Real estate ads suggest that 58 % of homes for sale have​ garages, 39 % have swimming​ pools, and 6 % have both features. What is the probability that a home for sale has ​a) a pool or a​ garage? ​b) neither a pool nor a​ garage? ​c) a pool but no​ garage?

Answer:

2100

Step-by-step explanation:

50*18=900

900+1,200=2100

To find the population in 2018, we calculate the total increase from 2000 to 2018 by multiplying the yearly increase of 50 people by 18 years, resulting in an additional 900 people. Adding this to the initial population of 1,200 people gives us a total population of 2,100 people in 2018.

If a population is recorded at 1,200 in the year 2000 and increases at a steady rate of 50 people each year, we can calculate the population in 2018 using a linear growth model. First, we need to determine the number of years between 2000 and 2018, which is 18 years. Next, we multiply the annual increase (50 people) by the number of years (18) to find the total increase over this period.

The calculation would be as follows:

We then add this total increase to the initial population to get the population in 2018:

The population in 2018 would be 2,100 people.

Answer:

$100 * (1 + 6.8%/12)^216 + $100*(1+6.8%/12)^215 + ... + $100*(1+6.8%/12)^1  

Now note that  

x + x^2 + x^3 + ... + x^N = x ( 1 + x + ... + x^(N-1) )  

= x ( (x^N -1)/(x-1) )  

Here, x = 1+6.8%1 = 1.00566666 and N = 216, so  

$100 * ( 1.00566666 ( 1.00566666^216 -1) / 0.00566666 )  

= $ 42398.33  

The total interest earned is $42,398 - $21,600 = $20,798

Step-by-step explanation:

The generic formula used in this compound interest calculator is V = P(1+r/n)^(nt)

V = the future value of the investment

P = the principal investment amount

r = the annual interest rate

n = the number of times that interest is compounded per year

t = the number of years the money is invested for

Answer:

The amount of water released during the​ week-long flood was 15,180,400,000 cubic feet per second.

Step-by-step explanation:

How many seconds are there in a week?

Each minute has 60 seconds

Each hour has 60 minutes

Each day has 24 hours

Each week has 7 days. So

60*60*24*7 = 604,800

A week has 604,800 seconds.

Water was released at a rate of 25,100 cubic feet per second.

In a week(604,800 seconds)

604,800*25,100 = 15,180,400,000

The amount of water released during the​ week-long flood was 15,180,400,000 cubic feet per second.

Answer:

0.48

Step-by-step explanation:

Probability that the first person chooses a hotel

⁵C₁

[tex]^5C_1=\frac{5!}{(5-1)!1!}\\=\frac{120}{24}=5[/tex]

Probability that the second person chooses a different hotel

⁴C₁

[tex]^4C_1=\frac{4!}{(4-1)!1!}\\=\frac{24}{6}=4[/tex]

because the choice of hotels has reduced by 1 as one hotel is occupied by the first person

Probability that the second person chooses a different hotel

³C₁

[tex]^3C_1=\frac{3!}{(3-1)!1!}\\=\frac{6}{2}=3[/tex]

because the choice of hotels has reduced by 2 as two different hotels are occupied by the first person and second person

∴ The favorable outcomes are =⁵C₁×⁴C₁׳C₁=5×4×3=60

The total number of outcomes=5³=125

∴Probability that they each check into a different hotel=60/125=0.48

Answer:

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

Step-by-step explanation:

We are to compare the means of two samples. Since only sample std deviations are used, we have to use t test for this hypothesis

H0: Means are equal

Ha: Means are not equal

(Two tailed test at 5% )

Difference between means [tex]M1-M2 = -2587[/tex]

Std deviation combined = 3856

Std error for difference = 460.88

t statistic[tex]= -2587/460.88=-5.613[/tex]

p value =0

Since p <0.05 reject null hypothesis.

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

Final answer:

To calculate the monetary value of Bill Mason's job offers, you subtract the taxes from the salary and add the nontaxable benefits. Job 1 results in a total monetary value of $36,500, while Job 2 is higher, with a total monetary value of $37,425. Therefore, Job 2 offers a higher total monetary value.

