calculate simple interest and maturity value

Principal is. 6,000. interest rate 5%. the time is 15. what is the solution

Answer: 0.2205

Step-by-step explanation:

Given : Technology Services department at Lahey Electronics revealed company employees receive an average of "2.7" non-work-related e-mails per hour.

i.e. [tex]\lambda = 2.7[/tex]

If the arrival of these e-mails is approximated by the Poisson distribution.

Then , the required probability is given by :-

[tex]P(X=x)=\dfrac{\lambda^xe^{-\lambda}}{x!}\\\\P(X=3)=\dfrac{(2.7)^3e^{-2.7}}{3!}\\\\=0.22046768454\approx0.2205[/tex]

Hence, the probability Linda Lahey, company president, received exactly 3 non-work-related e-mails between 4 P.M. and 5 P.M. yesterday =0.2205

To determine the probability that Linda Lahey received exactly 3 non-work-related e-mails in one hour based on a Poisson distribution with an average rate of 2.7 e-mails per hour, we apply the Poisson formula. This calculation offers a precise way to understand the likelihood of such an event occurring within a set timeframe.

Given that, on average, company employees receive 2.7 non-work-related e-mails per hour, we can use the Poisson formula to calculate this probability.

To find the probability of receiving exactly k events in a fixed interval of time, we use the formula:
P(X = k) = (λ^k * e^-λ) / k!
where λ is the average rate (2.7 emails per hour in this case), k is the number of events (3 emails), and e is the base of the natural logarithm (approximately 2.71828).

Plugging in the values, we calculate the probability as follows:
P(X = 3) = (2.7^3 * e^-2.7) / 3!
This calculation gives us the specific probability that Linda Lahey received exactly 3 non-work-related e-mails in one hour.

Answer:

[tex]6ft[/tex] length on the east and west sides

[tex]12ft[/tex] length on the north and south sides

Step-by-step explanation:

Using x for the length of the east side (and is equal to the length  of the west side) and y for the length of the north side (and is equal to the length  of the south side), the equation that gives the total price equalized to 480 is:

[tex]20x+20x+10y+10y=480[/tex]

[tex]40x+20y=480[/tex]

Solving for y

[tex]y=\frac{-40x+480}{20}[/tex]

[tex]y=-2x+24[/tex]

The area of the garden is [tex]A=xy[/tex], to find the largest, substitute y in the formula of the area

[tex]A=x(-2x+24)=-2x^2+24x[/tex]

For the optimization, find the largest area, is needed the critical point. To find this point, derive A and equalize the derivative to zero:

[tex]A'=-4x+24=0[/tex]

Solve for x:

[tex]-4x=-24[/tex]

[tex]x=\frac{-24}{-4}[/tex]

[tex]x=6[/tex]

To see if x=6 is a maximum or a minimum, derive A' and substitute with x=6

[tex]A''=-4[/tex]

In this case, the second derivative of A doesn't depend on x, and it has a negative value, meaning the value found is a maximum. Using x=6 to find y

[tex]y=-2x+24[/tex]

[tex]y=-2(6)+24[/tex]

[tex]y=12[/tex]

The area is:

[tex]A=xy=6*12=72 ft^2[/tex]

Answer: (0.541, 0.819)

Step-by-step explanation:

The confidence interval for proportion is given by :-

[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]

Given : The proportion of children attended the school = [tex]p=\dfrac{51}{75}=0.68[/tex]

Significance level : [tex]\alpha=1-0.99=0.01[/tex]

Critical value : [tex]z_{\alpha/2}=z_{0.005}=\pm2.576[/tex]

Now, the 99% z ‑confidence interval for proportion will be :-

[tex]0.68\pm (2.576)\sqrt{\dfrac{0.68(1-0.68)}{75}}\approx0.68\pm 0.139\\\\=(0.68-0.139,0.68+0.139)=(0.541,\ 0.819)[/tex]

Hence, the 99% z ‑confidence interval for p, the proportion of all children enrolled in kindergarten who attended preschool = (0.541, 0.819)

Answer:

The correct option is D. a = -2 and b = 4.

