Rate) × 0.75 (Time) = $47.25. Therefore, Margo will pay $47.25 in interest.

Answer:

the estimated answer is A: 9.75.

(the actual answer is 9.66, so rounding up makes it 9.75)

Answer:

  z = (3/8)y - 11

Step-by-step explanation:

Undo what has been done to z, in reverse order. Here, 11 is added and the sum is multiplied by 8/3.

To undo the multiplication, multiply the equation by the inverse, 3/8:

  (3/8)(8/3)(z+11) = (3/8)y

  z +11 = (3/8)y . . . . . . . . . simplify

To undo the addition of 11, add the opposite of 11 to the equation.

  z +11 -11 = (3/8)y -11

  z = (3/8)y -11 . . . . . . . . . simplify

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

The average speed for this additional distance is 6.57 mi/h

The given parameters include;

To find:

The average speed formula is given as;

[tex]average\ speed = \frac{total \ distance }{total \ time} \\\\average\ speed = \frac{16.5 \ miles \ +\ 5.5 \ miles}{\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} } \\\\13.8 \ mi/h = \frac{(16.5 \ miles \ +\ 5.5 \ miles)}{(\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} ) }\\\\13.8 = \frac{(16.5 \ +\ 5.5 )}{(\frac{16.5 }{21.8 }\ \ +\ \ \frac{5.5 }{v} ) }\\\\13.8 = \frac{22 }{0.757\ \ +\ \ \frac{5.5 }{v} }\\\\13.8 (0.757\ \ +\ \ \frac{5.5 }{v} ) = 22\\\\[/tex]

[tex]10.446 + \frac{75.9}{v} = 22\\\\\frac{75.9}{v} = 22-10.446\\\\\frac{75.9}{v} = 11.554\\\\v = \frac{75.9}{11.554} \\\\v = 6.57 \ mi/h[/tex]

Thus, the average speed for this additional distance is 6.57 mi/h

Learn more here: https://brainly.com/question/17289046

The runner must run at an average speed of 6.57 mi/h to achieve the overall average speed of 13.8 mi/h for the entire biathlon race.

To solve for the average running speed needed to achieve an overall average speed of 13.8 mi/h in the biathlon, we will use the concept of weighted averages. The total time for both cycling and running can be expressed in terms of the distances traveled and the speeds for each segment.

First, we calculate the time taken to complete the cycling portion, which is 16.5 miles at an average speed of 21.8 mi/h. The formula for time is distance divided by speed:

Next, we need to find the running speed. Let's call the average running speed 'R'. Using the same formula for time:

The overall average speed for the entire race is the total distance divided by the total time. The total distance is 16.5 miles + 5.5 miles = 22 miles. Let 'T' represent the total time for the race.

Average speed for the entire race = Total distance / Total time = 22 miles / T

Given that the overall average speed must be 13.8 mi/h, we set up the equation:

13.8 mi/h = 22 miles / T

Now we can express 'T' as the sum of the cycling time and running time:

T = Time (cycling) + Time (running) = 0.7565 hours + (5.5 miles / R)

Substitute this into our average speed equation:

13.8 mi/h = 22 miles / (0.7565 hours + (5.5 miles / R))

Now, we solve for 'R', which represents the average running speed. We cross multiply and isolate 'R' to obtain:

(13.8 mi/h) (0.7565 hours + (5.5 miles / R)) = 22 miles

This simplifies to:

10.4469 hours + 75.9 miles / R = 22 miles

Subtract 10.4469 hours from both sides:

75.9 miles / R = 11.5531 miles

Finally, divide 75.9 by 11.5531 to find the average running speed 'R':

R = 6.57 mi/h

The runner must run at an average speed of 6.57 mi/h to have an overall average speed of 13.8 mi/h for the biathlon.

Answer:

Claim :The proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 36%.

n =  170

x = 56

We will use one sample proportion test  

[tex]\widehat{p}=\frac{x}{n}[/tex]

[tex]\widehat{p}=\frac{56}{170}[/tex]

[tex]\widehat{p}=0.3294[/tex]

The proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 36%.

[tex]H_0:p \neq 0.36 \\H_a:p= 0.36[/tex]

Formula of test statistic =[tex]\frac{\widehat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

                                       [tex]=\frac{0.3294-0.36}{\sqrt{\frac{0.36(1-0.36)}{170}}}[/tex]

                                        =−0.8311

Now refer the p value from the z table

P-Value is .202987 (Calculated by online calculator)

Level of significance α = 0.05

Since p value < α

So we reject the null hypothesis .

Hence the claim is true

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.