CPhill

1) A rope is 120 feet long. Starting from one end, make a red mark every 3 feet, make a blue mark every 4 feet, and cut the rope at every mark. (Where there’s both a red mark and a blue mark, we only make one cut, not two.) How many pieces of rope do we have now?

To see the answer....note that....in the first 12 feet  we make cut at the  3, 4, 6, 8, 9 and 12 foot marks

So...every 12 feet....we have 6 pieces of rope

So....we have   (10) (6)  =  60 pieces in total 

Eh...I left out 60....the other multiple of 4, 5

So

(25 + 45) - 5  =  40

And 

100 - 40  =  60  !!!

Here's my best effort :

74.   cos ( θ + k π)  =

cosθ   cos kπ   -  sin θ  sin π  =

cos θ cos kπ

If  k is even  then     cos kπ  =  1 ...... so cos θ * (1)  =   (-1)^kcosθ   

If k is odd   then    cos kπ  =  -1 ...... so cos θ * (-1)  =   (-1)^kcosθ

Oops...sorry....you wanted the ones that remain  !!!

That would be  100  - 41   = 59

3)  If all multiples of 4 and all multiples of 5 are removed from the set of integers from 1 through 100, how many integers remain?

We have 25  multiples  of 4   and  20 multiples of 5

However....we've  "double-counted"  the multiples of 4 and 5   =  20, 40, 80  and 100

So

[ 25 + 20 ] -  4   =     41

1) In a survey of 100  students who watch television, 21 watch American Idol, 39 watch Lost, and 8 watch both. How many of the students surveyed watch at least one of the two shows?

(21 - 8)  =  13  = Those that watch "American Idol" only

(39 - 8)  =  31 =  Those that watch "Lost" only

So....the number that watch at least one  is  

13 + 8 +  31  =

52

Mmmmm....must be a "ghost in the machine"......consider it "un-flagged"

Nice, Geno   !!!

Good job, CC!!!

Focus  =  (-8, -3)

Directrix  x = - 2

This parabola opens to the left because the focus is to the left of the directrix

For this reason....we can throw out answers B and D.....these parabolas open up/down

The  vertex is  [ (-8 - 2)/2, -3)  =  ( -10/2, -3)  =  (-5, -3)

We can stop here

"C"   has  a vertex of (-5, -3)   and the presence of the " - "  in front of the "12" indicates that this one opens to the left