Step-by-step explanation:

Consider the provided equation:

[tex]y=a(x-2)^2+b\ \text{and}\ y=5[/tex]

The vertex form of a quadratic is:

[tex]y= a(x-h)^2+k[/tex]

Where, (h,k) is the vertex and the quadratic opens up if 'a' is positive and opens down if 'a' is negative.

Now consider the provided option A. a = 1 and b = -4.

Since the value of a is positive the graph opens up and having vertex (2,-4). Thus graph will intersect the line y = 5.

Refer the figure 1:

Now consider the option B. a = 2 and b = 5.

Since the value of a is positive the graph opens up and having vertex (2,5). Thus graph will intersect the line y = 5.

Refer the figure 2:

Now consider the option C. a = -1 and b = 6.

Since the value of a is negative the graph opens down and having vertex (2,6). Thus graph will intersect the line y = 5.

Refer the figure 3:

Now consider the option D. a = -2 and b = 4.

Since the value of a is negative the graph opens down and having vertex (2,4). Thus graph will not intersect the line y = 5.

Refer the figure 4:

Hence, the correct option is D. a = -2 and b = 4.

Answer:  (A U B) n (B n C') = {5, 8, 11}.

Step-by-step explanation:  We are given the following sets :

U = {4, 5, 6, 7, 8, 9, 10, 11},

A = {5, 7, 9},

B = {4, 5, 8, 11}

and

C = {4, 6, 10}.

We are to find the following :

(A U B) n (B n C')

We know that for any two sets A and B,

A ∪ B contains all the elements present in set A or set B or both,

A ∩ B contains all the elements present in both A and B,

A - B contains all those elements which are present in A but not B

and

A' contains all the elements present in the universal set U but not A.

We will be suing the following rule of set of theory :

A ∩ B' = A - B.

Therefore, we have

[tex](A\cup B)\cap(B\cap C')\\\\=(A\cup B)\cap (B-C)\\\\=(\{5,7,9\}\cup\{4,5,8,11\})\cap (\{4,5,8,11\}-\{4,6,10\})\\\\=\{4,5,7,8,9,11\}\cap\{5,8,11\}\\\\=\{5,8,11\}.[/tex]

Thus,  (A U B) n (B n C') = {5, 8, 11}.

Final answer:

To solve the set operation (A U B) n (B n C'), we first find the union of A and B, then the complement of C, and the intersection of B and C'. The final step is to intersect the results of (A U B) and (B n C'), which gives us {4, 5, 8, 11}.

Explanation:

The question involves operations on sets, specifically union, intersection, and complement. We have a universal set U and subsets A, B, and C. The objective is to find the result of (A U B) n (B n C'), which involves set union (U), set intersection (n), and the complement of a set (').

First, let's find the union of sets A and B: A U B = {s, 7, 9, 4, 5, 8, 11}.

Next, we need to find the complement of set C, which is C' = {4, 5, 7, 8, 9, 11} as these are the elements of U that are not in C.

Then, identify the intersection of sets B and C': B n C' = {4, 5, 8, 11}, because these elements are common to both B and C'.

Finally, we find the intersection of the two results: (A U B) n (B n C') = {4, 5, 8, 11}.

Answer:

Step-by-step explanation:

∠C and ∠F are vertical angles, so are congruent. Then any angle complementary to one of those will also be complementary to the other.

Likewise, ∠B and ∠E are vertical angles and congruent. Any angle complementary to one of them will also be complementary to the other. Here, ∠E and ∠F are listed as complementary, so we know ∠B and ∠F will be also.

Answer:2 and 4

Step-by-step explanation:

^B and ^F

^F and ^E

Answer:

B

Step-by-step explanation:

Let x = acres of corn and

     y = acres of soybean

If the cost of corn is 60 per acre, we represent that as 60x.

If the cost of soybean is 80 per acre, we represent that as 80y.  

