Suppose 60% of all college professors like tennis, 65% like bridge, and 50% like chess; 45% like any given pair of recreations.

(a) Should you be suspicious if told 20% like all three recreations?
(b) What is the smallest percentage who could like all three recreations?

Answer:

60, yes.

Step-by-step explanation:

Yes, it has symmetry if you rotate it. Because it is a hexagon, the degree of rotation is 360 (circle of rotation) divided by 6 sides, because it takes that many degrees to rotate till the next side enters the position of the previous side. 360/6 is equal to 60.

Answer : Yes, it has rotational symmetry. The angle of rotation is, [tex]60^o[/tex]

Step-by-step explanation :

Rotational symmetry : It is a shape that has Rotational Symmetry when it still the same after the rotation.

The given figure is hexagon and it has rotational symmetry because it still the same after the rotation.

As there are 6 sides of hexagon geometry and the degree of rotation is, [tex]360^o[/tex].

So, the angle of rotation = [tex]\frac{360^o}{6}=60^o[/tex]

Sandy would be able to make 5,333 earrings with a gold bar of 1,000 grams.

Given that;

Sandy has 2 grams of gold, and each earring requires 3/16 grams of gold.

Now for the number of earrings she can make, divide the total amount of gold she has by the amount of gold needed for each earring.

[tex]\text {Number of earrings} = \dfrac{\text {Total gold} }{\text {Gold per earring} }[/tex]

[tex]\text {Number of earrings} = \dfrac{\text {2} }{\text {3/16} }[/tex]

To divide by a fraction, multiply by its reciprocal:

[tex]\text {Number of earrings} = \text {2} \times \dfrac{16}{3}[/tex]

Now, let's simplify the calculation:

[tex]\text {Number of earrings} = \dfrac{32}{3}[/tex]

Therefore, Sandy can make 10 earrings with 2 grams of gold, leaving 2 grams remaining.

When she had a gold bar of 1,000 grams, use the same approach to find out how many earrings she can make:

[tex]\text {Number of earrings} = \dfrac{\text {1000} }{\text {3/16} }[/tex]

[tex]\text {Number of earrings} = 5333[/tex]

So, Sandy would be able to make 5,333 earrings with a gold bar of 1,000 grams.

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Sandy could make about 5,333 earrings from 1,000 grams of gold by dividing 1,000 grams by the weight of gold in each earring (3/16 grams), which equates to multiplying 1,000 by the reciprocal of 3/16

To solve this, we need to figure out how many times 3/16 grams goes into 1,000 grams. To do this, we divide 1,000 by 3/16.

However, when dividing by a fraction, it's often easier to multiply by its reciprocal (flip the fraction) instead. So we'll turn 1,000 into 1,000/1, and multiply by 16/3 (the reciprocal of 3/16).

Therefore, Sandy could make about 5,333 earrings from 1,000 grams of gold.

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Answer:

11.2

Step-by-step explanation:

tan( angle ) = opposite / adjacent

Answer:

Andre got better paid when he mowed for neighbor's and uncle's lawn which is $20/hr than Community center's lawn which is $15/hr.

Step-by-step explanation:

Given:

Andre mowed a neighbor's lawn for 1/2 hour and earned $10.

For 1/2 hour = $10

So for 1 hour = Money earned in 1 hour.

By using Unitary method we get;

Money earned in 1 hour = [tex]\frac{10}{\frac{1}{2}} = \frac{10\times2}{1} = \$20[/tex]

Hence Andre hourly rate to mowed neighbor's lawn is $20.

Also Given:

Andre mowed a Uncle's lawn for 3/2 hour and earned $30.

For 3/2 hour = $30

So for 1 hour = Money earned in 1 hour.

By using Unitary method we get;

Money earned in 1 hour = [tex]\frac{30}{\frac{3}{2}} = \frac{30\times2}{3} = \$20[/tex]

Hence Andre hourly rate to mowed Uncle's lawn is $20.

Also Given:

Andre mowed Community center's lawn for 2 hour and earned $30.

