Question

HELP WITH POLYNOMIALS PLEASE!! This is a pre-calc project about designing a roller coaster, and I just can't figure it out! Here are the requirements: The amusement park you are designing for gave you the following coaster requirements: - your coaster ride must have at least 3 relative maxima and/or minima - the ride length must be at least 4 minutes - the coaster ride starts at 250 feet - the ride dives below the ground into a tunnel at least once. There must be at least one double root, 2 real roots, and imaginary roots. Help please!!!AND IT ALSO HAS TO GO BACK TO 250 FEET

Answer 1

ok from the requirements we know that polynomial must be at least a 6th degree
double root -->2 , 2 real, 2 imaginary equal 6 total roots

f(0) = 250
f(T) = 250
where T is time it takes to complete roller coaster, T>4

also the double root and 2 real roots must be within the interval 0<t<T

a general format would look like this:
\[f(t) = a(t-d)^2 (t-h)(t-k)\]
--> a*d^2*h*k = 250

Answer 2

oops i forgot about the imaginary roots
\[f(t) = a(t^2 +i)(t-d)^2 (t-h)(t-k)\]

Answer 3

because there are a lot of unknowns, i would pre-determine length of ride and real roots

example:
T = 5
h = 1
k = 2
d = 4
\[f(0) = a(0+i)(0-4)^2(0-1)(0-2) = 250\]
\[f(5) = a(25+i)(5-4)^2 (5-1)(5-2) = 250\]
\[\rightarrow 32 a i = 250\]
\[\rightarrow 12a (25+i) = 250\]
solve for a and i

Answer 4

\[a = \frac{25}{48}\]
\[i = 15\]

Answer 5

@HelpMewithpolynomials