Explanation:

Bill Mason is considering two job offers, each with a different combination of salary and nontaxable benefits. To determine the monetary value for both jobs, considering a 25 percent tax rate, we must calculate the Net Annual Income for each job separately. Net Annual Income is the salary after taxes have been subtracted. Nonetheless, nontaxable benefits do not affect the Net Annual Income as they do not get taxed.

Calculations for Job 1:

Gross Salary: $41,300

Taxable Income (Tax Rate 25%): 0.25 x $41,300 = $10,325

Net Salary after Tax: $41,300 - $10,325 = $30,975

Nontaxable Benefits: $5,525

Total Monetary Value: Net Salary + Benefits = $30,975 + $5,525 = $36,500

Calculations for Job 2:

Gross Salary: $40,400

Taxable Income (Tax Rate 25%): 0.25 x $40,400 = $10,100

Net Salary after Tax: $40,400 - $10,100 = $30,300

Nontaxable Benefits: $7,125

Total Monetary Value: Net Salary + Benefits = $30,300 + $7,125 = $37,425

The final answer for the total monetary value of Job 1 is $36,500 and for Job 2 is $37,425. As per the given calculations, Job 2 offers a higher total monetary value compared to Job 1 when taking into account the salary after tax plus nontaxable benefits. This explanation should provide clarity on how to assess job offers based on their financial merits.

Answer: [tex](0.52,\ 0.68)[/tex]

Step-by-step explanation:

The formula for a [tex]\alpha[/tex]- level confidence interval for the population proportion:-

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

Given : n = 225 ; p = 0.60 ; [tex]\alpha= 1-0.99=0.01[/tex]

By using the given information , the confidence interval for the population proportion:-

[tex]0.6\pm z_{0.005}\times\sqrt{\dfrac{0.6(1-0.6)}{225}}\\\\=0.6\pm(2.576)(0.0327)\\\\=0.6\pm0.08\\\\=0.6-0.08,\ 0.6+0.08\\\\=(0.52,\ 0.68)[/tex]

Hence, the resulting confidence interval is [tex](0.52,\ 0.68)[/tex] .

To construct a 99% confidence interval for the population proportion, we can use the sample proportion, critical value, and standard error. The resulting confidence interval is (0.516, 0.684).

To calculate a confidence interval for a population proportion, we can use the formula:

CI = sample proportion ± (critical value)(standard error)

Given that the sample proportion is 60% (0.6) and the confidence level is 99%, we need to find the critical value corresponding to a 99% confidence level. Using a normal distribution, the critical value is approximately 2.576. The standard error can be calculated using the formula:

SE = √((sample proportion)(1 - sample proportion) / sample size)

Let's assume the sample size is 225. Plugging these values into the formula, we can calculate the standard error:

SE = √((0.6)(1 - 0.6) / 225) ≈ √(0.24 / 225) ≈ √0.001067 ≈ 0.0326

Now we can calculate the confidence interval:

CI = 0.6 ± (2.576)(0.0326) ≈ 0.6 ± 0.084

Therefore, the 99% confidence interval for the population proportion is approximately (0.516, 0.684).

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Answer:

General Solution is [tex]y=x^{3}+cx^{2}[/tex] and the particular solution is  [tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]

Step-by-step explanation:

[tex]x\frac{\mathrm{dy} }{\mathrm{d} x}=x^{3}+3y\\\\Rearranging \\\\x\frac{\mathrm{dy} }{\mathrm{d} x}-3y=x^{3}\\\\\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{3y}{x}=x^{2}[/tex]

This is a linear diffrential equation of type

[tex]\frac{\mathrm{d} y}{\mathrm{d} x}+p(x)y=q(x)[/tex]..................(i)

here [tex]p(x)=\frac{-2}{x}[/tex]

[tex]q(x)=x^{2}[/tex]

