Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed less than 100 milliseconds 81% of the time, but only 20% of changes to a previous item can be completed in less than this time. If 30% of transactions are changes, what is the probability that a transaction can be completed in less than 100 milliseconds? Round your answer to two decimal places (e.g. 98.76).

In logic, own symbols are used in order to be able to represent the relations between propositions in a general and independent way to the proposition, in order to be able to find the relationship process that operates in the communicated message, the propositional logic.

For this purpose there are, among others, the following logical operators: conjunction (and) ∧, disjunction (or) ∨, denial (not) ¬,  conditional (if - then) ⇒ and double conditional (if and only if, iff) ⇔.

So for this case we have:

Answer

a) Sam had pizza last night if and only if Chris finished her homework.

p⇔q

b) Pat watched the news this morning iff Sam did not have pizza last night.

r⇔¬p

c) Pat watched the news this morning if and only if Chris finished her homework and Sam did not have pizza last night.

r⇔(q∧¬p)

d) In order for Pat to watch the news this morning, it is necessary and sufficient that Sam had pizza last night and Chris finished her homework.

r⇔(p∧q)

e) q ⇔ r

Chris finished his homework if and only if Pat watched the news this morning

f) p ⇔ (q ∧ r)

Sam had pizza last night if and only if Chris finished his homework and Pat watched the news this morning

g) (¬p) ⇔ (q ∨ r)

Sam didn't have pizza last night if and only if Chris finished his homework or Pat watched the news this morning

h) r ⇔ (p ∨ q)

Pat watched the news this morning if Sam had pizza last night or Chris finished his homework

Answer:

Step-by-step explanation:

Write expressions for the number of students in each major. Then write the equation needed to relate them the way the problem statement says they are related.

For year y, the number of students in each major is ...

1) Engineering is twice Business:

  120 +22y = 2(105 -4y) . . . . . Engineering is double Business in year y

  120 +22y = 210 -8y . . . . . . . eliminate parentheses

  120 +30y = 210  . . . . . . . . . . add 8y

  4 + y = 7 . . . . . . . . . . . . . . . . divide by 30

  y = 3  . . . . . . . . subtract 4

In 3 years there will be 2 times as many students majoring in Engineering than in Business.

__

2) Engineering is twice Biology:

  120 +22y = 2(98 +6y) . . . . Engineering is double Biology in year y

  120 +22y = 196 +12y . . . . . eliminate parentheses

  120 +10y = 196 . . . . . . . . . . subtract 12y

  10y = 76 . . . . . . . . . . . . . . . .subtract 120

  y = 7.6 . . . . . . divide by 10

In 8 years there will be 2 times as many students majoring in Engineering than in Biology.

Answer:

Ans. Effective annual rate=1.8928%

Annual Compound semi-annually=1.8839%

Step-by-step explanation:

Hi, this is the formula to find the effective annual rate for this zero-coupon bond.

[tex]EffectiveAnnualRate=\sqrt[n]{\frac{FaceValue}{Price} } -1[/tex]

n= years to maturity

That is:

[tex]EffectiveAnnualRate=\sqrt[8]{\frac{50,000}{43,035} } -1=0.018928[/tex]

Means that the effective interest rate is 1.8928% effective annual

Now, let´s find the compound interest rate.

First, we have to turn this rate effective semi-annually

[tex]Semi-AnnualRate=(1+0.018928)^{\frac{1}{2} }  -1=0.00942[/tex]

0.942% effective semi annual

To turn this into a semi-annual, compounded semi-annually, we just have to multiply by 2, so we get.

1.8839% compounded semi-annually

Best of luck

Answer:

(a) Set A = {1,2,3}

    Set B = {1,2,4,5}

(b) Set A Ç B = {1,2}

     Set A e B = {1,2}

Step-by-step explanation:

As per the question,

Given data :

A ∪ B = read as A union B = {1,2,3,4,5}

A ∩ B = read as A intersection B =  {1,2}

(a) An example of sets A and B such that A ∩ B = {1,2} and A U B = {1,2,3,4,5).