For 2 hour = $30

So for 1 hour = Money earned in 1 hour.

By using Unitary method we get;

Money earned in 1 hour = [tex]\frac{30}{2}=\$15[/tex]

Hence Andre hourly rate to mowed Community Center's lawn is $15.

Now Since we can see that when he mowed neighbor and uncle's lawn he was paid at rate of $20 and when he mowed Community Center's lawn he got paid at rate of $15.

Hence Andre got better paid when he mowed for neighbor's and uncle's lawn which is $20/hr than Community center's lawn which is $15/hr.

Answer:

A. 14%.

B. [tex]\frac{43}{50}[/tex]

Step-by-step explanation:

We have been given that a running back was the MVP (most valuable player) in 0.14 of the first 50 Super Bowls.

A. To find the percent, when the MVPs were running backs, we need to convert 0.14 into percent by multiplying by 100 as:

[tex]0.14\times 100=14\%[/tex]

Therefore, 14 percent of the MVPs were running backs.

B. To find the fraction of the MVPs were not running backs, we will subtract 0.14 from 1 to find the MVPs, who were not running backs. Finally, we will convert the answer into fraction as:

[tex]1-0.14=0.86[/tex]

Now, we will multiply and divide 0.86 by 100 as:

[tex]0.86\times \frac{100}{100}=\frac{86}{100}[/tex]

Reduce the fraction by dividing numerator and denominator by 2:

[tex]\frac{43}{50}[/tex]

Therefore, [tex]\frac{43}{50}[/tex] of the MVPs were not running backs.

14% of the MVPs in the first 50 Super Bowls were running backs. Yet, 43 out of 50 MVPs (or 86%) were not running backs.

The running back was the MVP in 0.14 of the first 50 Super Bowls according to the question. To find the percent of the MVPs that were running backs, we simply convert the 0.14 to percentage by multiplying it by 100. Hence, 14% of the MVPs were running backs.

For the fraction of the MVPs that were not running backs, we need to calculate the remaining part not covered by the running backs. Given as 1 (entirety) minus 0.14 gives 0.86. In fraction terms, this is same as 86/100 which simplifies to 43/50.

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Answer:

The answer to your question is x = 17.32; y = 8.67

Step-by-step explanation:

Process

1.- Use trigonometric functions to find x and y

   a) sin Ф = [tex]\frac{opposite side}{hypotenuse}[/tex]

      Ф = 60°

      opposite side = 15

      hypotenuse = ?

     [tex]hypotenuse = \frac{opposite side}{sin \alpha }[/tex]

     [tex]hypotenuse = \frac{15}{sin 60}[/tex]

     [tex]hypotenuse = \frac{15}{0.87}[/tex]

            hypotenuse = 17.32

     b) cosФ = [tex]\frac{y}{hypotenuse}[/tex]

         [tex]y = hypotenuse x cos 60[/tex]

         [tex]y = 17.32 x 0.5[/tex]

                y = 8.67

Answer:

[tex]24.68\ ft[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

Let

x ----> ladder height in ft

Applying the Pythagorean Theorem

[tex]25^2=4^2+x^2[/tex]

solve for x

[tex]625=16+x^2[/tex]

[tex]x^2=625-16[/tex]

[tex]x^2=609[/tex]

[tex]x=\sqrt{609}\ ft[/tex]

[tex]x=24.68\ ft[/tex]

The ladder reaches up 24.7 ft on the garage wall.

The ladder reaches up 24 ft on the garage wall.

To find the height the ladder reaches, we can use the Pythagorean theorem, where the ladder is the hypotenuse:

Height² + Base² = Hypotenuse²

Height² + 4² = 25²

Height² + 16 = 625

Height² = 609

Height = √609

Height =24.67 ≈ 24.7 ft

Answer:

In general if speed limit is not posted it will be 25 mph in residential area  

Step-by-step explanation:

When speed limit is not posted then it is 25 mph in residential area or school zones

But its also depend on the density of population in that area is density is very low then speed limit can go up to 35 mph and if density of population is high then speed limit will be restricted up to 25 mph

So in general if speed limit is not posted it will be 25 mph in residential area  

Answer:

  b.  Miguel ends up with $35 less than he started with

Step-by-step explanation:

He spends $35 × 4 = $140.