The solution of equation i is given by

[tex]y\times e^{\int p(x)dx}=\int  e^{\int p(x)dx}\times q(x)dx[/tex]

we have [tex]e^{\int p(x)dx}=e^{\int \frac{-2}{x}dx}\\\\e^{\int \frac{-2}{x}dx}=e^{-2ln(x)}\\\\=e^{ln(x^{-2})}\\\\=\frac{1}{x^{2} } \\\\\because e^{ln(f(x))}=f(x)]\\\\Thus\\\\e^{\int p(x)dx}=\frac{1}{x^{2}}[/tex]

Thus the solution becomes

[tex]\tfrac{y}{x^{2}}=\int \frac{1}{x^{2}}\times x^{2}dx\\\\\tfrac{y}{x^{2}}=\int 1dx\\\\\tfrac{y}{x^{2}}=x+c[/tex][tex]y=x^{3}+cx^{2[/tex]

This is the general solution now to find the particular solution we put value of x=2 for which y=6

we have [tex]6=8+4c[/tex]

Thus solving for c we get c = -1/2

Thus particular solution becomes

[tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]

Answer: (a) 0.8641

(b) 0.1359

Step-by-step explanation:

Given : The monthly worldwide average number of airplane crashes of commercial airlines [tex]\lambda= 3.5[/tex]

We use the Poisson  distribution for the given situation.

The Poisson distribution formula for probability is given by :-

[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]

a) The probability that there will be at least 2 such accidents in the next month is given by :-

[tex]P(X\geq2)=1-(P(X=1)+P(X=0))\\\\=1-(\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!})\\\\=1-(0.1358882254)=0.8641117746\approx0.8641[/tex]

b) The probability that there will be at most 1 accident in the next month is given by :-

[tex]P(X\leq1)=(P(X=1)+P(X=0))\\\\=\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!}\\\\=0.1358882254\approx0.1359[/tex]

Answer:[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]

Step-by-step explanation:

Given a total of 12 diesel engines

out of which 2 are defective

Company have to choose 5 good engines to ship.

Therefore no of ways in good engines are selected is [tex]^{10}C_5[/tex]

no of ways in which any 5 engines are selected from a total of 12 engines is

[tex]^{12}C_5[/tex]

Therefore the required probability =[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]

=[tex]\frac{7}{22}[/tex]=0.318

Answer:

2 ties of 4 feet and 1 tie of 5 feet

2 ties of 6 feet and 1 tie of 9 feet

Step-by-step explanation:

Given data

2 ties = 4 feet

2 ties = 6 feet

1 tie = 9 feet

1 tie = 5 feet

to find out

what are the total possible dimensions for each set of flower beds

solution

there are many combination but

the best possible combination are for 1st triangular

when we use 2 ties of 4 feet and 1 tie of 5 feet

and

for 2nd triangular

when we use 2 ties of 6 feet and 1 tie of 9 feet

only these are best combination

Answer:

Step-by-step explanation:

Given that in the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College.

Table is prepared as follows

                            Bus coll.     Lib Arts coll      Educ. coll        Total

Observed                  90                120                 90                  300

Expected p.c.             35                 35                 30                  100

Expected number     105                105                 90                 300

Percent*300/100

Hence expected frequency for business college = 105

Answer: a) 0.2   b) 0.6

c) The event of selecting red tulip is not likely to occur.

The event of selecting purple tulip is likely to occur.

Step-by-step explanation:

Given : Total number of tulips = 100

The number of red tulips = 20

The number of purple tulips =60

The probability that a randomly selected tulip bulb is​ red :-

[tex]\dfrac{\text{Number of red tulips}}{\text{Total tulips}}\\\\=\dfrac{20}{100}=0.2[/tex]

Since 0.2 is less than 0.5.

It means that the event of selecting red tulip is not likely to occur.

The probability that a randomly selected tulip bulb is​ purple :-

[tex]\dfrac{\text{Number of purple tulips}}{\text{Total tulips}}\\\\=\dfrac{60}{100}=0.6[/tex]

Since 0.6 is more than 0.5.