So, first draw the Venn-diagram, From below Venn diagram, One of the possibility for set A and set B is:

Set A = {1,2,3}

Set B = {1,2,4,5}

(b) An example of sets A and B such that A Ç B and A e B,

Set A Ç B read as common elements of set A in Set B.

Therefore,

Set A Ç B = {1,2}

Set A e B implies that which element/elements of A is/are present in set B.

Therefore,

Set A e B = {1,2}

Answer:

4:3

Step-by-step explanation:

You count up all the red marbles which equals 4 and put them on one side, then you add up all the rest of the marbles same color or not which equals 3 and put it on the other side of the 4

Step-by-step explanation:

.454 kilograms= 1 pound.

Multiply .454 by 160

4.54

160

--------

000

2724 0<--Place marker

45400<--Double Place Marker

- - - - - - - -

72640 <-- Add

To find decimal point, count decimal place (Number of digits after the decimal on both numbers you multiply together) In this case, it's 2 ( 5 and 4 in 4.54) So, you count two spaces from right to left in your answer and tah dah!

72.640 (Zero isn't needed- just a placemarker)

Hope I was helpful :)

Answer:

(a) [tex]\frac{dy}{(2y+1)}=tdt[/tex] (b) [tex]y=\frac{e^{t^2}+e^{2c}-1}{2}[/tex]

Step-by-step explanation:

(1) We have given [tex]\frac{dy}{dt}+ycost=0[/tex]

[tex]\frac{dy}{dt}=-ycost[/tex]

[tex]\frac{dy}{y}=-costdt[/tex]

Integrating both side

[tex]lny=-sint+c[/tex]

[tex]y=e^{-sint}+e^{-c}[/tex]

(2) [tex]\frac{dy}{dt}-2ty=t[/tex]

[tex]\frac{dy}{dt}=2ty+t[/tex]

[tex]\frac{dy}{dt}=t(2y+1)[/tex]

[tex]\frac{dy}{(2y+1)}=tdt[/tex]

On integrating both side  

[tex]\frac{ln(2y+1)}{2}=\frac{t^2}{2}+c[/tex]

[tex]ln(2y+1)={t^2}+2c[/tex]

[tex]2y+1=e^{t^2}+e^{2c}[/tex]

[tex]y=\frac{e^{t^2}+e^{2c}-1}{2}[/tex]

The domain of the function represented in the graph is 30 SXS 60 or 30 to 60. Hence option C is correct.

Given that,

Five-year-old students at an elementary school were involved.

They were given a 30-yard head start in a race.

The graph represents the distance run by the average student in 30 seconds.

The graph is related to the age of the runner.

Since we can see that,

The graph starts from 30 yards and ends at 60 yards

Therefore,

The domain of the function represented in the graph is 30 SXS 60 or 30 to 60, which is option C.

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Answer:

[tex]\frac{7}{6}[/tex]

Step-by-step explanation:

[tex]-4\frac{1}{2} + 5\frac{2}{3} =[/tex]

For the first term you have to multiply 2x(-4) and add 1, for the second term you have to multiply 3x5 and add 2.

[tex]\frac{2(-4)+1}{2} + \frac{3(5)+2}{3} =[/tex]

[tex]-\frac{9}{2} +\frac{17}{3} =[/tex]

Now you need to find the lowest common multiple between the denominators, just cross multiply as it is shown:

[tex]-\frac{9}{2} (\frac{3}{3} )+\frac{17}{3} (\frac{2}{2} )=[/tex]

[tex]-\frac{27}{6} +\frac{34}{6} = \frac{7}{6}[/tex]

finally you get the result by doing a substraction = 7/6, or 1[tex]\frac{1}{6}[/tex]

Answer with Step-by-step explanation:

We are given that two functions f and g are decreasing function for all real numbers .