He earns $21 × 5 = $105.

His net increase is $105 -140 = -$35.

Miguel ends up with $35 less than he started with.

The probability that a randomly chosen adult from the survey has no desire to take skiing lessons is 57/80, or as a decimal, 0.7125.

Out of a sample of 400 adults, 115 expressed an interest in taking skiing lessons. Therefore, to find the number of adults who have no interest in skiing lessons, we subtract the number interested (115) from the total number surveyed (400).

The calculation is as follows: 400 - 115 = 285 adults who have no interest in taking skiing lessons. The probability that a randomly selected adult has no desire to take skiing lessons is the number of adults with no interest divided by the total number surveyed. This gives us:

       Probability = 285 / 400

        Probability = 57 / 80

If we want to express this as a decimal rounded to the nearest millionth, we perform the division:

Probability = 0.7125

This result is already rounded to the fourth decimal place, which is more precise than rounding to the nearest millionth.

Answer:

11 hours 15 mins

Step-by-step explanation:

Wilma & Betty both have a pool that holds 10500 gallons of water each.

Wilma's garden hose fills at the rate of 700 gallons per hour.

Betty's garden hose fills at a rate of 400 gallons per hour.

Volume = Rate * time

Time = Volume /rate

The time it will take for Wilma's pool to be filled = 10500/700

= 15 hours

The time it will take for Betty's pool to be filled = 10500/400

= 26.25hours

= 26 hours 15 minutes

it will take Betty (26.25 - 15) to fill her pool than Wilma.

= 11.25hours

= 11 hours 15 mins

Answer: GLOBAL WARMING FOESNT EXIST

Step-by-step explanation: STOP RIDING BIKES MORE CARS

Answer:

Step-by-step explanation:

Abby's car gets approximately 24 miles per gallon. This means that for every 24 mile that her car covers, it uses 1 gallon of gas. She is planning a 1200 mile trip. This means that the number of gallons of gas that she would need would be 1200/24 = 50 gallons.

One average price of 1 gallon of gas is $4.20. The total amount that she would spend in buying 50 gallons of gas would be

50 × 4.2 = $210

Abby needs 50 gallons of gas for her 1200-mile trip. At $4.20 per gallon, the total cost will be $210. This calculation helps Abby budget for her fuel expenses.

Abby needs to calculate the amount of fuel and the cost for a 1200-mile trip with her car that gets approximately 24 miles per gallon. Here's how to find out:

To find the number of gallons Abby needs, divide the total trip distance by her car's miles per gallon (MPG):

1200 miles / 24 MPG = 50 gallons

Multiply the number of gallons by the cost per gallon:

50 gallons x $4.20 per gallon = $210

Abby should plan to buy 50 gallons of gas for her 1200-mile trip, costing about $210 at $4.20 per gallon.

Answer:

[tex]12+17+y+z[/tex] or [tex]29+y+z[/tex].

Step-by-step explanation:

We have been given that Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two days. One expression for the total miles he will run is [tex]12+y+17+z[/tex].

The Commutative Property of Addition states that we can add numbers in any order. For example a and b be two numbers.

According to Commutative Property of Addition [tex]a+b=b+a[/tex].

Similarly, we can write an equivalent expression to our given expression as:

[tex]12+17+y+z[/tex]

We can simplify our expression as:

[tex]29+y+z[/tex].

Therefore, our required expression would be [tex]12+17+y+z[/tex] or [tex]29+y+z[/tex].

