Rom

check that you've got it set to use whatever units you are, radians or degrees

There were 20% more oranges than grapes. 

\(o = (1+0.2)g\) 

When 126 fruits were added, the oranges were increased by 10% and the grapes increased by 9%

\((1+0.1)o + (1+0.09)g = o + g + 126\) 

Plug the first equation into the second and simplify and solve for g

Then use that value to calculate o

Finally find o + g

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)?

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

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

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

\(P[1] = \dbinom{2}{1}(0.48)(0.52) = 2 (0.48)(0.52) = 0.4992\\ \text{The table at the link got this wrong by a factor of 2}\\ \text{because they only accounted for one of the 2 paths to get to 1 girl/1 boy}\)

\(\text{The expression seems to vary from }\dfrac 3 2\\ \text{which corresponds to an equilateral triangle to }2\\ \text{which corresponds to an isosceles triangle with one leg having zero length}\\ \dfrac 3 2 \leq \dfrac{a}{b+c}+ \dfrac{b}{a+c}+\dfrac{c}{a+b} < 2\)

\(3! \sqrt{49} = 42\)