We have to prove that fog is increasing.

Increasing Function: If [tex]x_1 \leq x_2[/tex]

Then, [tex]f(x_1)\leq f(x_2)[/tex]

Decreasing function: if [tex]x_1\leq x_2[/tex]

Then, [tex]f(x_1)\geq f(x_2)[/tex]

Suppose [tex]f(x)=-x[/tex]

[tex]g(x)=-2x[/tex]

fog(x)=f(g(x))=f(-2x)=-(-2x)=2x

Therefore, f and g are decreasing function for all real numbers  then fog is  increasing.

Hence, proved

Answer:

one

Step-by-step explanation:

I think, there were 4 lines and the two lines are reaching each other

Answer:

[tex]f''(x)=f''(-x)[/tex]

Step-by-step explanation:

A function satisfying the equation [tex]f(x)=f(-x)[/tex] is said to be an even function. This denomination comes from the fact that the same relation is satisfied for functions of the form [tex]x^{n}[/tex] with [tex]n[/tex] even. Observe that if [tex]f[/tex] is twice differentiable we can derivate using the chaing rule as follows:

[tex]f(x)=f(-x)[/tex] implies [tex]f'(x)=f'(-x)\cdot (-1)=-f'(-x)[/tex]

Applying the chain rule again we have:

[tex]f'(x)=-f'(-x)[/tex] implies [tex]f''(x)=-f''(-x)\cdot (-1)=f''(-x)[/tex]

So we have that function [tex]f''(x)[/tex] is also an even function.

Answer:

The system will have no solution when [tex]k = -4[/tex] and [tex]h \neq 7.5[/tex].

Step-by-step explanation:

We can find these values by the Gauss-Jordan Elimination method.

The Gauss-Jordan elimination method is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.

We have the following system:

[tex]-3x - 3y = h[/tex]

[tex]-4x + ky = 10[/tex]

This system has the following augmented matrix:

[tex]\left[\begin{array}{ccc}-3&-3&h\\-4&k&10\end{array}\right][/tex]

The first thing i am going to do is, to help the row reducing:

[tex]L_{1} = -\frac{L_{1}}{3}[/tex]

Now we have

[tex]\left[\begin{array}{ccc}1&1&-\frac{h}{3}\\-4&k&10\end{array}\right][/tex]

Now I want to reduce the first row, so I do:

[tex]L_{2} = L_{2} + 4L_{1}[/tex]

So:

[tex]\left[\begin{array}{ccc}1&1&-\frac{h}{3}\\0&k+4&10 - \frac{4h}{3}\end{array}\right][/tex]

From the second line, we have

[tex](k+4)y = 10- \frac{4h}{3}[/tex]

The system will have no solution when there is a value dividing 0, so, there are two conditions:

[tex]k+4 = 0[/tex] and [tex]10 - \frac{4h}{3} \neq 0[/tex]

[tex]k+4 = 0[/tex]

[tex]k = -4[/tex]

...

[tex]10 - \frac{4h}{3} \neq 0[/tex]

[tex]\frac{4h}{3} \neq 10[/tex]

[tex]4h \neq 30[/tex]

[tex]h \neq \frac{30}{4}[/tex]

[tex]h \neq 7.5[/tex]

The system will have no solution when [tex]k = -4[/tex] and [tex]h \neq 7.5[/tex].

