A normal distribution is observed from the times to complete an obstacle course. The mean is 69 seconds and the standard deviation is 6 seconds. Using the Empirical Rule, what is the probability that a randomly selected finishing time is greater than 87 seconds?

In 6 months, they will have same amount of money.

Step-by-step explanation:

Let,

x be the number of months

Given,

Joe's savings = $200

Per month deposit = $50

J(x)=200+50x   Eqn 1

Kathy savings = $50

Per month deposit = $75

K(x)=50+75x   Eqn 2

For same amount;

J(x)=K(x)

[tex]200+50x=50+75x\\50x-75x=50-200\\-25x=-150[/tex]

Dividing both sides by -25;

[tex]\frac{-25x}{-25}=\frac{-150}{-25}\\x=6[/tex]

In 6 months, they will have same amount of money.

Keywords: functions, division

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Answer:

The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].

Step-by-step explanation:

Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".  

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"  

If we assume that we have [tex]3[/tex] groups and on each group from [tex]j=1,\dots,15[/tex] we have [tex]15[/tex] individuals on each group we can define the following formulas of variation:  

[tex]SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2 [/tex]  

[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]  

[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]  

And we have this property  

[tex]SST=SS_{between}+SS_{within}[/tex]  

The degrees of freedom for the numerator on this case is given by [tex]df_{num}=df_{within}=k-1=3-1=2[/tex] where k =3 represent the number of groups.

The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].

And the total degrees of freedom would be [tex]df=N-1=3*15 -1 =44[/tex]

And the F statistic to compare the means would have 2 degrees of freedom on the numerator and 42 for the denominator.  

Answer:

  1.  9 < s < 17

  2.  5 < MN < 19

  3.  AD > BD

Step-by-step explanation:

1. The triangle inequality tells you the sum of any two sides of a triangle must exceed the length of the other side. (Some versions say, "must be not less than ..." rather than "must exceed.") In practice, this means two things:

Here, we can use the latter fact to write the desired inequality. The difference of the given sides is 13 -4 = 9; their sum is 13 +4 = 17. The third side must lie between 9 and 17. If that side length is designated "s", then ...

  9 < s < 17

(If you don't mind a "triangle" that looks like a line segment, you can use ≤ instead of <.)

__

2. Same as (1) using different numbers.

  12 -7 < MN < 12 +7

  5 < MN < 19

__

3. Side CD is congruent to itself, and side CA is shown congruent to side CB. This means the requirements of the Hinge Theorem are met. That theorem tells you the longer side is opposite the greater angle:

  AD > BD

Answer:

  The substitution and addition properties of equality.

Step-by-step explanation:

The substitution property tells you that equals may be substituted for each other at any time.

The addition property of equality tells you that the same quantity can be added to both sides of an equation without violating the equal sign.

So, if you start with the equation ...

  a = b

and you add c to both sides (addition property), you get

  a + c = b + c

and if c = d, this can become (substitution property) ...

  a + c = b + d . . . . . d substituted for c

In other words, we have added the equations

  a = b

  c = d

to get ...

  a + c = b + d

The addition and substitution properties of equality make this valid.

it is b cuz im smart and i know t

Answer:

The probability that only 1 letter will be put into the envelope with its correct address is [tex]\frac{1}{3}[/tex]

Step-by-step explanation:

Given:

Number of Letters=4

Number of addresses= 4

To Find:

The probability that only 1 letter will be put into the envelope with its correct address=?

Solution:

Let us assume first letter goes in correct envelope and others go in wrong envelopes, then

=> Probability putting the first letter in correct envelope =[tex]\frac{1}{4}[/tex]

=>  Probability putting the second letter in correct envelope =[tex]\frac{2}{3}[/tex]

=>  Probability putting the third letter in correct envelope= [tex]\frac{1}{2}[/tex]

=> Probability putting the fourth letter in correct envelope = 1;  

( only 1 wrong addressed envelope is left);

This event can occur with other 3 envelopes too.

Hence total prob. = [tex]4\times(\frac{1}{4}\times\frac{2}{3}\times\frac{1}{2}\times1)[/tex]

=> [tex]\frac{1}{3}[/tex]

Answer:

  8.1 hours

Step-by-step explanation:

A model of the fraction remaining can be ...

  f = (1/2)^(t/37) . . . . t in hours

So, for the fraction remaining being 86%, we can solve for t using ...

