Finding critical numbers for this ridiculous calculus problem?

[lemres]

This is my first time ever asking the internet for help on a math question. I feel really stumped here.

So this is the question.

"Economic data is quite variable and very complicated, making it difficult to model. The models we use here are rough approximations of the Unemployment Rate and Gross Domestic Product (GDP).
The unemployment rate can be modeled by: U(t) = 5cos(2t/pi)/[2+sin(2t/pi)] + 6 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

The GDP growth rate can be modeled by: G(t) = 5cos(2t/pi+5)+2 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

If the GDP growth rate is increasing, the economy is expanding and if it is decreasing, the economy is contracting. A recession occurs when GDP growth rate is negative.

Ok so the question is "During what years was the U.S economy in a recession?".
I'm assuming you use the GDP equation for this question. This is what I got:
G'(t) = cos(2t/pi+5) * t
cos(2t/pi+5) * t = 0 (Divided both sides by cos(2t/pi+5)
t=0

I honestly don't know what to do with cos(2t/pi+5). I thought you could bring t out of it but it doesn't seem like it would work. I felt this was too easy though so I have doubts about this being the right answer.
Help would be much appreciated. Thanks in advance!

P.S - The title of this thread is for another question that I have that's about finding crit. numbers. I hope any of you don't mind helping me with another weird question. -_-

 

[Ishuda]

This is my first time ever asking the internet for help on a math question. I feel really stumped here.

So this is the question.

"Economic data is quite variable and very complicated, making it difficult to model. The models we use here are rough approximations of the Unemployment Rate and Gross Domestic Product (GDP).
The unemployment rate can be modeled by: U(t) = 5cos(2t/pi)/[2+sin(2t/pi)] + 6 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

The GDP growth rate can be modeled by: G(t) = 5cos(2t/pi+5)+2 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

If the GDP growth rate is increasing, the economy is expanding and if it is decreasing, the economy is contracting. A recession occurs when GDP growth rate is negative.

Ok so the question is "During what years was the U.S economy in a recession?".
I'm assuming you use the GDP equation for this question. This is what I got:
G'(t) = cos(2t/pi+5) * t
cos(2t/pi+5) * t = 0 (Divided both sides by cos(2t/pi+5)
t=0

I honestly don't know what to do with cos(2t/pi+5). I thought you could bring t out of it but it doesn't seem like it would work. I felt this was too easy though so I have doubts about this being the right answer.
Help would be much appreciated. Thanks in advance!

P.S - The title of this thread is for another question that I have that's about finding crit. numbers. I hope any of you don't mind helping me with another weird question. -_-

First, about posts - see
http://www.freemathhelp.com/forum/threads/41538-Read-Before-Posting!!
There is no problem making multiple posts and asking multiple questions, just please make it one question per thread and make (as) sure (as you can) that it is in the correct section

On to the other question: "During what years was the U.S economy in a recession?" You are correct that you would use the GDP formula
\(\displaystyle G(t)\space =\space 5\space cos(\frac{2\space t}{\pi}+5)\space +\space 2\)
and you do need the derivative [A recession occurs when GDP growth rate is negative]. However, it looks like you did not do the derivative correctly if you got
G'(t) = cos(\(\displaystyle \frac{2\space t}{\pi}\space +5\)) * t

 

[HallsofIvy]

This is my first time ever asking the internet for help on a math question. I feel really stumped here.

So this is the question.

"Economic data is quite variable and very complicated, making it difficult to model. The models we use here are rough approximations of the Unemployment Rate and Gross Domestic Product (GDP).
The unemployment rate can be modeled by: U(t) = 5cos(2t/pi)/[2+sin(2t/pi)] + 6 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

The GDP growth rate can be modeled by: G(t) = 5cos(2t/pi+5)+2 where t is in years and t=0 represents January 1, 1985.
0 < or equal to t < or equal to 25

If the GDP growth rate is increasing, the economy is expanding and if it is decreasing, the economy is contracting. A recession occurs when GDP growth rate is negative.

Ok so the question is "During what years was the U.S economy in a recession?".
I'm assuming you use the GDP equation for this question.

Why would that be an "assumption"? Just above is you are told "A recession occurs when GDP growth rate is negative".

This is what I got:
G'(t) = cos(2t/pi+5) * t

What? You are told that G(t)= 5 cos(2t/pi+ 5)+ 2. The derivative of that is NOT what you give here- it is G'(t)= -(2/pi) sin(2t/pi+ 5).

cos(2t/pi+5) * t = 0 (Divided both sides by cos(2t/pi+5)
t=0

You always have to be careful when multiplying by a variable. You cannot divide by 0 so this will not give you solutions were cos(2t/pi+ 5)= 0.

If AB= 0 then either A= 0 or B= 0. Of course, as I pointed out before, you have the wrong derivative. You want to find t such that -(2/pi)sin(2t/pi+ 5) is negative. Since this is a continuous function, places where it is positive and places where it I negative must be separate by points where it is 0 so you can start by solving -(2/pi)sin(2t/pi+ 5)= 0 then checking to see in which of the intervals determined it is negative.

I honestly don't know what to do with cos(2t/pi+5). I thought you could bring t out of it but it doesn't seem like it would work. I felt this was too easy though so I have doubts about this being the right answer.
Help would be much appreciated. Thanks in advance!

P.S - The title of this thread is for another question that I have that's about finding crit. numbers. I hope any of you don't mind helping me with another weird question. -_-

 

[lemres]

Why would that be an "assumption"? Just above is you are told "A recession occurs when GDP growth rate is negative".

There's actually another equation the problem gives me that also has something to do with years so I wasn't sure.

What? You are told that G(t)= 5 cos(2t/pi+ 5)+ 2. The derivative of that is NOT what you give here- it is G'(t)= -(2/pi) sin(2t/pi+ 5).


You always have to be careful when multiplying by a variable. You cannot divide by 0 so this will not give you solutions were cos(2t/pi+ 5)= 0.

If AB= 0 then either A= 0 or B= 0. Of course, as I pointed out before, you have the wrong derivative. You want to find t such that -(2/pi)sin(2t/pi+ 5) is negative. Since this is a continuous function, places where it is positive and places where it I negative must be separate by points where it is 0 so you can start by solving -(2/pi)sin(2t/pi+ 5)= 0 then checking to see in which of the intervals determined it is negative.

Wow I can't believe I made such a dumb mistake... Thanks for that haha.
I'm still confused though because if I'm supposed to be solving for t, how do I get t by itself and out of sin?
Or do I use the sum formula? I think I'm overthinking this problem.