Answer

-16

Step By Step explanation

Answer:

[tex] - 12 - 3 \times ( - 8 + ( { - 4)}^{2} - 6) + 2[/tex]

[tex] - 12 - 3 \times ( - 8 + 16 - 6) + 2[/tex]

[tex] - 12 - 3 \times (8 - 6) + 2[/tex]

[tex] - 12 - 3 \times 2 + 2[/tex]

[tex] - 12 - 6 + 2[/tex]

[tex] - 12 - 4[/tex]

[tex] - 16[/tex]

Answer:

%82.5

Step-by-step explanation:

From point 1 we know that Moe´s grade just before taking the final exam represents 60% of the final grade. Then, using the information in the point 2 we can compute Moe´s final grade as follows:

[tex]FG=0.40*FE+0.60*0.45[/tex],

where FG is Moe´s Final Grade and FE is Moe´s final exam grade. Then,

[tex]\frac{ FG-0.60*0.45}{0.40}=FE[/tex].

So, in order to receive the passing grade average of 60% for the class Moe needs to obtain in his exam:

[tex]FE=\frac{ 0.60-0.60*0.45}{0.40}=0.825[/tex]

That is, he need al least %82.5 to obtain a passing grade.

Answer:

Domain : D{-∞,∞} the reals.

Step-by-step explanation:

The function is plotted in the image.

[tex]  f(x) = -3.15 * x + 723.45 [tex]

the linear functions usually have a domain from - infinite to infinite, the domain when is a piece wise function or discontinuous, the domain is defined in the pieces where is defined.

In this case there is no restriction so the function is continuous.

Answer:  The proof is done below.

Step-by-step explanation:  Given that A is a 3 x 3 matrix such that det (A) = 9.

We are to prove the following :

[tex]det(3A^{-1})=3.[/tex]

For a non-singular matrix B of order n, we have two two properties of its determinant :

[tex](i)~det(B^{-1})=\dfrac{1}{det(B)},\\\\\\(ii)~det(kB)=k^ndet(B),~\textup{k is a scalar.}[/tex]

Therefore, we get

[tex]det(A^{-1})=\dfrac{1}{det(A)}=\dfrac{1}{9},[/tex]

and so,

[tex]det(3A^{-1})~~~~~~~[\textup{since A is of order 3}]\\\\=3^3det(A^{-1})\\\\=27\times\dfrac{1}{9}\\\\=3.[/tex]

Hence proved.

Answer:

Step-by-step explanation:

We know that multiplication product of two given positive real number will always be positive real number and if one of the real number is negative, the multiplication product will always be negative.

so for the given condition, if [tex]x > 0[/tex] and [tex]y > 0[/tex], [tex]x[/tex] and [tex]y[/tex] are both positive real numbers

hence their multiplication product [tex]xy[/tex] will also be a positive number.

∴ [tex]xy > 0[/tex]

Answer:

a) Yes.

b) Yes.

Step-by-step explanation:

Meredith and Hillary both want coffe, but they don't know if the other two people do, therefore they can't know if everyone want coffee. If they didn't want coffee, their answer would have been just "no".

Aly knows that she doesn't want coffee, therefore she knows that not everyone wants coffee.

Hillary and Meredith said 'I don't know' which implies they don't know if everyone wants coffee because they themselves do not want it. Aly confirmed that not everyone wants coffee. Therefore, neither Hillary nor Meredith got a coffee.

We have a logical puzzle where Hillary, Meredith, and Aly are deciding whether they want coffee. The key to solving this puzzle is understanding the implications of their statements to the waiter's question: "Does everyone want coffee?"

Hillary says, "I don't know." This means Hillary cannot be sure that everyone wants coffee, so there are two possibilities: either she does not want coffee or she doesn't know the preferences of the others. Meredith also responds with "I don't know," implying the same possibilities for her.

Finally, Aly states, "Not everyone wants coffee." This is the definitive answer that tells us at least one person does not want coffee. Since Aly knows for sure that not everyone wants coffee, it implies that either she does not want coffee herself or knows of someone else who doesn’t. Given that Hillary and Meredith both said they did not know, they could not have communicated their preference to Aly.

Therefore:

Answer:

P(B) = 0.30

Step-by-step explanation:

This is a probability problem that can be modeled by a diagram of Venn.