Answer:

The temperature in the morning was [tex]-10.5^o[/tex]

Step-by-step explanation:

we know that

1) The high temperature in Fairbanks, Alaska was 15.7 degrees

That night it fell 38.4 degrees

so

The temperature at night was

[tex]15.7^o-38.4^o=-22.7^o[/tex]

2) The next morning, it rose 12.2 degrees.

so

The temperature in the morning was

[tex]-22.7^o+12.2=-10.5^o[/tex]

Answer:The percent of increase of the area of the deck is 12.5%

Step-by-step explanation:

The initial area of the deck is 160 square feet. You and your family have decided to increase the area of your deck to 180. The amount by which the area of the deck was increased is is 180 - 160 = 20 square feet.

The percent of increase of the area of the deck would be

Increase / initial area × 100.

20/160 × 100 = 12.5℅

Answer:

(-1,3), (0,-1)

(-1-3)/(0+1)= -4/1= -4

the answer is d

Answer:

Step-by-step explanation:

[tex]\bold{METHOD\ 1:}\\\\\text{look at the picture}\ \#1\\\\\text{The fomula of a slope:}\\\\m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\text{Form the graph wi hate points:}\\\\(-1, 3)\ \text{and}\ (0, -1).\\\\\text{Substitute:}\\\\m=\dfrac{-1-3}{0-(-1)}=\dfrac{-4}{1}=-4[/tex]

[tex]\bold{METHOD\ 2:}\\\\\text{look at the picture}\ \#2\\\\slope=\dfrac{rise}{run}\\\\rise=-4\\run=1\\\\slope=\dfrac{-4}{1}=-4[/tex]

Answer:

We fail to reject H₀ as there is insufficient evidence at 0.5% level of significance to conclude  that the mean hours of TV watched per day differs from the claim.

Step-by-step explanation:

This is a two-tailed test.

We first need to calculate the test statistic. The test statistic is calculated as follows:

Z_calc = X - μ₀ / (s /√n)

where

Therefore,

Z_calc = (3.02 - 3) / (2.64 /√(1326))

           = 0.2759

Now we have to calculate the z-value. The z-value is calculated as follows:

z_α/2 = z_(0.05/2) = z_0.025

Using the p-value method:

P = 1 - α/2

  = 1 - 0.025

  = 0.975

Thus, using the positive z-table, you will find that the z-value is

1.96.

Therefore, we reject H₀ if | Z_calc | > z_(α/2)

Thus, since

| Z_calc | < 1.96, we fail to reject H₀ as there is insufficient evidence at 0.5% level of significance to conclude  that the mean hours of TV watched per day differs from the claim.

Using a statistical hypothesis test, we do not find sufficient evidence to conclude that the mean number of hours per day people spend watching TV in 2012 differs from the sociologist's claim of 3 hours.

The question pertains to evaluating a claim about population mean using a sample mean. Given the data, we can perform hypothesis testing. The null hypothesis (H0) is that the mean number of hours spent watching TV is 3, and the alternative hypothesis (H1) is that the mean is not 3. The sample mean is 3.02, standard deviation is 2.64, and the sample size is 1326.

Firstly, we need to calculate the standard error: SE = standard deviation/sample size square root = 2.64/ sqrt(1326) = 0.072. Then, we calculate the test statistic (Z): Z = (sample mean - population mean)/SE = (3.02-3)/0.072 = 0.278. Given α=0.05, the Z value for a two-tailed test is +/- 1.96.

The computed Z value of 0.278 falls within the acceptance region (-1.96 < Z < 1.96), so we do not reject the null hypothesis. Therefore, the data do not provide sufficient evidence to conclude that the mean hours of TV watched per day differs from the sociologist's claim.

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Answer:

The ad is about the price of a house which is sufficient for two families to live in it along with the contact details to purchase this house.

According to the ad, the total cost of the home is $190,000. It is sufficient for two families. It may be double story as well. To purchase this house, one can call at the given number which is 212-123-4567. This can be the original number or the format by the editor to show original number. Actual area of the house is not mentioned in the ad.