The addition of these 2 grains cannot go over 4800; that means that it can be 4800 on the dot, but it cannot be more.  So "less than or equal to" is our sign.  Putting that all together:

60x + 80y ≤ 4800

Answer:

[tex]f (x) = x (x + 5) (x-9)[/tex]

Step-by-step explanation:

The zeros of the polynomial are all the values of x for which the function [tex]f (x) = 0[/tex]

In this case we know that the zeros are:

[tex]x = 9,\  x-9 =0[/tex]

[tex]x = 0[/tex]

[tex]x = -5[/tex], [tex]x + 5 = 0[/tex]

Now we can write the polynomial as a product of its factors

[tex]f (x) = x (x + 5) (x-9)[/tex]

Note that the polynomial is of degree 3 because the greatest exponent of the variable x that results from multiplying the factors of f (x) is 3

The polynomial f(x) of degree 3 that has the zeros 9, 0, and -5 can be found by setting up and multiplying the factors (x-9), (x-0), and (x+5). The resulting polynomial f(x) is therefore x(x - 9)(x + 5).

To find a polynomial f(x) of degree 3 that has the given zeros, you use the fact that the zeros (or roots) of a polynomial are the values that make the polynomial equal to zero. In this case, the zeros are 9, 0, and -5. Consequently, the factors of the polynomial are (x-9), (x-0), and (x+5).

Now multiply these factors together to get the polynomial. The result is:

f(x) = x(x - 9)(x + 5).

This is the polynomial of degree 3 with the given zeros.

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Answer:

7

Step-by-step explanation:

[tex]x^2-y^2[/tex] is a difference of squares.

When factoring a difference of squares, you can use this formula [tex]u^2-v^2=(u-v)(u+v)[/tex].

So [tex]x^2-y^2[/tex] can be factored as [tex](x-y)(x+y)[/tex].

So back to the problem:

[tex]x^2-y^2=56[/tex]

Rewriting with a factored left hand side:

[tex](x-y)(x+y)=56[/tex]

We are given x-y=4 so rewriting again with this substitution:

[tex]4(x+y)=56[/tex]

Dividing both sides by 4:

[tex](x+y)=14[/tex]

So we have x+y equals 14.

We are asked to find the average of x and y which is (x+y)/2.

So since x+y=14 , then (x+y)/2=14/2=7.

Answer:

3200 people

Step-by-step explanation:

p = The frequency of the dominant gene

q = The frequency of the recessive gene

[tex]q^2=\frac{400}{10000}\\\Rightarrow q^2=0.04\\\Rightarrow q=0.2[/tex]

p+q = 1

⇒p = 1-q

⇒p = 1-0.2

⇒p = 0.8

Hardy-Weinberg formula

p² + 2pq + q² = 1

Now for heterozygous trait

2pq = 2×0.8×0.2 = 0.32

Multiplying with the population

0.32×10000 = 3200

∴ 3200 people would be expected to be heterozygous for this trait.

According to the Hardy-Weinberg formula, the expected number of people heterozygous for the eye color trait can be calculated as 768 in a population of 10,000. This calculation takes into account the dominance of the brown eye color trait and the frequency of blue-eyed individuals.

In this scenario, we are considering a single gene controlling the trait for eye color, with brown eyes being completely dominant to blue eyes.

Using the Hardy-Weinberg formula, we can calculate the expected frequency of each genotype in the population. The formula is: p^2 + 2pq + q^2 = 1.

We are given that 400 people have blue eyes in a population of 10,000. Therefore, the frequency of the recessive allele (q) can be calculated as the square root of the frequency of the blue-eyed individuals, which is 400/10,000 = 0.04.

Since brown eyes are completely dominant, the frequency of the dominant allele (p) can be calculated as 1 - q, which is 1 - 0.04 = 0.96.

Now we can calculate the expected number of heterozygous individuals (2pq): 2 * 0.96 * 0.04 * 10,000 = 768.

Therefore, we would expect 768 people to be heterozygous for the eye color trait in this population.

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Answer:

The population would be 781.

The population is growing with the rate of 12.50 persons/yr.