We have the following probabilities:

[tex]P(A) = P_{A} + P(A \cap B) = 0.50[/tex]

In which [tex]P_{A}[/tex] is the probability that only A happens.

[tex]P(B) = P_{B} + P(A \cap B) = P_{B} + 0.15[/tex]

To find P(B), first we have to find [tex]P_{B}[/tex], that is the probability that only B happens.

Finding [tex]P_{B}[/tex]:

The problem states that P(A OR B) = 0.65. This is the probability that at least one of this events happening. Mathematically, it means that:

[tex]1) P_{A} + P(A \cap B) + P_{B} = 0.65[/tex]

The problem states that P(A) = 0.5 and [tex]P(A \cap B) = 0.15[/tex]. So we can find [tex]P_{A}[/tex].

[tex]P(A) = P_{A} + P(A \cap B)[/tex]

[tex]0.5 = P_{A} + 0.15[/tex]

[tex]P_{A} = 0.35[/tex]

Replacing it in equation 1)

[tex]P_{A} + P(A \cap B) + P_{B} = 0.65[/tex]

[tex]0.35 + 0.15 + P_{B} = 0.65[/tex]

[tex]P_{B} = 0.65 - 0.35 - 0.15[/tex]

[tex]P_{B} = 0.15[/tex]

Since

[tex]P(B) = P_{B} + P(A \cap B)[/tex]

[tex]P(B) = 0.15 + 0.15[/tex]

[tex]P(B) = 0.30[/tex]

Answer:

A 1.4% of the total connectors are expected to fail during the warranty period.

Step-by-step explanation:

Let's assume a population of 1000 connectors (to make the math easiest) and let's analize the dry connectors.  

Of the 1000 connectors, 900 are kept dry. and of that number, 9 are the ones that fails during the warranty period.  (90% of 1000 is 900. 1% of 900 is 9)

Of the 1000 connectors, 100 are wet. And of that number, 5 are connectors that will fail during the warranty period. (10% of 1000 is 100, and 5% of 100 is 5)

So overall we have 14 connectors that will fail from 1000 connectors.

That is a 1.4% of the total samples.

Answer:

a) Mode: 2 Median: 3 Mean: 4.6

b) Mode: 7 Median: 8 Mean: 9.6

c) Just added 5 to values. General below.

Step-by-step explanation: 2, 2, 3, 6, 10

a) Mode: 2 (Most apperances)

Median: 3 (odd data, middle number)

Mean: (2+2+3+6+10)/5 = 23/5 = 4.6

b) + 5

Data: 7,7,8,11,15

Mode: 7 (Most apperances)

Median: 8 (odd data, middle number)

Mean: (7+7+8+11+15)/5 = 48/5 = 9.6

c) The results from (b) is (a) + 5

In general: Let's add x to the same data provided:

2+x, 2+x, 3+x, 6+x, 10+x,

For the mode, it does not matter, the number with most apperances will continue to be the mode + x

For the median, same thing. It is just the median + x

For the mean, same thing. For the set of 5 numbers:

(2+x + 2+x + 3+x + 6+x + 10+x)/5 =

(23+5x)/5

23/5 + 5x/5 =

23/5 + x

For example, If it was 6 numbers, we would add 6 times that number and divide it by 6, adding x to the mean.

To compute the mode, median, and mean of a data set, count the frequency of each number, arrange the data in order, and find the middle value. Adding the same constant to each data value affects the mean but does not change the mode or median.

To compute the mode, median, and mean of the data set {2, 2, 3, 6, 10}, we can follow these steps:

After adding 5 to each data value, the new data set becomes {7, 7, 8, 11, 15}.

In general, when the same constant is added to each data value in a set, the mode remains unchanged, the median remains unchanged, and the mean is affected by adding the constant to each value. The mean increases when the constant is positive and decreases when the constant is negative.

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Answer:  2399760

Step-by-step explanation:

The concept we use here is Partial derangement.