  0.86 = 0.5^(t/37)

  log(0.86) = (t/37)log(0.5)

  t = 37·log(0.86)/log(0.5) ≈ 8.0509 ≈ 8.1 . . . hours

It takes about 8.1 hours to decay to 86% of the original concentration.

The 11 kilometer road will be 1.1 cm long on the map

Step-by-step explanation:

Scale is used on maps to show large locations or roads as small representatives of the larger objects.

The scale factor is usually in proportion to the original length or dimensions.

Given that

1 cm = 10 km

Then, we will divide the number of kilometers by 10 to find the length of road on map

11 km on map = [tex]\frac{11}{10}[/tex]

Hence,

The 11 kilometer road will be 1.1 cm long on the map

Keywords: Maps, Scales

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Answer:

15

Step-by-step explanation:

to find area you need to do height times base and divide by 2 or multiply it by 1/2.

10 x 3 = 30

30/2 = 15

Answer:

-75.955 kilometers

Step-by-step explanation:

multiply the speed by the time to get distance

the spacecraft is descending, so change in height will most likely be answered as a negative number

13.81 × 5.5 = 75.955

The required change in the height of the spacecraft will be 76 kilometers down.

Given that, an astronaut is returning to earth in a spacecraft. If the spacecraft is descending at a rate of 13.81 kilometers per minute, what will its change be in height after 5 1/2 minutes is to be determined.

Distance is defined as the object traveling at a particular speed in time from one point to another.

Here,
Speed = 13.81 km/s
Time =  5 1 /2 minute = 5+0.5 minute = 5.5 minutes
Distance traveled = speed * time
Distance traveled = 13.81 * 5.5 ≈ 76 km (down)

Thus, the required change in the height of the spacecraft will be 76 kilometers down.

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Answer:

Plot the point (3, -5)

Step-by-step explanation:

Recall that the general number a+bi can be plotted on a complex plane with the x axis as the real part and the y axis as the imaginary part.

In short,

a = real part = x

b = imaginary part = y

So (a,b) = (x,y)

In this case, a = 3 and b = -5 which is how I got (3, -5).

It might help to rewrite 3 - 5i into 3 + (-5)i

Answer:

25%.

Step-by-step explanation:

Let E be the event that the dart lands inside the triangle.

We have been given that a rectangular board has an area of 648 square centimeters. The triangular part of the board has an area of 162 square centimeters.

We know that probability of an event represents the chance that an event will happen.

[tex]\text{Probability}=\frac{\text{Favorable no. of events}}{\text{Total number of possible outcomes}}[/tex]

[tex]\text{Probability that dart lands inside the triangle}=\frac{\text{Area of triangle}}{\text{Area of rectangle}}[/tex]

[tex]\text{Probability that dart lands inside the triangle}=\frac{162}{648}[/tex]

[tex]\text{Probability that dart lands inside the triangle}=0.25[/tex]

Convert into percentage:

[tex]0.25\times 100\%=25\%[/tex]

Therefore, the probability that dart lands inside the triangle is 25%.

Answer:

25%

Step-by-step explanation:

Answer:

  $9486.68

Step-by-step explanation:

The future value of a annuity formula can be used:

  FV = P((1+r)^n -1)/r

  750000 = P(1.06^30 -1)/0.06

  P = 750000(.06)/(1.06^30 -1) = 9486.68

Adam has to contribute $9,486.68 each year to reach his goal.

Adam can calculate how much he needs to contribute to his IRA each year using the future value of an annuity formula. By substituting the known values into the rearranged formula, he can find the required annual payment.

To find out how much Adam needs to contribute each year, we can use the formula for the future value of an annuity. The future value (FV) of an annuity formula is:

FV=P*[((1+r)^n -1)/r]

Where:

In this question, we want to find P, so we rearrange the formula as follows:

P = FV / [((1+r)^n -1)/r]

Substituting the given values:

P= $750,000 / [((1+0.06)^30 -1)/0.06]

By calculating the above expression, we will get the amount Adam needs to contribute annually to reach his goal.