Step-by-step explanation:

Since, the formula for calculating the population, increasing with a rate per period,

[tex]A=P(1+r)^{n}[/tex]

Where, P is the initial population,

r is the rate per period,

n is the number of period,

t is the total years,

Here, P = 500, r = 25 % = 0.25, n = 2 ( the number of '10 year period' of in 20 years is 2 )

Hence, the population in 20 years would be,

[tex]A=500(1+0.25)^2=500(1.25)^2=781.25\approx 781[/tex]

Now, the rate of increasing per 10 year is 25 %,

⇒ The rate of increasing per year is 2.5 %,

Thus, the growing people per year = 2.5 % of 500 = 0.025 × 500 = 12.50

Hence, the population is growing at 12.50 person per year.

Answer:

  29 months

Step-by-step explanation:

My TVM solver says that balance will be reached after about 29 monthly payments.

This question pertains to compound interest. You are depositing $270 monthly into an account with a monthly compound interest rate of 4.8%. By using the compound interest formula with logarithmic adjustments for monthly deposits, you can determine how long it will take you to save $8200.

The subject of the question is how long it would take to save up $8,200 for a boat by making $270 monthly deposits into an account that has a monthly compound interest rate of 4.8%. This is a question of compound interest. The formula for compound interest is A = P (1 + r/n)^(nt), where A is the total amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal form), n is the number of times that interest is compounded per year, and t is the time the money is invested for, in years.

In this case, we require to find 't' when we have A = $8200 , P = $270 (deposited every month), r = 4.8% (in decimal form, it becomes 0.048) and n = 12 (compounded monthly). However, as $270 is getting compounded every month, a slightly adjusted formula to calculate the number of months, t is required which is t = [log(A/P)] /[n * log(1 + r/n)]. By substituting A = $8200 and P = $270 and other values to this formula, we can find the time needed. This would require logarithmic math which is done usually in high school math courses or higher.

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Answer: The sum of five consecutive even numbers for this sequence is 20n.

Step-by-step explanation:

Since we have given that

Number of consecutive even numbers = 5

Middle value = 4n

Since there are 5 consecutive even numbers:

4n-4,4n-2,4n,4n+2,4n+4

So, Sum of five consecutive even numbers would be

[tex]4n-4+4n-2+4n+4n+2+4n+4\\\\=20n[/tex]

Hence, the sum of five consecutive even numbers for this sequence is 20n.

Answer:

The number of combinations are made when one person taken at a time out of four person=4.

Step-by-step explanation:

We are given that a six person committee composed of Alice,Ben,Connie, Dolph,Egbert, and Francisco.

We have to select three persons out of six persons one is chairperson,secretary and treasurer.

We have to find the number of combinations of different officer are made when two persons Dolph and Francisco must hold office.

Now, if two persons Dolph and Francisco must hold the office then we have to select only one member out of 4 persons.

Therefore ,using combination formula

[tex]\binom{n}{r}[/tex]=[tex]\frac{n!}{r!(n-r)!}[/tex]

We have n=4 and r=1 then

The number of combination of different officer are made =[tex]\binom{4}{1}[/tex]

The number of combination of different officer are made=[tex]\frac{4!}{1!(4-1)!}[/tex]

The number of combination of different officer are made=[tex]\frac{4\times 3!}{3!}[/tex]

The number of combination of different officer are made=4

Hence, the number of combinations are made when one person taken at a time out of four person=4.

Answer: 4

Answer:

Height of the mountain is 5108.80 feet.

Step-by-step explanation:

From the figure attached, h is the height of a mountain AB.

At a point C angle of elevation of the mountain is 24°

Now survey team gets closer to the mountain by 1000 feet then angle of elevation is 26°.