It means that the event of selecting purple tulip is likely to occur.

Final answer:

The probability of selecting a red tulip bulb is 20%, and the probability of selecting a purple tulip bulb is 60%. These probabilities reflect the likelihood of picking a bulb of a particular color at random from the bag.

Explanation:

The question involves calculating the probability of selecting a red or purple tulip bulb from a bag.

Probability of Selecting a Red Tulip Bulb

The probability, P(Red), is calculated by dividing the number of red bulbs by the total number of bulbs:

P(Red) = Number of Red Bulbs / Total Number of Bulbs = 20 / 100 = 0.2

Probability of Selecting a Purple Tulip Bulb

Similarly, the probability, P(Purple), is:

P(Purple) = Number of Purple Bulbs / Total Number of Bulbs = 60 / 100 = 0.6

Interpretation of Probabilities

These probabilities indicate that there is a 20% chance of selecting a red bulb and a 60% chance of selecting a purple bulb from the bag. The higher the probability, the more likely it is to select a bulb of that color at random.

Answer: 38

Step-by-step explanation:

                 i.e. [tex]P-Q=n(P)-n(P\cap Q)[/tex]

Let A be the number of students who correctly answered the first question and B be the number of students who correctly answered the second question .

Given : [tex]n(A)=76[/tex]

[tex]n(B)=60[/tex]

[tex]n(A\cap B)=38[/tex]

Then the number of students who answered the first question correctly, but not the second is given by :-

[tex]A-B=n(A)-n(A\cap B)\\\\=76-38=38[/tex]

Hence, the number of students who answered the first question correctly, but not the second is 38.

Answer: 34650

Step-by-step explanation:

The number of permutations of n objects, where one object is repeated [tex]n_1[/tex] times , another is repeated [tex]n_2[/tex] times and so on is :

[tex]\dfrac{n!}{n_1!n_2!....n_k!}[/tex]

Given : The number of letters in string MISSISSIPPI = 11

Here I is repeated 4 times, S is repeated 4 times and P is repeated 2 times.

Then , the number of different strings can be made from the letters in MISSISSIPPI, using all letters is given by :-

[tex]\dfrac{11!}{4!4!2!}=34650[/tex]

Therefore , there are 34650 different strings can be made from the letters in MISSISSIPPI.

To find the number of different strings that can be formed from the letters in MISSISSIPPI, we use the formula for permutations of a multi set. The total is 34,650 unique permutations.

The question asks how many different strings can be made from the letters in MISSISSIPPI, using all letters.

This is a problem of calculating permutations of a multi set.

The word MISSISSIPPI has 11 letters with the following counts of each letter: M occurs 1 time, I occurs 4 times, S occurs 4 times, and P occurs 2 times.

To find the total number of unique permutations, we use the formula for the permutations of a multiset:

The formula is:

Permutations = n! / (n1! * n2! * ... * nk!),

where n is the total number of items to arrange, and n1, n2, ..., nk are the counts of each distinct item.

For MISSISSIPPI:

Total permutations = 11! / (1! * 4! * 4! * 2!) = 34650

Therefore, there are 34,650 different strings that can be made from the letters in MISSISSIPPI using all the letters.

Answer:

Step-by-step explanation:

Given lines in parametric form

line [tex]L_1[/tex]

[tex]\frac{x}{3}=\frac{y-1}{-2} =\frac{z-2}{-3}[/tex]

direction vector of [tex]L_1 v_1=<3,-2,-3 >[/tex]

Line [tex]L_2[/tex]

direction vector of [tex]L_2 v_2=<-9,6,9 >[/tex]

therefore

[tex]v_2=-3v_1[/tex]

thus lines are parallel.