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Answer:

5 (strawberries / hours)

Step-by-step explanation:

calculation fro morning

strawberries / minutes  x   minutes / hours =   strawberries / hours

 so after adding the value in above equation

3/4* 60/1  = 45 strawberries / hours

 calculation in the afternoon

strawberries / minutes  x   minutes / hours =   strawberries / hours

 2/3 x 60/1  =  40 strawberries / hours

 so now by calculating difference between morning and afternoon packing rates,  you can easily calculate

 45-40 = 5 (strawberries / hours)

Answer:Problem 1. Let G be a group and let H, K be two subgroups of G. Dene the set HK = {hk : h ∈ H,k ∈ K}.

a) Prove that if both H and K are normal then H ∩ K is also a normal subgroup of G.

b) Prove that if H is normal then H ∩ K is a normal subgroup of K.

c) Prove that if H is normal then HK = KH and HK is a subgroup of G.

d) Prove that if both H and K are normal then HK is a normal subgroup of G.

e) What is HK when G = D16, H = {I,S}, K = {I,T2,T4,T6}? Can you give geometric description of HK?

Solution: a) We know that H ∩ K is a subgroup (Problem 3a) of homework 33). In order to prove that it is a normal subgroup let g ∈ G and h ∈ H ∩ K. Thus h ∈ H and h ∈ K. Since both H and K are normal, we have ghg−1 ∈ H and ghg−1 ∈ K. Consequently, ghg−1 ∈ H ∩ K, which proves that H ∩ K is a normal subgroup.

b) Suppose that H G. Let K ∈ k and h ∈ H ∩ K. Then khk−1 ∈ H (since H is normal in G) and khk−1 ∈ K (since both h and k are in K), so khk−1 ∈ H ∩ K. This proves that H ∩ K K.

c) Let x ∈ HK. Then x = hk for some h ∈ H and k ∈ K. Note that x = hk = k(k−1hk). Since k ∈ K and k−1hk ∈ H (here we use the assumption that H G), we see that x ∈ KH. This shows that HK ⊆ KH. To see the opposite inclusion, consider y ∈ KH, so y = kh for some h ∈ H and k ∈ K. Thus y = (khk−1)k ∈ HK, which proves that KH ⊆ HK and therefoere HK = KH. To prove that HK is a subgroup note that e = e · e ∈ HK. If a,b ∈ HK then a = hk and b = h1k1 for some h,h1 ∈ H and k,k1 ∈ K. Thus ab = hkh1k1. Since HK = KH and kh1 ∈ KH, we have kh1 = h2k2 for some k2 ∈ K, h2 ∈ H. Consequently,

ab = h(kh1)k1 = h(h2k2)k1 = (hh2)(k2k1) ∈ HK

(since hh2 ∈ H and k2k1 ∈ K). Thus HK is closed under multiplication. Finally,

Step-by-step explanation:

Hey! How are you? My name is Maria, 19 years old. Yesterday broke up with a guy, looking for casual sex.

Write me here and I will give you my phone number - *pofsex.com*

My nickname - Lovely

The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is calculated to be 0.54.

To find the probability that a woman has none of the three risk factors, given that she does not have risk factor A, we will use the given probabilities and apply the principles of probability. Let's denote the probabilities of having only the risk factors as P(A), P(B), and P(C), the probabilities of having exactly two risk factors as P(A and B), P(A and C), and P(B and C), and the probability of having all three risk factors as P(A and B and C).

Given:

P(A) = P(B) = P(C) = 0.1
P(A and B) = P(A and C) = P(B and C) = 0.12
P(A and B and C | A and B) = 1/3

We can calculate P(A and B and C) using the conditional probability:

P(A and B and C) = P(A and B) × P(A and B and C | A and B) = 0.12 × 1/3 = 0.04

To find the probability of not having A, denoted as P(A'), we can use the complement rule:

P(A') = 1 - P(A) - P(A and B) - P(A and C) - P(A and B and C) = 1 - 0.1 - 0.12 - 0.12 - 0.04 = 0.62

Since P(A') includes probabilities of women with neither of the risk factors or only with B or C, we need to subtract the probabilities of having only risk factors B and C:

P(None | A') = P(A') - P(B) - P(C) + P(B and C) = 0.62 - 0.1 - 0.1 + 0.12 = 0.54

The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is thus 0.54.