Now from ΔABC,

tan24 = [tex]\frac{h}{x+1000}[/tex]

0.445 = [tex]\frac{h}{x+1000}[/tex]

h = 0.445(x + 1000)------(1)

From ΔABD,

tan26 = [tex]\frac{h}{x}[/tex]

0.4877 = [tex]\frac{h}{x}[/tex]

h = 0.4877x -----(2)

Now we equation 1 and equation 2

0.4452(x + 1000) = 0.4877x

0.4877x - 0.4452x = 1000(0.4452)

0.0425x = 445.20

x = [tex]\frac{445.20}{0.0425}[/tex]

x = 10475.29 feet

Now we plug in the value of x in equation 2.

h = (10475.29)×(0.4877)

h = 5108.80 feet

Therefore, height of the mountain is 5108.80 feet

Answer:

=10km/h

Step-by-step explanation:

Let motor boat speed be represented by x and current y

The speed upstream = Motor boats speed - rate of current

=x-y

The net speed down stream = Motor boats speed + rate of current

=x+y

Let us find the speed upstream =distance/ time taken

=150km/5hrs

=30km/h

Speed down stream= 150km/3h

=50 km/h

The problem forms simultaneous equations.

x-y=30

x+y=50

Using elimination method we solve the equations.

Add the two equations to eliminate y.

2x=80

x=40

Current, y= 50-x

=10km/h

Answer:

1) [tex]40\ \frac{km}{h}[/tex]

2) [tex]10\ \frac{km}{h}[/tex]

Step-by-step explanation:

 Let' call "b" the speed of the motorboat and "c" the speed of the current.

We know that:

[tex]V=\frac{d}{t}[/tex]

Where "V" is the speed, "d" is distance and "t" is time.

Then:

[tex]d=V*t[/tex]

We know that distance traveled upstream is 150 km and the time is 5 hours. Then, we set up the folllowing equation:

 [tex]5(b-c)=150[/tex]  (Remember that in the trip upstream the speed of the river is opposite to the motorboat)

For the return trip:

 [tex]3(b+c)=150[/tex]  

 By solving the system of equations, we get:

- Make both equations equal to each other and solve for "c".

[tex]5(b-c)=3(b+c)\\\\5b-5c=3b+3c\\\\5b-3b=3c+5c\\\\2b=8c\\\\c=\frac{b}{4}[/tex]

- Substitute "c" into the any original equation and solve for "b":

[tex]5(b-\frac{b}{4})=150\\\\\frac{3}{4}b=30\\\\b=40\ \frac{km}{h}[/tex]

- Substitute "b" into [tex]c=\frac{b}{4}[/tex]:

[tex]c=\frac{40}{4}\\\\c=10\ \frac{km}{h}[/tex]

Answer:

Step-by-step explanation:

Using linear differential equation method:

\frac{\mathrm{d} y}{\mathrm{d} x}+3y=e^5^x

I.F.= [tex]e^{\int {Q} \, dx }[/tex]

I.F.=[tex]e^{\int {3} \, dx }[/tex]

I.F.=[tex]e^{3x}[/tex]

y(x)=[tex]\frac{1}{e^{3x}}[\int {e^{5x}} \, dx+c][/tex]

y(x)=[tex]\frac{e^{2x}}{5}+e^{-3x}\times c[/tex]

substituting x=2

c=[tex]\frac{25-e^4}{5e^{-6}}[/tex]

Now

y=[tex]\frac{e^{2x}}{5}+e^{-3x}\times \frac{25-e^4}{5e^{-6}}[/tex]

Step-by-step explanation:

The probability of getting tails each time is 1/2.  Seven times in a row, the probability is:

P = (1/2)^7

P = 0.0078125

the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

The probability of obtaining seven tails in a row when flipping a coin can be calculated using the principle of independent events in probability. Each flip of the coin is an independent event with two possible outcomes: heads or tails.

Probability can be calculated by using the formula:

[tex]P = \frac{number\ of\ desired\ outcomes}{total\ outcomes}[/tex]

here the desired outcome is 1 as we only need tails so we can say that:

[tex]P(tails) = \frac{1}{2} = 0.5[/tex]

To find the probability of obtaining seven tails in a row, you need to multiply the probability of getting tails on each individual flip:

[tex]Probability (7\ tails\ in\ a\ row) = 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5\\\\Probability (7\ tails\ in\ a\ row) = (0.5)^7 \approx 0.0078125[/tex]

Therefore, the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

Answer:

He invest for 2 years.