(ii)distance between two lines is

[tex]L_2[/tex] is given by

[tex]\frac{x-3}{-9}=\frac{y-1}{6} =\frac{z-4}{9}[/tex]=s

[tex]\frac{x-3}{-3}=\frac{y-1}{2} =\frac{z-4}{3}[/tex]=3s

[tex]\left | \frac{\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1 \\ a_1&b_1&c_1 \\ a_2&b_2&c_2\end{vmatrix}}{\sqrt{\left ( a_1b_2-a_2b_1 \right )^2+\left ( b_1c_2-b_2c_1 \right )^2+\left ( c_1a_2-c_2a_1 \right )^2}}\right |[/tex]

where [tex]a_1[/tex]=3

[tex]b_1[/tex]=-2

[tex]c_1[/tex]=-3

[tex]a_2[/tex]=-9

[tex]b_2[/tex]=6

[tex]c_2[/tex]=9

distance(d)=0 units since value of the matrix

[tex]\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1\\ a_1&b_1&c_1 \\a_2&b_2 &c_2 \end{vmatrix}[/tex]

is zero

Step-by-step explanation:

write the equation in slope intercept form

7x + 4y - 20=0

find the slope of corresponding line.

????

then

find the y intercept of corresponding line

(x,y)=???.

Answer:

$933.12

Step-by-step explanation:

This is a composite figure:  an upper rectangle and a lower one.  We need to find the volumes of each one individually, add the volumes together, then multiply the total volume by .02

The upper rectangle has a volume of:

V = 72×12×24

V = 20,736 cubic inches

The lower rectangle has a volume of:

V = 72×36×10

V = 25,920 cubic inches

The sum of the two volumes is

V = 46,656 inches cubed

Multiply that by .02:

46,656(.02) = $933.12

Answer:

Number of adults = 7

Number of children = 16

Step-by-step explanation:

Tickets for a play cost 2 pounds for a child, and 4 pounds for an adult.

Let x number of adults and y number of children.

1 child ticket cost = 2 pound

y children ticket cost = 2y pound

1 adult ticket cost = 4 pound

x adults ticket cost = 4x pound

Total number of ticket sales were 60 pounds

Therefore, 4x + 2y = 60  ------------- (1)

One adult brought 4 children with him and the remaining adults each brought 2 children with them.

Remaining number of adult whose brought 2 children = x-1

Number children = 2(x-1)

Total number of children = 2(x-1)+4

Therefore, y=2x+2 ---------------------(2)

System of equation,

 2x + y = 30

-2x + y = 2

Using augmented matrix to solve system of equation.

[tex]\begin{bmatrix}2&1&\ |30\\-2&1&|2\end{bmatrix}\\\\R_2\rightarrow R_2+R_1\\\\\begin{bmatrix}2&1& |30\\0&2&|32\end{bmatrix}\\\\R_2\rightarrow\dfrac{1}{2}R_2\\[/tex]

[tex]\begin{bmatrix}2&1&\ |30\\0&1&|16\end{bmatrix}\\\\R_1\rightarrow R_1-R_2\\\\\begin{bmatrix}2&0&\ |14\\0&1&|16\end{bmatrix}\\\\\\[/tex]

[tex]R_1\rightarrow \dfrac{1}{2}R_1\\\\\begin{bmatrix}1&0&|7\\0&1&|16\end{bmatrix}\\\\[/tex]

Now, we find the value of variable.

[tex]x=7\text{ and }y=16[/tex]

Hence, Number of adults are 7 and Number of children are 16.

Answer:

We conclude that population mean is equal to 100 at the α=0.01 level of​ significance.

Step-by-step explanation:

Given information:

Sample size, n=40

Sample mean=104.3

sample standard deviation, s=18.2

We need to check whether the population mean greater than 100 at the α=0.01 level of​ significance.

Null hypothesis:

[tex]H_0:\mu=100[/tex]

Alternative hypothesis:

[tex]H_1:\mu>100[/tex]

Let as assume that the data follow the normal distribution. It is a right tailed test.

The formula for z score is

[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]z=\frac{104.3-100}{\frac{18.2}{\sqrt{40}}}[/tex]

[tex]z=1.494263[/tex]

[tex]z\approx 1.49[/tex]

Using the standard normal table the p-value at z=1.49 and 0.01 level of significance is 0.068112.

