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(a): 273+-22=273-22=251 Kelvin

(b): 273+0=273 Kelvin

(c): 273+-32=241 Kelvin

Plugging in the values, we get: \(-2--4.1+5=-2+4.1+5=2.1+5=7.1.\)

Both negatives will make positive, so: \(-29+11=-18.\)

This is just: \(-14-10=-24.\)

Count 9 dots from -6, and you'll land on: \(-6+9=3\) .

All multiples of 2 =50 / 2 = 25

All multiples of 3 that are odd =50/2 =16/2 =8

All multiples of 2  +  All multiples of 3 that are odd

25 + 8 =33. Therefore, the remaining numbers in the set are:

50 - 33 = 17 - Numbers remain in the set.

[1, 5, 7, 11,  13,  17, 19,  23, 25, 29, 31, 35, 37, 41, 43, 47, 49]

First this problem is equivalent  to "Let m be a positive integer such that m>=6,How many different values can gcd(m,m+6) attain?"

Let's proof gcd(m,m+6)<7

Suppose p is factor of m and satisfy p>=7 and m/p=q,that m=p*q.

Then,we can write m+6 as p*q+6

(m+6)%p=(p*q+6)%p=6,that is p can not be factor of m+6

Therefore,gcd(m,m+6)<=6,acctually are the divisors of 6.

Why 4 and 5 can't be attain from gcd(m,m+6)? You can proof it by using above procedure.

1 can be gcd(m,m+6),for example 7 and 13

2 can be gcd(m,m+6),for exampe 8 and 14

3 can be gcd(m,m+6),for exampe 9 and 15

6 can be gcd(m,m+6),for exampe 6 and 12

(I am not a master on number theory.This is just a little thought on this one.By the way,is there any abstract algebra to do this?) 

Let's find the equation of the line that passes through  (0, 3)  and  (-8, 0) .

slope  =  [0 - 3] / [-8 - 0]   =   -3 / -8   =   3 / 8

The y-intercept is the y value when  x  is  0 .

We know that the line goes through the point (0, 3) , so when  x  is  0 ,  y is  3 .

The y-intercept is  3 .

The equation of the line in slope-intercept form is...

y  =  (3/8)x + 3

We also know that the line passes through the point  (t, 5) .

So we know it makes a true equation when  x  is  t  and  y  is  5 .

So let's plug in  t  for  x  and  5  for  y , and solve for  t .

5  =  (3/8)t + 3

                              Subtract  3  from both sides of the equation.

2  =  (3/8)t

                              Multiply both sides by  8/3 .

2 * 8/3  =  t

16/3  =  t

t  =  16/3

Here's an edit of terte's graph with the line  y = (3/8)x + 3  and the point  (16/3, 5)  added:

f(x)  =  g(g(x)) - g(x)        Plug in  3  for  x .

f(3)  =  g(g(3)) - g(3)

g(x)  =  2x - 1         Now let's find  g(3) .  Plug in  3  for  x .

g(3)  =  2(3) - 1

g(3)  =  6 - 1

g(3)  =  5

f(3)  =  g(g(3)) - g(3)       Substitute  5  in for  g(3) .

f(3)  =  g(5) - 5

g(x)  =  2x - 1        Now let's find  g(5) .  Plug in  5  for  x .

g(5)  =  2(5) - 1

g(5)  =  10 - 1

g(5)  =  9

f(3)  =  g(5) - 5                Substitute  9  in for  g(5) .

f(3)  =  9 - 5

f(3)  =  4

(This question a little chanellge to me. Difinitely fun!)

The first anwser is right,but I think my anwser is more specific.

given f(3)=5 

f(9)=f(3)+2=5+2=7

f(27)=f(9)+2=7+2=9

f(81)=f(27)+2=9+2=11

\({f}^{-1}(11)={f}^{-1}(f(81))=81\)

We can first graph it:https://www.desmos.com/calculator/zwhxypzkrb

Thanks for your solutions! They really helped!

The answer is 81

you can see every time we multiply x by 3, the y goes up by 2

so 11-5=6

6/2=3

3^1*3^3=3^4=81

The answer is 4

g(g(x))=9

g(x)=5

9-5=4

I believe that the answer is C

In order to find the probability of a certain event, we need to set up the fraction; \(\frac{p(successful)}{p(possible)}\). Let's see how many numbers are successful:

Less than or equal to 5: \(1, 2, 3, 4, 5\)

Multiple of 7: \(7, 14, 21, 28\).

These nine numbers are successful, so your probability would be \(\frac{9}{30} = \frac{3}{10}\).

- Daisy

Less than or equal to 5....there are 5 cards  ...1,2,3,4,5

Multiple of 7....there are 4 cards   ....  7, 14, 21 , 28

So....  [5  + 4 ] / 30   = 9 / 30  =   3 /10

Problem 1: \(3600\) seconds are in \(1\) hour, so we can do \(\frac{4508}{3600} ≈ 1.25\) hours. 

Problem 2: \(258\) nanometers is equal to \(2.579999999e-9\) hectometers, which is about \(2.58e-9\) hectometers.

Problem 3: In order to convert micrograms to miligrams, we have to move to decimal point \(3\) times to the left. This gives us \(0.0081\) miligrams.

Problem 4: Remember: K H D B D C M(Kilo, Hecto, Deca, Basic, Deci, Centi, and Mili). This tells us that in order to convert kiloliters to centiliters, we need to move the decimal point \(5\) times to the right. This gives us \(640000\) centiliters. 

Problem 5: To convert kilograms to grams, we have to move the decimal point \(3\) times to the right. This will give us \(83000\) grams. 

- Daisy

it did not work

I don't really see much to explain here, aside from the fact that the horizontal axis would be the model of the car and the vertical axis would be the average miles per gallon of gas. Therefore, your answer would be B. model of car.

- Daisy

We can check each country:

Australia; From 1950 - 2000, there is an increase, and from 2000 - 2050 there is also an increase. So this country will expect an increase in population.

Hungary; While there is a slight increase from 1950 - 2000, there is also a slight decrease from 2000 - 2050.

Canada; From 1950 - 2000, there is an increase, and from 2000 - 2050 there is also an increase. So this country will expect an increase in population.

S. Korea; From 1950 - 2000, there is an increase, but there is a decrease from 2000 - 2050.

Egypt; From 1950 - 2000, there is an increase, and from 2000 - 2050 there is also an increase. So this country will expect an increase in population.

Out of these 5 countries, 3(Australia, Canada, and Egypt) will expect an increase in population from 1905 to 2000 and from 2000 to 2050. 

- Daisy 

This is an irreducible quadratic  [ cannot be factored  ]

You are correct, RP....63  mph  is the answer  !!!

Here's another way to figure it

Assuming that  the data  points are linear...let  ( x , y)  = ( 4 , 252)  and  (6 , 378)

The slope between these will be the rate of change

[ 378  -252  ] / [ 6  - 4  ]   =  126  / 2   = 63 miles  /  1  hr   

Just as RP  found....!!!!

This quadratic, \(z^2 - z - 3 = 0\) is a bit trickier. For this one, we need to use the quadratic formula, which is \(\frac{-b ± \sqrt{b^2 - 4ac}}{2a}\).

When we plug in the values of \(a, b,\) and \(c\), we get \(\frac{-(-1) ± \sqrt{(-1)^2 - 4(1)(-3)}}{2(1)} = \frac{1±\sqrt{13}}{2}\). This means \(z = \frac{1}{2} + \frac{1}{2}\sqrt{13}\) or \(z = \frac{1}{2} + \frac{-1}{2}\sqrt{13}\).

- Daisy