Step-by-step explanation:

Given : Suppose you have $1,950 in your savings account at the end of a certain period of time. You invested $1,700 at a 6.88% simple annual interest rate.

To find : How long, in years, did you invest your money?

Solution :

Applying simple interest formula,

[tex]A=P(1+r)^t[/tex]

Where, A is the amount A=$1950

P is the principal P=$1700

r is the interest rate r=6.88%=0.0688

t is the time

Substitute the values in the formula,

[tex]1950=1700(1+0.0688)^t[/tex]

[tex]\frac{1950}{1700}=(1.0688)^t[/tex]

[tex]1.147=(1.0688)^t[/tex]

Taking log both side,

[tex]\log(1.147)=\log ((1.0688)^t)[/tex]

Applying logarithmic formula, [tex]\log a^x=x\log a[/tex]

[tex]\log(1.147)=t\log (1.0688)[/tex]

[tex]t=\frac{\log(1.147)}{\log (1.0688)}[/tex]

[tex]t=2.06[/tex]

Approximately, He invest for 2 years.

Answer: The value of b = 0.99

The probability that a day requires more water than is stored in city reservoirs is 0.161.

Step-by-step explanation:

Given : Water use in the summer is normally distributed with

[tex]\mu=310.4\text{ million gallons per day}[/tex]

Standard deviation : [tex]\sigma=40 \text{ million gallons per day}[/tex]

Let x be the combined storage capacity requires by the reservoir on a random day.

Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{350-310.4}{40}=0.99[/tex]

The probability that a day requires more water than is stored in city reservoirs is  :

[tex]P(x>350)=P(z>0.99)=1-P(z<0.99)\\\\=1-0.8389129=0.1610871\approx0.161[/tex]    

Hence, the probability that a day requires more water than is stored in city reservoirs is 0.161

Answer:

The standard form of required line is x-y=7.

Step-by-step explanation:

The standard form of a line is

[tex]ax+by=c[/tex]

Where, a,b,c are integers with no factor common to all three and a≥0.

The give equation of line is

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

Here a=1 and b=1.

The slope of a standard line is

[tex]m=\frac{-a}{b}[/tex]

[tex]m_1=\frac{-1}{1}=-1[/tex]

The product of slops of two perpendicular lines is -1.

[tex]m_1\cdot m_2=-1[/tex]

[tex](-1)\cdot m_2=-1[/tex]

[tex]m_2=1[/tex]

The slope of required line is 1.

The point slope form of a line is

[tex]y-y_1=m(x-x_1)[/tex]

Where, m is slope.

The slope of required line is 1 and it passes through the point (1,-6). So, the equation of required line is

[tex]y-(-6)=1(x-1)[/tex]

[tex]y+6=x-1[/tex]

Add 1 on each side.

[tex]y+7=x[/tex]

Subtract y from both the sides.

[tex]7=x-y[/tex]

Therefore the standard form of required line is x-y=7.

Answer: There is 49.28% of cumulative markup for the department at the end of April.

Step-by-step explanation:

Since we have given that

Price of opening inventory = $170,000

Mark up rate = 48%

Amount of mark up is given by

[tex]\dfrac{48}{100}\times 170000\\\\=\$81600[/tex]

Price of additional merchandise = $80000

Mark up rate = 52%

Amount of mark up is given by

[tex]\dfrac{52}{100}\times 80000\\\\=\$41600[/tex]

So, total mark up would be

$81600 + $41600 = 123200

So, the cumulative markup percentage for the department at the end of April is given by

[tex]\dfrac{123200}{170000+80000}\times 100=\dfrac{123200}{250000}\times 100=49.28\%[/tex]

Hence, there is 49.28% of cumulative markup for the department at the end of April.

The scientist can determine the amount of Solution A and Solution B required by setting up and solving a system of two linear equations representing the total solution volume and the total salt amount.