Answer:  A

Step-by-step explanation:

The sum to infinity of a geometric series is

S (∞ ) = \frac{a}{1-r} ( - 1 < r < 1 )

where a is the first term 8 and r is the common ratio, hence

S(∞ ) = {8}{1-\{1}{2} } = {8}{1}{2} } = 16

Answer:

Step-by-step explanation:

32 i think

Answer:

Mean : [tex]\mu=8[/tex]

Variance : [tex]\sigma^2=7.92[/tex]

Standard deviation =  [tex]\sigma=2.81[/tex]

Step-by-step explanation:

We know that , in Binary Distribution having parameters p (probability of getting success in each trial) and n (Total number trials) , the mean and variance is given by:-

Mean : [tex]\mu=np[/tex]

Variance : [tex]\sigma^2=np(1-p)[/tex]

We are given that ,

Total social security recipients  : n=800

The probability of social security recipients are too young to vote : p=1%= 0.01

Here success is getting social security recipients are too young to vote .

Then, the mean, variance and standard deviation of the number of recipients who are too young to vote will be :-

Mean : [tex]\mu=800\times0.01=8[/tex]

Variance : [tex]\sigma^2=800\times 0.01(1-0.01)=8\times0.99=7.92[/tex]

Standard deviation =  [tex]\sigma=\sqrt{\sigma^2}=\sqrt{7.92}=2.81424945589\approx2.81[/tex]

Hence, the mean, variance and standard deviation of the number of recipients who are too young to vote :

Mean : [tex]\mu=8[/tex]

Variance : [tex]\sigma^2=7.92[/tex]

Standard deviation =  [tex]\sigma=2.81[/tex]

In a sample of 800 social security recipients, with 1% being too young to vote, the mean is 8, the variance is 7.92, and the standard deviation is approximately 2.81.

The study indicates that 1% of social security recipients are too young to vote. When sampling 800 social security recipients, we treat the number of recipients too young to vote as a binomial random variable (since each recipient is either too young or not, with a fixed probability of being too young).

To find the mean of the binomial distribution, we use the formula:
Mean = n * p

Where n is the sample size (800) and p is the probability of success (0.01).

Mean = 800 * 0.01 = 8

The variance of the binomial distribution is given by the formula:
Variance = n * p * (1 - p)

Variance = 800 * 0.01 * (1 - 0.01) = 7.92

To calculate the standard deviation, we take the square root of the variance.

Standard Deviation = √(Variance) = √(7.92) = 2.81

Answer:

It takes less time sending 5 letters the traditional way with a probability of 36.7%.

Step-by-step explanation:

First we must take into account that:

- The traditional method is distributed X ~ Poisson(L = 1)

- The new method is distributed X ~ Poisson(L = 5)

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

Where L is the intensity in which the events happen in a time unit and x is the number of events.

To solve the problem we must calculate the probability of events  (to send 5 letters) in a unit of time for both methods, so:

- For the traditional method:

[tex]P(X=5)=\frac{1^{5}e^{-1}}{1!}\\\\P(X=5) = 0.367[/tex]

- For the new method:

[tex]P(X=5)=\frac{5^{5}e^{-5}}{5!}\\\\P(X=5) = 0.175[/tex]

According to this calculations we have a higher probability of sending 5 letters with the traditional method in a unit of time, that is 36.7%. Whereas sending 5 letters with the new method is less probable in a unit of time. In other words, we have more events per unit of time with the traditional method.

Answer:

Step-by-step explanation:

v*2+(v-14)*2=400

2v+2v-28=400

4v=400+28

4v=428

v=107 km/h

speed of slowest car=107-14=93 km/h

Answer:

Step-by-step explanation:

Assuming the same number of bowling and mini golf games are played, let x represent the total number of games played, either bowling or mini golf. let y represent the total cost of bowling. Let z represent the total cost of golfing

Bowling cost $2 to rent a club plus $5 per game. It means that the cost, y for x bowling games will be

y = 2 + 5x

Mini golf cost $5 to rent a club, plus $4 per game. It means that the cost, y for x mini golf games will be

z = 5 + 4x

For the total cost to be the same, we will equate both equations(y = zl

2 + 5x = 5 + 4x

5x - 4x = 5 - 2

x = 3

There would be 3 games before total cost would be the same

To find a line parallel to another, determine the slope of the original line, which is 2 in this case, and then use the point-slope form with a given point and the same slope to find the equation, resulting in y = 2x + 9.