(0.068112 > 0.01) p-value is greater than α, so we accept the null hypothesis.

Therefore, we conclude that population mean is equal to 100 at the α=0.01 level of​ significance.

Answer: 0.0336

Step-by-step explanation:

Given : The distribution of cholesterol levels in teenage boys is approximately normal with mean :[tex]\mu= 170[/tex]

Standard deviation : [tex]\sigma= 30[/tex]

The formula for z-score is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=225

[tex]z=\dfrac{225-170}{30}=1.83[/tex]

The p-value =[tex]P(z>1.83)=1-P(z<1.83)[/tex]

[tex]=1-0.966375=0.033625\approx0.0336[/tex]

The probability that a teenage boy has a cholesterol level greater than 225 =0.0336

Answer:

2/9

Step-by-step explanation:

We are given Susan (S) is 3 times as old as Paul (P) so S=3P.

And Paul (P) is 3/2 times as old as Mary (M) so P=(3/2)M.

What fraction of Susan's age is Mary?

So if S=3P and P=(3/2)M, then by substitution we have

[tex]S=3(\frac{3}{2})M[/tex]

[tex]S=\frac{9}{2}M[/tex]

Multiply both sides by the reciprocal of 9/2 resulting in:

[tex]\frac{2}{9}S=M[/tex]

This says Mary is two-ninths the age of Susan.

Answer:

b) 2/9.

Step-by-step explanation:

Let Mary's age be x , then:

Paul's age is (3/2)x.

Susan's age =  3(3/2)x = 9/2 x.

So Mary is  2/9 the age of Susan.

Answer:

The seasons would become constant. It would be equinox throughout the year.

Step-by-step explanation:

The earth would be in a state of constant equinox i.e., the length of day and night would be same in a particular place.

The season of a place would be what it is when it is normally titled at equinox.

The animal and plant life which depend on the seasons would be affected.

Snow would only occur at parts where it normally snows at equinoxes.

Answer:

Alfred has 250 socks in his collection.

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

Based on the information given to us we can see that out of All his Brown socks he has 20% in the wash and the rest are in his drawer. Meaning 80% of the Brown socks are in his drawer. So we first need to find how many Brown socks are in the wash. We can solve this using the Rule of Three property as shown in the picture below.

120 drawer   ⇒   80%

      x wash    ⇒  20%

[tex]\frac{120*20}{80} = 30[/tex]

Now that we have the amount of Brown socks in the washer we can add that to the amount in the drawer to find the total amount of Brown socks.

[tex]Br = 120+30\\Br = 150[/tex]

So we now know that there are a total of 150 Brown socks. Since the question states that the Brown socks are 60% of the total we can use the Rule of Three to find the total.

150 Brown   ⇒   60%

    T  Total   ⇒   100%

[tex]\frac{150*100}{60} = T[/tex]

[tex]250 socks = T[/tex]

Finally, we can see that Alfred has 250 socks in his collection.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Alfred owns a total of 250 socks. This was derived by first finding out that Alfred has 150 brown socks, which represent 60% of his total socks' collection. Thus, the total number of socks Alfred owns is 250.

The question deals with the calculation of a total number of socks owned by Alfred.

If we know that 120 brown socks represent 80% of all of Alfred's brown socks (because 20% of them are in the wash), we can calculate the total number of brown socks. To do this, divide 120 by 0.8, which equals 150. Hence, Alfred has 150 brown socks.

We know from the problem that the brown socks account for 60% of all his socks. Hence the total number of socks (brown and black) is calculated by dividing the total number of brown socks (150) by 0.6. After performing this percentage calculation, we find that Alfred owns 250 socks.

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Answer:

The probability is [tex]\frac{1}{9}[/tex]

Step-by-step explanation:

There are a total of 9 die in the box after she added the "cheat" die. Since there is only 1 "cheat" die in the box and chooses a die at random then the probability of her having chosen the cheat die is [tex]\frac{1}{9}[/tex] . The fact that she rolled two sixes did not affect when she choose the die therefore the probability remains as [tex]\frac{number.of.cheat.die}{total.dice}[/tex].