Lets let the amount of Solution A the scientist will use be x and the amount of Solution B she will also use be y. We know that x + y = 110 ounces because her final mix should be 110 ounces. Also, we know that 0.4x + 0.65y = 0.55*(x+y) = 60.5 because the amount of salt from Solution A and Solution B should add up to the amount of salt in the final mixture. Solving this system of linear equations to obtain the values for x and y, gives the required amounts of Solution A and Solution B needed.

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Answer:

The correct option is C. Fredo

Step-by-step explanation:

In a Mario Puzo's fictional novel named The Godfather, Frederico Corleone or Fredo is a fictional character. In the novel, Fredo's father is killed by the assasins. Witnissing his father being shot, Fredo goes into a shock.

To protect and aid Fredo's recovery, his elder brother Sonny, sends him to Las Vegas.

Therefore, Fredo is sent to Las Vegas

[tex]\bf cos^2(x)+2cos(x)+1=0\implies \stackrel{\textit{let's notice, this is simply }ax^2+bx+c=0}{[cos(x)]^2+2cos(x)+1=0} \\[2em] [cos(x)+1][cos(x)+1]=0 \\\\[-0.35em] ~\dotfill\\\\ cos(x)+1=0\implies cos(x)=-1\implies x=cos^{-1}(-1)\implies \stackrel{\textit{for the range }[0,2\pi ]}{x=\pi } \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{all solutions}}{x=\pi +2\pi n~~,~~ n \in \mathbb{Z}}~\hfill[/tex]

Final answer:

The probability of the first letter being 'A' or 'Z' in the word ARIZONA is 0.143, the probability of it being a vowel is 0.429, and for the letter 'H', which is not present in the word, the probability is 0.

Explanation:

The probability that the first letter is 'A' in a random arrangement of the letters in the word ARIZONA is simply the number of 'A's divided by the total number of letters. Since there is one 'A' out of seven letters, the probability is 1/7, which in decimal form is approximately 0.143, rounded to the nearest thousandth.

Similarly, for the letter 'Z', since there's one 'Z' in the word ARIZONA, the probability is also 1/7, which is about 0.143 when rounded to the nearest thousandth.

The probability that the first letter is a vowel (A, I, or O in ARIZONA) involves adding the probabilities of each individual vowel being the first letter. There are three vowels out of seven letters, so the probability is 3/7, which is approximately 0.429, rounded to the nearest thousandth.

Since the letter 'H' is not in the word ARIZONA, the probability that the first letter is 'H' is 0.

Answer:

1. No

2. 7

3. b=54

Step-by-step explanation:

1. We can answer this by assuming a number.

Let our number be 23

Multiplying its digits = 6

Multiplying the result with initial number = 6 * 23 = 138

So it is not possible to get 1716 as a result by thinking of a natural number and applying the operation mentioned in the question.

2. What is the largest prime factor of the factorial 49! ?

First of all we have to define prime factors:

Prime factors are the prime numbers that can be multiplied together to equal the original number.

The factors of 49 are: 1, 7, 49

7 is the largest prime factor of 49

3. The GCD(a, b) = 18, LCM(a, b) = 108. If a=36, findb.

We will use the relationship:

[tex]GCD * LCM = a*b\\18*108=36b\\1944=36b\\b= \frac{1944}{36} \\b=54[/tex]

..

Answer: 0.125

Step-by-step explanation:

Given: A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed in interval (50,52).

∴ The probability density function of X will be :-

[tex]f(x)=\dfrac{1}{b-a}=\dfrac{1}{52-50}=\dfrac{1}{2}[/tex]

The required probability will be:-

[tex]P(51.25<x<51.5)=\int^{51.5}_{51.25}f(x)\ dx\\\\=\dfrac{1}{2}\int^{51.5}_{51.25}\ dx\\\\=\dfrac{1}{2}[x]^{51.5}_{51.25}\\\\=\dfrac{1}{2}(51.5-51.25)=\dfrac{0.25}{2}=0.125[/tex]

Hence, the probability that a given class period runs between 51.25 and 51.5 minutes =0.125