To find an equation of the line that is parallel to another, we must first determine the slope of the given line. The line passing through the points (-3, -4) and (1, 4) has a slope calculated by the formula ∆y/∆x = (4 - (-4))/(1 - (-3)) = 8/4 = 2.

Since parallel lines have the same slope, our new line will also have a slope of 2. We can use the point-slope form of a line's equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Substituting our known point (-1, 7) and the slope 2, we get y - 7 = 2(x - (-1)). Simplified, the equation of our parallel line is y = 2x + 9.

Answer: 1.15 pounds

Step-by-step explanation:

For uniform distribution.

The standard deviation is  :

[tex]\sigma=\sqrt{\dfrac{(b-a)^2}{12}}[/tex]

, where a = Lower limit of interval [a,b].

b = Upper limit of interval [a,b].

Given : The changes in water weight are uniformly distributed between minus two and plus two pounds in a day.

i.e. Interval =  [-2 , +2]

Here , a= -2 and b= 2

Then, the standard deviation is  :

[tex]\sigma=\sqrt{\dfrac{(2-(-2))^2}{12}}[/tex]

[tex]\sigma=\sqrt{\dfrac{(2+2)^2}{12}}[/tex]

[tex]\sigma=\sqrt{\dfrac{16}{12}}=\sqrt{1.3333}=1.15468610453\approx1.15[/tex]

Hence, the standard deviation of your weight over a day =  1.15 pounds

The standard deviation of the uniform distribution representing an adult's daily change in weight due to water is around 1.155 pounds.

The question is about the standard deviation of the adult weight changes due to gain or loss in water content which is uniformly distributed between minus two and plus two pounds in a day.

To calculate the standard deviation for this uniform distribution, you need to follow these steps:

So, the standard deviation of your weight changes over a day due to the water flux is approximately 1.155 pounds.

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Answer: 70676

Step-by-step explanation:

First we draw a diagram representing the problem, which can be found in the picture uploaded,

Point a is the point of the plane, you can see where the angle of depression is imputed in the diagram, point C is the point where the gateway arch is, and drawing a vertical line to the ground from the point of the plane, point Blank is where that vertical line touches the ground

So we can tell the the angle of depression from the plane to the arch is the same as the angel of elevation from the arch to the plane

And we are to look for the distance between B and C which is labeled x in the diagram

So looking at the right angle triangle made from this question, we can see we have the opposite length which the angle of elevation from the arch is looking at, and we are looking for the adjacent length, so we use SOH, CAH, TOA, to solve

Choosing TOA which means

Tan(angle) = (opposite length)/(adjacent length)

Tan 23 = 30000/x

Multiplying both sides by x

xtan23 = 30000

Dividing both sides by tan23

x = 30000/tan23

x = 70675.57

Approximately 70676

Answer:

A) [tex]E(w)=450+0.10w[/tex]

B) $1200

Step-by-step explanation:

Let w represent employee's weekly sales and [tex]E(w)[/tex] be total weekly earnings.

We have been given that a phone store employee earns a salary of $450 per week plus 10% commission on her weekly sales.

A) The total weekly earnings of employee would be weekly salary plus 10% of weekly sales.

10% of weekly sales, w, would be [tex]\frac{10}{100}*w=0.10w[/tex]

[tex]E(w)=450+0.10w[/tex]

Therefore, the function [tex]E(w)=450+0.10w[/tex] models the employee's weekly earnings.

B) To find the weekly sales in the week, when employee earned $570, we will substitute [tex]E(w)=570[/tex] in our formula and solve for w as:

[tex]570=450+0.10w[/tex]

[tex]570-450=450-450+0.10w[/tex]

[tex]120=0.10w[/tex]

[tex]0.10w=120[/tex]

[tex]\frac{0.10w}{0.10}=\frac{120}{0.10}[/tex]

[tex]x=1200[/tex]

Therefore, the amount of employee's weekly sales was $1200.

To solve -3x + 3 < 6 and find the value of x, subtract 3 from both sides and divide by -3. The solution is x > -1.

To solve the inequality -3x + 3 < 6, we want to isolate the variable x. First, subtract 3 from both sides to get -3x < 3. Then, divide both sides by -3 (remember to flip the inequality sign when dividing by a negative) to obtain x > -1. Therefore, the solution is x > -1.