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

We have [tex]\lambda(21)=6[/tex], where [tex]\lambda[/tex] is the Carmichael function. So we have

[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\pmod{21}[/tex]

The powers of 5 modulo 6 follow a periodic pattern

[tex]5^1\equiv5\pmod6[/tex]

[tex]5^2\equiv25\equiv1\pmod6[/tex]

[tex]5^3\equiv1\cdot5\equiv5\pmod6[/tex]

[tex]5^4\equiv5^2\equiv1\pmod6[/tex]

and so on, with odd powers of 5 equivalent to 5 modulo 6. So

[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\equiv10^5\pmod{21}[/tex]

The rest is easy to deal with. We have

[tex]10^2\equiv16\pmod{21}[/tex]

[tex]10^3\equiv160\equiv13\pmod{21}[/tex]

[tex]10^4\equiv130\equiv4\pmod{21}[/tex]

[tex]10^5\equiv40\equiv19\pmod{21}[/tex]

and so the answer is 19.

Answer:

System A has 2 real solutions.

System B has 0 real solutions.

System C has 1 real solutions.

Step-by-step explanation:

If the graph of system of equation intersect each other at n points then the system of equation has n real solutions.

System A:

[tex]x^2+y^2=17[/tex]               .... (1)

[tex]y=-\frac{1}{2}x[/tex]             .... (2)

Plot the graph of these equations.

From graph (1) it is clear that the graph of equation (1) and (2), intersect each other at two points, (-3.688,1.844) and (3.688,-1.844).

Therefore, System A has two real solutions.

System B:

[tex]y=x^2-7x+10[/tex]               .... (3)

[tex]y=-6x+5[/tex]             .... (4)

Plot the graph of these equations.

From graph (2) it is clear that the graph of equation (3) and (4), never intersect each other.

Therefore, System B has 0 real solutions.

System C:

[tex]y=-2x^2+9[/tex]               .... (5)

[tex]8x-y=-17[/tex]             .... (6)

Plot the graph of these equations.

From graph (3) it is clear that the graph of equation (5) and (6), intersect each other at one points, (-2,1).

Therefore, System C has 1 real solutions.

Answer:

You will requiere 0.34375 oz of the solution for 22 ounces of water.

Step-by-step explanation:

We should start by matching the units: in the example we have 2 gallons of water, while the question refers to 22 ounces of water.

So, by the equivalence we are given, we now know that 2 gallons of water are equal to 256 ounces.

From this point the question can be solved using an arithmetic method known as cross-multiply (sometimes refered as the Rule of Three). This consist of wrinting an equation of the form:

[tex]\frac{a}{b} = \frac{c}{x}\\[/tex]

Where b is the amount of solution required for a ounces of water, and x (unknown variable) is the amount of solution for c ounces of water.

So we have the next equation:

[tex]\frac{256}{4} =\frac{22}{x}[/tex]

We apply the correspondent arithmetic rules so we can calculate the x as follows:

[tex]x=\frac{(4)(22)}{256}[/tex]

[tex]x=\frac{88}{256}[/tex]

[tex]x=0.34375[/tex]

This way we can reply that 0.34375 oz of solution are required for 22 oz of water.

We are asked to prove by the method of mathematical induction that:

      3n(n+1) is divisible by 6 for all positive integers.

[tex]3n(n+1)=3\times 1(1+1)\\\\i.e.\\\\3n(n+1)=3\times 2\\\\i.e.\\\\3n(n+1)=6[/tex]

which is divisible by 6.

Hence, the result is true for n=1

i.e. 3k(k+1) is divisible by 6.