Answer: it would be A

Answer:

B.a vertical shift of 9 units up

Step-by-step explanation:

Given [tex]f(x) = 4x-2\\g(x) = 4x+7[/tex]

[tex]g (x) = f (x) + k[/tex]

It means shifting [tex]f (x)\ k[/tex] unit vertically.

Now, we will find the value of [tex]k[/tex] for the given function

[tex]g(x) = 4x+7\\\\add\ 2\ and\ subtract\ 2\\\\g(x) = 4x+7+2-2\\g(x) = 4x-2+9\\\\We\ have\ f(x)=4x-2\\\So,\ g(x)=f(x)+9[/tex]

[tex]k=9[/tex]

Hence, vertical shift of 9 units.

Answer:

C. a horizontal shift of 9 units left

Step-by-step explanation:

Look at this helpful chart:

Vertical Translations

translation up k units: g(x) = f(x) + k, where k > 0

translation down k units: g(x) = f(x) – k, where k > 0

Horizontal Translations

translation left k units:  g(x) = f(x + k), where k > 0

translation right k units:  g(x) = f(x – k), where k > 0

The change happening in Martina's graph is therefore a horizontal translation to the left.

Answer:

C) I and III only

Step-by-step explanation:

Let full pool is denoted by O

Days Hose x takes to fill pool O = a

Pool filled in one day x = O/a

Days Hose y takes to fill pool O = b

Pool filled in one day y = O/b

Days Hose z takes to fill pool O = c

Pool filled in one day z = O/c

It is given that

                         a>b>c

[tex]a>b>c>d\\\implies x<y<z<(x+y+z)\\[/tex]

Days if if x+y+z fill the pool together = d

1 day if x+y+z fill the pool together [tex]=O(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=\frac{O}{d}---(1)[/tex]

I) d < c

d are days when hose x, y, z are used together where as c are days when only z is used so number of days when three hoses are used together must be less than c when only z hose is used. So d < c

III) [tex]\frac{c}{3}<d<\frac{a}{3}[/tex]

Using (1)

[tex]\frac{bc+ac+ab}{abc}=\frac{1}{d}\\\\d=\frac{abc}{ab+bc+ca}\\\\As\quad(a>b>c)\\(ab+bc+ca)<3ab\\\\d=\frac{abc}{ab+bc+ca}>\frac{abc}{3ab}\\\\d>\frac{c}{3}[/tex]

Similarly

[tex]\frac{bc+ac+ab}{abc}=\frac{1}{d}\\\\d=\frac{abc}{ab+bc+ca}\\\\As\quad a>b>c\\(ab+bc+ca)>3bc\\\\d=\frac{abc}{ab+bc+ca}<\frac{abc}{3bc}\\\\d<\frac{a}{3}[/tex]

So,

[tex]\frac{c}{3}<d<\frac{a}{3}[/tex]

Answer:

x  =  15  hours  

y  =  25  hours

Step-by-step explanation:

Lets call  "x"  number of working hours of Clark family sprinklers

and   " y "  number of working hours of Thomas family sprinklers

Then they both worked in such way that

x  +  y  =  40   (1)

On the other hand the total water output  1100 lts were supplied according to:

35 lts/h *  y (hours)   +  15 lts/h *  x (hours)  =  1100 lts

Then  

35*y   +  15*x  =  1100   (2)

Equations 1 and 2 become a system of two equations with two uknown variables  x  and  y

We solve that system

y   =  40  -  x        and     35* ( 40 - x ) +  15* x    =  1100

1400  -  35x   +  15x   = 1100    ⇒    -  20x  = -300

x  =  15  hours

And

y  =  40  -  15   =  25  hours

Answer: 52/57

Step-by-step explanation:please see attachment for explanation

Final answer:

The question involves calculating the probability of selecting at least 2 business majors from a group, using complementary probability and combinations. The probability that at least 2 of the 4 students selected are business majors is approximately [tex]\(0.1608\).[/tex]

Explanation:

To find the probability that at least 2 of the 4 students selected are business majors, we can calculate the probability of exactly 2, 3, and 4 students being business majors, and then sum these probabilities.

