/sci/ - Science & Math

eng here, I have a task for the math nerds.

I need a flexible model to describe a polynomial but the constraints are fucking me up.

here is the problem.

x-axis is the day in hours (or minutes), y-axis is temperature.
every day has individual length, every day has individual temperatures.
the idea is to have a general model for temperature development over one day by providing a polynome.
every day starts with sunset(x=0) for us.
I have the length, tmin and tmax of the days.
(the aim is to describe time ranges of a day above a certain temperature - but thats not important here)

x= 0h (or 0 mins) is the minimum temperature (which is roughly around sunset and to make it easier we put it exactly there)
x= 24h (or 1440 mins) is again the same minimum temperature.
both points are local minima and also the value tmin.

this part is easy.
difficult to me is what comes next.

at a certain length (that I know but varies) the maximum temperature is reached. its always 75% of the length of a day to make it easier here.
for example when a day has 10 hours of light tmax will be reached at x = 0.75*10h = 7.5h (or 0.75*10*60 mins), when the day is 8h it is 6h etc.
this point needs to be the value tmax AND a local maximum.

this gives me 6 equations for a polynome.
y(x=0h) = tmin
y'(x=0h) = 0
y(x=24h) = tmin
y'(x=24h) = 0
y(x=0.75*length)=tmax
y'(x=0.75*length)=0

>>1858909
"you are swimming in a race across a lake and back. swimmers must swim to, and then back from, a buoy placed 2640ft. from the center of the start/finish line. you start the race 100ft from the center of the start/finish line as shown.
/
/ |
/ | 2640ft
/ |
/ __ |
100ft

a. you swim to the buoy at a steady rate of 0.7 foot per second. Write a set of parametric equations for your path (i got this part)
b. Use the equation to determine how long it takes you to reach the buoy
c. If you continue to swim at a stteady rate of 0.7 foot per second straight back to the center of the start/finish line, how long will it take for you to complete the race?

>mfw part b

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