It says that for m things , the number of ways to arrange them such that k things are not in their fixed position is given by :-

[tex]m!-^kC_1(m-1)!+^kC_2(m-2)!-^kC_3(m-3)!+........[/tex]

Given digits : 0,1,2,3,4,5,6,7,8,9

Prime numbers = 2,3,5,7

Now  by Partial derangement the number of ways to arrange 10 numbers such that none of 4 prime numbers is in its original position will be :_

[tex]10!-^4C_1(9)!+^4C_2(8)!-^4C_3(7)!+^4C_4(6)!\\\\=3628800-(4)(362880)+\dfrac{4!}{2!2!}(40320)-(4)(5040)+(1)(720)\\\\=3628800-1451520+241920-20160+720\\\\=2399760[/tex]

Hence, the number of  ways can the digits 0,1,2,3,4,5,6,7,8,9 be arranged so that no prime number is in its original position = 2399760

Answer:

62 inches

Step-by-step explanation:

Let x be the height ( in inches ) of Ivan,

∵ Marcia is 5 inches shorter than Ivan,

height of Marcia = x - 5,

Marcia is  twice as tall as Sally-Jo,

height of sally-jo = [tex]\frac{x-5}{2}[/tex]

Mom is 2 inches taller than Ivan.

⇒ height of mom = x + 2,

Mom is 3 inches shorter than Dad,

height of dad = x + 2 + 3 = x + 5,

So, mean height of family is,

[tex]\frac{x+x-5+\frac{x-5}{2}+x+2+x+5}{5}[/tex]

[tex]=\frac{2x+2x-10+x-5+2x+4+2x+10}{10}[/tex]

[tex]=\frac{9x-1}{10}[/tex]

According to the question,

x + 5 = 74 ( 1 ft = 12 in )

x = 69

Hence, mean height of the family = [tex]\frac{9\times 69-1}{10}[/tex]

[tex]=\frac{620}{10}[/tex]

= 62 inches

Answer: 49 people

Step-by-step explanation:

First you need to separate the people that read both from the people that read the daily news. Do this by subtracting 171 - 40 = 131.

Now you can subtract the total amount of people that only read the Daily Mews from the total amount of people polled (198-131=67)

From here you need to subtract the amount of people that read neither from the remaining total. (67-18=49)

this is where you get the 49 people out of 198 that read the Sun Gazette

The number of people who read the Sun Gazette is calculated by subtracting the number of people who only read the Daily News and those who read neither from the total polled. The result is 49 Sun Gazette readers.

This problem involves the mathematics concept of set theory, specifically union and intersection of sets. Here, the total number of people polled are considered as the 'Universal Set'. The 'Daily News readers' and the 'Sun Gazette readers' are the two subsets of this Universal set.

The data given can be interpreted as follows:

Since 40 people read both, they are being counted twice in the 171 (Daily News readers) figure. Hence, we subtract 40 from 171.

The number of people who only read the Daily News = 171 - 40 = 131

We subtract this number and those who read neither from the total polled to find the number of Sun Gazette readers.

The number of Sun Gazette readers = 198 - 131 (only Daily News readers) - 18 (neither) = 49 Sun Gazette readers

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Answer:

[tex]8+27=17+18=35 cm=0.35 meter[/tex]

Step-by-step explanation:

[tex]8+27=35[/tex] cm

= [tex]17+18=35[/tex] cm as [tex]35-18=17[/tex] cm

Now as the final answer is in meters, so, we will convert 35 cm in meters.

100 cm = 1 meter

So, 35 cm = [tex]\frac{35}{100}=0.35[/tex] meters

Therefore, we can write the final expression as:

[tex]8+27=17+18=35 cm=0.35 meter[/tex]

Answer:

182 of these adults did not do any of these three activities last Friday night.

Step-by-step explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents the adults that watched TV

-The set B represents the adults that hung out with friends.

