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Sandy will get $6120 more if her salary is increased by 12%

\(sa(1+0.12)=sa+6120\) 

Sandy's salary is 15% less of her colleague Sonia's salary.

\(sa = (1-0.15)so\) 

First solve for sa using the first equation then plug that value into the second equation

and solve for so

\(f(t) = \cos\left(\dfrac{\pi t}{6}\right)\\ \dfrac{df}{dt}(t) = -\sin\left(\dfrac{\pi t}{6}\right)\cdot \dfrac \pi 6\\ \dfrac{df}{dt}(1) = -\sin\left(\dfrac{\pi}{6}\right)\cdot \dfrac \pi 6 = -\dfrac 1 2 \dfrac \pi 6 = -\dfrac{\pi}{12}\)

What did you get?

Problem 1 you are correct

Problem 3 you are not correct (what is 540/2?)

Problem 2 - come on... it's almost trivial.  What is f(-2)?  what is g(-2)?

\(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) = \\ \dfrac 1 2 + \cos(\alpha)\sin(\beta)\\ \cos(\alpha)\sin(\beta) = \sin(\alpha+\beta)-\dfrac 1 2\)

\(\text{You should know the range of }\sin(\alpha +\beta)\\ \text{So what is the range of }\sin(\alpha +\beta)-\dfrac 1 2\)

Hyperbolas and parabolas are not tangents. That makes no sense.

I assume you mean that they only touch each other in one place. Which means they have only one simultaneous solution.

I would try making x^2 the subject of the formula for the first equation.

then

substitute this value of x^2 inot the second equation.

then

solve the new equation as an quadratic.  You may then have 2 values for y  (in terms of m)

and

etc

Try that and get back with what you have done.

 No one answer over me. Thanks 

1) Average rate of change is change in f(x) value divided by change in x values.   

This is the gradient of the line joining the end points.

Please no one answer further on question 1

Who cares  ?  

When you ask too many questions you draw negative attention to yourself.

A very nice answer and a very nice acknowledgment response.   

Thanks guys  ( In Australia guys includes males as well as females )

Yeah, that is right. In general, for two points, the slope is:\(\frac{y-2-y_1}{x_2-x_1}.\)

THANKS TERTRE    

Tertre, I have written a more full answer.   (I started before you but finished after)

I could not get it from your hint.  EDIT : (Maybe I could, I have not thought about your hint very much yet)

Even what I have done is only one scenario.  I have not shown that it will always be the case.

I have not been able to get my head around the general case myself.    

Anyway, I will leave mine hidden for a bit and see if the guest comes back and interacts with you.   :)

40 boys and 28 girls stand in a circle, hand in hand, all facing inward.

Exactly 18 boys give their right hand to a girl. How many boys give their left hand to a girl?

If you had the   King at one 'end' of the the table with the Queen on his right    KQ                  

then you have all the maids to the right of the queen (there are 10 of them)      FFFFFFFFFF                      

then you have the married couples  with the wife always on the right. (17 of them )    MF MF MF MF MF MF MF MF MF MF MF MF MF MF MF MF MF

then you have all the bachellors with the king to the right of the last one    (   22 of them)   M .....    M  

Which boys have girls on their left ?

Not the king

All 17 of the couples do

One of the bachelors does  (lucky him)

So that is 18.

It probably always works out to 18.        

OHHHHH I GET IT!!!

I never would have thought of splitting it into two equations very brilliant!!

Hint: When does x^y=1? Split this into cases, and try to look for what the power must be equal to.

Hint: 22 boys give their right hand to a boy.

I know, it's a hint, but it's pretty big, as you can get the answer from there.

That is a really tricky question.  I wonder if there really is just one answer???

Yes, it looks like homework questions.

\(P[\text{Cubs win the series}] = \sum \limits_{l=0}^3~\left(\dfrac 3 5\right)^4 \left(\dfrac 2 5\right)^l = \dfrac{16443}{78125} \approx 0.21 = 21\%\)

thanks :)