First, let's find the probability of selecting exactly 2 business majors and 2 non-business majors:

1. Probability of selecting 2 business majors: [tex]\( \frac{{14 \choose 2}}{{21 \choose 4}} \)[/tex]

2. Probability of selecting 2 non-business majors:[tex]\( \frac{{7 \choose 2}}{{21 \choose 4}} \)[/tex]

Then, we can find the probability of selecting exactly 3 business majors and 1 non-business major:

1. Probability of selecting 3 business majors: [tex]\( \frac{{14 \choose 3}}{{21 \choose 4}} \)[/tex]

2. Probability of selecting 1 non-business major: [tex]\( \frac{{7 \choose 1}}{{21 \choose 4}} \)[/tex]

Finally, we find the probability of selecting all 4 business majors:

1. Probability of selecting 4 business majors: [tex]\( \frac{{14 \choose 4}}{{21 \choose 4}} \)[/tex]

Now, we sum up these probabilities:

[tex]\[\text{Probability of at least 2 business majors} = \text{Probability of selecting exactly 2} + \text{Probability of selecting exactly 3} + \text{Probability of selecting all 4}\][/tex]

[tex]\[\text{Probability of at least 2 business majors} = \left( \frac{{14 \choose 2}}{{21 \choose 4}} \times \frac{{7 \choose 2}}{{21 \choose 4}} \right) + \left( \frac{{14 \choose 3}}{{21 \choose 4}} \times \frac{{7 \choose 1}}{{21 \choose 4}} \right) + \left( \frac{{14 \choose 4}}{{21 \choose 4}} \right)\][/tex]

[tex]\[= \left( \frac{{91}}{{5985}} \times \frac{{21}}{{5985}} \right) + \left( \frac{{364}}{{5985}} \times \frac{{7}}{{5985}} \right) + \left( \frac{{1001}}{{5985}} \right)\][/tex]

[tex]\[= \left( \frac{{1911}}{{5985^2}} \right) + \left( \frac{{2548}}{{5985^2}} \right) + \left( \frac{{1001}}{{5985}} \right)\][/tex]

[tex]\[= \frac{{1911 + 2548 + 1001}}{{5985^2}}\][/tex]

[tex]\[= \frac{{5460}}{{5985^2}}\][/tex]

[tex]\[\approx 0.1608\][/tex]

So, the probability that at least 2 of the 4 students selected are business majors is approximately [tex]\(0.1608\).[/tex]

Answer:

E(x) = 15 minutes

Step-by-step explanation:

The random variable X (waiting time)  has a uniform distribution between the interval [0,30], because it is just as likely that you arrive in any time and then your waiting time is minimum 0 minutes and maximum 30 minutes

The expected value  of a random variable uniform is:

E(x) = [tex]\frac{a+b}{2}[/tex]

      Where a and b are the interval's extremes

Thus

E(x) = [tex]\frac{0+30}{2}[/tex]

E(x) = 15 minutes

Step-by-step explanation:

The equation of the line in Slope-Intercept form is:

[tex]y=mx+b[/tex]

Where "m" is the slope of the line and "b" is the y-intercept.

So, the lines of this system are:

1) [tex]y= 2x + 1[/tex]

Where:

[tex]m=2\\b=1[/tex]

2) [tex]y=2x- 2[/tex]

Where:

[tex]m=2\\b=-2[/tex]

Notice that:  

- Since both lines have the same slope, they are parallel.

- The symbol of the first inequality is [tex]\geq[/tex] . This indicates that the line is solid and the shaded region must be above the line.

- The symbol of the second inequality is [tex]\leq[/tex] . This indicates that the line is solid and the shaded region must be below the line.

Therefore, the  second graph represents the system of linear inequalities.

Answer:

b

Step-by-step explanation:

Answer:

The answers to all the questions are given below.