Let n=k+1

then

[tex]3n(n+1)=3(k+1)\times (k+1+1)\\\\i.e.\\\\3n(n+1)=3(k+1)(k+2)\\\\i.e.\\\\3n(n+1)=(3k+3)(k+2)\\\\i.e.\\\\3n(n+1)=3k(k+2)+3(k+2)\\\\i.e.\\\\3n(n+1)=3k^2+6k+3k+6\\\\i.e.\\\\3n(n+1)=3k^2+3k+6k+6\\\\i.e.\\\\3n(n+1)=3k(k+1)+6(k+1)[/tex]

Since, the first term:

[tex]3k(k+1)[/tex] is divisible by 6.

( As the result is true for n=k)

and the second term [tex]6(k+1)[/tex] is also divisible by 6.

Hence, the sum:

[tex]3k(k+1)+6(k+1)[/tex]  is divisible by 6.

Hence, the result is true for n=k+1

Hence, we may say that the result is true for all n where n belongs to positive integers.

To prove by induction that the expression 3n(n+1) is divisible by 6, we start with the base case of n=1, which is divisible by 6, and then show that if it holds for an integer k, it also holds for k+1. By factoring and using the induction hypothesis, we demonstrate the expression's divisibility by 6 for all positive integers.

Proof by Induction of Divisibility by 6

To prove by induction that 3n(n+1) is divisible by 6 for all positive integers, we follow two steps: the base case and the inductive step.

Base Case

Let's check for n=1:

Inductive Step

Assume the statement holds for a positive integer k, so 3k(k+1) is divisible by 6.

Now, we must show that 3(k+1)((k+1)+1) = 3(k+1)(k+2) is also divisible by 6.

Factoring out the common term we get:

Notice that (k+1) is an integer, hence 3(k+1) is divisible by 3. If k is even, then k+1 is odd, so 3(k+1) is still divisible by 3 but not necessarily by 6. However, if k is odd, k+1 is even and 3(k+1) is divisible by 6. Since the divisible by 3 part is always true, and the divisible by an additional factor of 2 part is true every other time, the sum 6m + 3(k+1) is divisible by 6 regardless of whether k is odd or even. This is because adding a multiple of 6 to either another multiple of 6 or a multiple of 3 always results in a multiple of 6.

Therefore, 3n(n + 1) is divisible by 6 for all positive integers n.

Answer:

80,00[tex]ft^{2}[/tex]

Step-by-step explanation:

According to my research, the formula for the Area of a rectangle is the following,

[tex]A = L*W[/tex]

Where

Since the building wall is acting as one side length of the rectangle. We are left with 1 length and 2 width sides. To maximize the Area of the parking lot we will need to equally divide the 800 ft of fencing between the Length and Width.

800 / 2 = 400ft

So We have 400 ft for the length and 400 ft for the width. Since the width has 2 sides we need to divide 60 by 2.

400/2 = 200 ft

Now we can calculate the maximum Area using the values above.

[tex]A = 400ft*200ft[/tex]

[tex]A = 80,000ft^{2}[/tex]

So the Maximum area we are able to create with 800 ft of fencing is 80,00[tex]ft^{2}[/tex]

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

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The largest possible parking lot, given 800 feet of fencing with one side not requiring a fence, would be a rectangular lot with dimensions of 400 feet by 200 feet.

We're dealing with a rectangular parking lot here where one of its sides is bordered by a building, so it doesn't need a fence. We have 800 feet of fence available for the three remaining sides. Let's denote the length of the rectangular parking lot by 'x' and the width by 'y'.

Because we only need to fence three sides, we can establish the following equation based on the total amount of fence available:

x + 2y = 800

To calculate the maximum possible area of a rectangle, we need to use the formula for the area of a rectangle, which is length times width (Area = x * y). However, we want to express the area in terms of a single variable. To do this, we can rearrange our fence equation to solve for y:

y = (800 - x) / 2

Now replace y in the area equation:

Area = x * (800 - x) / 2

For the area to be maximum, the derivative of the area with respect to x must be equal to zero. Differentiating and solving for x, we get the dimensions as x = 400 and y = 200.

So the largest possible parking lot would have dimensions of 400 feet by 200 feet.

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