-The set C represents the adults that ate pizza

-The set D represents the adults that did not do any of these three activities.

We have that:

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

In which a is the number of adults that only watched TV, [tex]A \cap B[/tex] is the number of adults that both watched TV and hung out with friends, [tex]A \cap C[/tex] is the number of adults that both watched TV and ate pizza, is the number of adults that both hung out with friends and ate pizza, and [tex]A \cap B \cap C[/tex] is the number of adults that did all these three activies.

By the same logic, we have:

[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]

[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]

This diagram has the following subsets:

[tex]a,b,c,D,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]

There were 510 adults suveyed. This means that:

[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]

We start finding the values from the intersection of three sets.

Solution:

43 watched TV, hung out with friends, and ate pizza:

[tex]A \cap B \cap C = 43[/tex]

47 hung out with friends and ate pizza, but did not watch TV:

[tex]B \cap C = 47[/tex]

29 watched TV and hung out with friends, but did not eat pizza:

[tex]A \cap B = 29[/tex]

28 watched TV and ate pizza, but did not hang out with friends:

[tex]A \cap C = 28[/tex]

161 ate pizza

[tex]C = 161[/tex]

[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]

[tex]161 = c + 28 + 47 + 43[/tex]

[tex]c = 43[/tex]

196 hung out with friends

[tex]B = 196[/tex]

[tex]196 = b + 47 + 29 + 43[/tex]

[tex]b = 77[/tex]

161 watched TV

[tex]A = 161[/tex]

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

[tex]161 = a + 29 + 28 + 43[/tex]

[tex]a = 61[/tex]

How may 18-24 year olds did not do any of these three activities last Friday night?

We can find the value of D from the following equation:

[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]

[tex]61 + 77 + 43 + D + 29 + 28 + 47 + 43 = 510[/tex]

[tex]D = 510 - 328[/tex]

[tex]D = 182[/tex]

182 of these adults did not do any of these three activities last Friday night.

To find the number of 18-24 year olds who did not do any of the three activities last Friday night, we can use the principle of inclusion-exclusion. By subtracting the number of people who did at least one of the activities from the total number of participants, we find that 455 individuals did not participate in watching TV, hanging out with friends, or eating pizza.

To find the number of 18-24 year olds who did not do any of the three activities (watch TV, hang out with friends, eat pizza), we need to subtract the number of people who did at least one of these activities from the total number of participants. We can use the principle of inclusion-exclusion to solve this problem.

Let's define:

From the given information, we know:

To find the number of people who did not do any of these activities, we can use the formula:

n(A' ∩ B' ∩ C') = n(U) - n(A) - n(B) - n(C) + n(A ∩ B) + n(A ∩ C) + n(B ∩ C) - n(A ∩ B ∩ C)

Substituting the known values, we have:

n(A' ∩ B' ∩ C') = 510 - 161 - 196 - 161 + 28 + 29 + 47 - 43

n(A' ∩ B' ∩ C') = 455

Therefore, there were 455 18-24 year olds who did not do any of the three activities last Friday night.

Answer:

Turning point has coordinates [tex]\left(\dfrac{27}{13},\dfrac{8}{13}\right)[/tex]

Step-by-step explanation:

Gandalf the Grey started in the Forest of Mirkwood at a point P(3, 0) and began walking in the direction of the vector [tex]\vec{v}=-3i+2j.[/tex] The coordinates of the vector v are (-3,2). Then he changed the direction at a right angle, so he was walking in the direction of the vector [tex]\vec{u}=2i+3j[/tex] (vectors u and v are perpendicular).