Step-by-step explanation:

8)The ratio of the numbers 27 feet and 54 yard

1 foot = 0.3333 yard

27 feet = 9 yard

Required ratio = [tex]\frac{9}{54}[/tex]  = 0.167

9)A 17 ounce bag costs $5.27.

Hence cost of 1 ounce bag = [tex]\frac{5.27}{17}[/tex] = $0.31

12)For trip of 240 miles ,it uses 8 gallons.

13)1 quart = 0.25 gallons

Hence the ratio would be [tex]\frac{1}{4}[/tex]

14)1 pound = 0.0005 tonnes.

Hence the ratio would be [tex]\frac{1}{2000}[/tex]

19)1 week = 7 days

So the ratio would be [tex]\frac{1}{9}[/tex]

These are the required ratios

Answer:

Jill is most correct and it should be $33

Step-by-step explanation:

Total number of seats = x =20000+13000+17000

x =50000

Total cost of seats as per the price;

a = 20*20000

a =400,000

b = 30*13000

b =390,000

c = 50*17000

c =850,000

Average  = (a + b + c) /x

Average = (400,000+390,000+850,000)/50,000

Average cost = 1,640,000/50,000

Average cost = $32.8

Answer:

the vertices (1,1,1), (1,0,0), (0,1,0) and (0,0,1) form a tetrahedron. The length of each side is √2

Step-by-step explanation:

The cube has 8 vertices: (0,0,0), (1,1,0), (0,1,0), (1,0,0), (0,0,1), (0,1,1), (1,0,1), (1,1,0) and (1,1,1). The first four of them are the vertices of the bottom square and the last four are the vertices of the upper square of the cube.

We will take two non-consecutive vertices from each square. For the upper one we take (1,1,1) as the problem suggests, and (0,0,1), which is not consecutive from (1,1,1) and its distance is √2. The non consecutive vertices from the bottom square respect to the vertex (1,1,1) are (0,0,0), (0,1,0) and (1,0,0).

We take (0,1,0) and (1,0,0) because (0,0,0) is consecutive from (0,0,1) hence its distance from it is not √2, but 1.

Note that we take (1,1,1), (0,0,1), (0,1,0) and (1,0,0). If we take any two vertices and compare them toguether we will notice that both of those vertices differ in two places and are equal in the other. In the places where they differ one has the value 1 and the other 0, so the distance between those vertices is √(1²+1²) = √2.

Thus, the vertices (1,1,1), (1,0,0), (0,1,0) and (0,0,1) form a tetrahedron.

Final answer:

Explaining how to find the vertices of a cube forming a regular tetrahedron and confirming why all cube edges have the same length.

Explanation:

To find the four vertices of a cube that form a regular tetrahedron starting with (1, 1, 1), we can consider the cube's diagonals. The vertices of the regular tetrahedron can be located at the center of each face of the cube, which are at coordinates (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2).

The length of an edge of a cube is the distance between two adjacent vertices. To calculate the edge length, we can use the distance formula. Since all edges of a cube connect two adjacent vertices, they have the same length due to the cube's symmetry.

Therefore, all edges of the cube have the same length because each connects two adjacent vertices with equal coordinates.

Only problem is with the simplifying.

We all know that 5/5 = 1, it is natural to  assume (x+a)/(x+a) is also 1, but in some cases where x+a=0, it is undefined. In this equation, where they simplify (x-2) and (x-6), you must say that x is not 2 nor 6 or, you just delete 0/0 which is undefined.

Therefore the only solution would be x=-1

The error that the student made in the rational equation simplification is that; She made 6 and -1 to be a solution but 6 is not a solution but only x = -1 because 6 makes the function undefined

         From the simplification of the rational equation, the solution the person got is; x = 6 or -1

         Now, when we put 6 for x in the rational equation, it is discovered that the denominator becomes zero for two of the expressions.

Now, when the denominator of a fraction is zero, that fraction is said to be undefined.

         Whereas when x = -1, we don't get an undefined function. Thus, the mistake the student made is that 6 is not a solution but only x = -1

Read more about Rational Equations at; https://brainly.com/question/8519709