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Answer:

Option 4 - 0.8508

Step-by-step explanation:

Given : A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses.

To find : Estimate the probability of getting at least 20% correct ?

Solution :

20% correct out of 60,

i.e. [tex]20\%\times 60=\frac{20}{100}\times 60=12[/tex]

Minimum of 12 correct out of 60 i.e. x=12

Each question has 4 possible answers of which one is correct.

i.e. probability of answering question correctly is [tex]p=\frac{1}{4}=0.25[/tex]

Total question n=60.

Using a binomial distribution,

[tex]P(X\geq 12)=1-P(X\leq 11)[/tex]

[tex]P(X\geq 12)=1-[P(X=0)+P(X=1)+P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)][/tex]

[tex]P(X\geq 12)=1- [^{60}C_0(0.25)^0(1-0.25)+^{60-0}+^{60}C_1(0.25)^1(1-0.25)^{60-1}+^{60-1}+^{60}C_2(0.25)^2(1-0.25)^{60-2}+^{60}C_3(0.25)^3(1-0.25)^{60-3}+^{60}C_4(0.25)^4(1-0.25)^{60-4}+^{60}C_5(0.25)^5(1-0.25)^{60-5}+^{60}C_6(0.25)^6(1-0.25)^{60-6}+^{60}C_7(0.25)^7(1-0.25)^{60-7}+^{60}C_8(0.25)^8(1-0.25)^{60-8}+^{60}C_9(0.25)^9(1-0.25)^{60-9}+^{60}C_{10}(0.25)^{10}(1-0.25)^{60-10}+^{60}C_{11}(0.25)^{11}(1-0.25)^{60-11}][/tex]

[tex]P(X\geq 12)\approx 0.8508 [/tex]

Therefore, option 4 is correct.

Answer:

  1.00 inches

Step-by-step explanation:

The distance from vertex to focus is "p" in the quadratic equation ...

  x^2 = 4py

In the given equation, p=1. Since units are inches, ...

the bulb should be placed 1.00 inches from the vertex.

Answer:

The answer to your question is (6, 0)

Step-by-step explanation:

Solve the system of equations by elimination

                                  x -  4y = 6             (I)

                                2x + 2y = 12            (II)

Multiply (II) by 2

                                  x  - 4y =   6

                                4x + 4y = 24

Simplify

                                5x  + 0  = 30

Find x

                                5x = 30

                                  x = 30/ 5

                                  x = 6

Find "y"

                              6 - 4y = 6

                              -4y = 6 - 6

                              -4y = 0

                                 y = 0/-4

                                 y = 0

Answer:

(6,0)

Step-by-step explanation:

Given equations are:

\[x - 4y = 6\]         -------------------- (1)

\[2x + 2y = 12\]    -------------------- (2)

Multiplying (1) by 2 :

\[2x - 8y = 12\]     -------------------- (3)

Calculating (2) - (3) :

\[2x + 2y -2x + 8y = 12 - 12\]

=> \[10y =0\]

=> \[y = 0\]

Substituting the value of y in (1):

\[ x  = 6 \]

So the required solution of the system of equations is x=6,y=0. This can be alternatively expressed in coordinate notation as (6,0).

To minimize the paper used for a poster with a specific print area and margin size, we derive a formula for total paper area, take its derivative with respect to the print width, solve for the width, and then find the matching height.

To find the dimensions of the page that will minimize the amount of paper used for a rectangular poster that contains 81 square inches of print with 5-inch margins on all sides, we need to set up a function to minimize. Let the width of the print area be x inches, and the height be y inches. Therefore, the total dimensions of the poster will be (x + 10) inches wide and (y + 10) inches high due to the margins on each side.

The area of print is given, so x*y = 81. We will minimize the total area of the page, A = (x+10)(y+10). Substituting the value of y from the print area equation, y =[tex]\frac{81}{x}[/tex], we get A(x) = (x+10)([tex]\frac{81}{x}[/tex]+10).

Now, to find the dimensions that minimize the paper used, we will take the derivative of A(x) with respect to x, set it to zero, and solve for x. From there, we can find the corresponding value of y to get the dimensions that will use the least amount of paper while still fitting the print area and margins.