Let B(x,y) be the turning point. Find vectors PB and BQ:

[tex]\overrightarrow{PB}=(x-3,y-0)\\ \\\overrightarrow {BQ}=(5-x,5-y)[/tex]

Note that vectors v and PB and vectors u and BQ are collinear, so

[tex]\dfrac{x-3}{-3}=\dfrac{y}{2}\\ \\\dfrac{5-x}{2}=\dfrac{5-y}{3}[/tex]

Hence

[tex]2(x-3)=-3y\Rightarrow 2x-6=-3y\\ \\3(5-x)=2(5-y)\Rightarrow 15-3x=10-2y[/tex]

Now solve the system of two equations:

[tex]\left\{\begin{array}{l}2x+3y=6\\ -3x+2y=-5\end{array}\right.[/tex]

Multiply the first equation by 3, the second equation by 2 and add them:

[tex]3(2x+3y)+2(-3x+2y)=3\cdot 6+2\cdot (-5)\\ \\6x+9y-6x+4y=18-10\\ \\13y=8\\ \\y=\dfrac{8}{13}[/tex]

Substitute it into the first equation:

[tex]2x+3\cdot \dfrac{8}{13}=6\\ \\2x=6-\dfrac{24}{13}=\dfrac{54}{13}\\ \\x=\dfrac{27}{13}[/tex]

Turning point has coordinates [tex]\left(\dfrac{27}{13},\dfrac{8}{13}\right)[/tex]

The coordinates of the point where Gandalf makes the turn are (5, 5).

To find the point where Gandalf makes a right angle turn, we need to find the intersection of the line formed by the vector v and the line connecting points P and Q. The equation of the line formed by the vector v is given by y = 2x - 3. The equation of the line connecting points P and Q is given by y = x. To find the intersection point, we can solve these two equations simultaneously. Substituting y = 2x - 3 into y = x, we get x = 5. Substituting x = 5 into y = x, we get y = 5. Therefore, the coordinates of the point where Gandalf makes the turn are (5, 5).

Answer:

The solutions of the quadratic equation are [tex]x_{1} = \frac{5 + \sqrt{13}}{6}, x_{2} = \frac{5 - \sqrt{13}}{6}[/tex]

Step-by-step explanation:

This is a second order polynomial, and we can find it's roots by the Bhaskara formula.

Explanation of the bhaskara formula:

Given a second order polynomial expressed by the following equation:

[tex]ax^{2} + bx + c, a\neq0[/tex].

This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = (x - x_{1})*(x - x_{2})[/tex], given by the following formulas:

[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}[/tex]

[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}[/tex]

[tex]\bigtriangleup = b^{2} - 4ac[/tex]

For this problem, we have to find [tex]x_{1} \text{and} x_{2}[/tex].

The polynomial is [tex]3x^{2} - 5x +1[/tex], so a = 3, b = -5, c = 1.

Solution

[tex]\bigtriangleup = b^{2} - 4ac = (-5)^{2} - 4*3*1 = 25 - 12 = 13[/tex]

[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a} = \frac{-(-5) + \sqrt{13}}{2*3} = \frac{5 + \sqrt{13}}{6}[/tex]

[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a} = \frac{-(-5) - \sqrt{13}}{2*3} = \frac{5 - \sqrt{13}}{6}[/tex]

The solutions of the quadratic equation are [tex]x_{1} = \frac{5 + \sqrt{13}}{6}, x_{2} = \frac{5 - \sqrt{13}}{6}[/tex]

Answer:

Age of Alice=25 years

Age of Jane=20 years

Step-by-step explanation:

We are given that the age of Jane is 80 % of the age Alice.

We have to find the age of Jane and Alice.

Let x be the age of Alice

According to question

Age of Jane=80% of Alice=80% of x=[tex]\frac{80}{100}\times x=\frac{4x}{5}[/tex]

[tex]x+\frac{4x}{5}=45[/tex]

[tex]\frac{5x+4x}{5}=45[/tex]

[tex]\frac{9x}{5}=45[/tex]

[tex]x=\frac{45\times 5}{9}=25[/tex]

Age of Alice=25 years

Age of Jane=[tex]\frac{4}{5}\times 25=20 years[/tex]

Age of Jane=20 years