Mathematical Modelling, Vectors and Parametric Equations

[Nile Anderson]

Homework Statement

Sorry to disappoint the math fanatics but no this is not a question that integrates all three topics at once but individual ones. I still need assistance though with the following more so in the reasoning behind them as I feel my logic is flawed.
https://scontent-mia1-1.xx.fbcdn.net/hphotos-xfa1/v/t34.0-12/11350393_1080048282023208_1468717877_n.jpg?oh=42c6757af8f078a9a8f525051260bde9&oe=5576CB04
https://scontent-mia1-1.xx.fbcdn.net/hphotos-xfa1/v/t34.0-12/11358829_1080048288689874_1045920915_n.jpg?oh=8a5bced9bb2de46109e19b8e50124969&oe=5575A90C
https://scontent-mia1-1.xx.fbcdn.net/hphotos-xpt1/v/t34.0-12/10003237_1080048292023207_149145089_n.jpg?oh=d49bf300e8c582b94ef1b70a47c9c616&oe=5575A4F2

Homework Equations

There are not really any relevant equations but look at my solutions below.

The Attempt at a Solution

I am pretty sure these are wrong
22D
28C
42C
43B

 

[Orodruin]

Please also state your reasoning behind your answers. Simply selecting on random does not really qualify as an attempt and it will be easier for everyone to see when/if you have the wrong idea.

 

[Nile Anderson]

Please also state your reasoning behind your answers. Simply selecting on random does not really qualify as an attempt and it will be easier for everyone to see when/if you have the wrong idea.

Sorry my good sir,
22) To be completely honest , the approach to this one is fuzzy for me as I have only done two variables at a time before this but i thought that i could square the x and y to get that x² + y² = r²cos²θ and then adding z squared would produce x² + y² +z² = r²cos²θ + r²sin²θ= r², is this correct , is there no simpler approach if so
28) This one was even trickier I imagined that is they were perpendicular then their dot product = 0 , but things got hairy when r.s came up , as I only had so far (r-s)(2r+3s)=0, I said r=s for one solution so r.s must me rs which makes no sense when I repeat it now .
42) I found the first derivative and assumed where it was greater than zero , the graph was increasing
43) I said it was 10 years between 1990 and 2000 and so it was zero to ten.

 

[Nile Anderson]

Please also state your reasoning behind your answers. Simply selecting on random does not really qualify as an attempt and it will be easier for everyone to see when/if you have the wrong idea.

Is this not satisfactory ?

 

[haruspex]

Your reasoning and answer for 22 and 42 are fine.
For 43, you are right, but your explanation is a bit vague. Note the exact statement of the start and end times.
For 28:

28) This one was even trickier I imagined that is they were perpendicular then their dot product = 0 , but things got hairy when r.s came up , as I only had so far (r-s)(2r+3s)=0.

Expand that and isolate r.s.

 

[Nile Anderson]

Your reasoning and answer for 22 and 42 are fine.
For 43, you are right, but your explanation is a bit vague. Note the exact statement of the start and end times.
For 28:

Expand that and isolate r.s.

Darn should have went with first instinct
(r-s)(2r+3s)=0
2r.r +3 s.r - 2s.r - 3s.s=0
2r² + s.r -3s²=0
s.r= 3s²-2r²
Making A the answer, Thanks Mr. Haruspex Sir
About the integral one though, it was just basically an intuitive guess , I still don't understand why I would start at 0 and why end at 10 , why that would not work out to 11 years versus the 10 from 1990 to 1999 ( Because it is not the end of 2000), I am pretty clueless about that one, no intuition whatsoever

 

[Orodruin]

Exactly how many years are between Jan 1, 1990, and Jan 1, 2000? You are allowed to use decimals.

 

[Nile Anderson]

10 years , yes ok but why do we start at zero , is that a general rule ?

 

[Orodruin]

No, this must be chosen according to how the time t=0 was selected, but this is not explicitly stated in the problem.