 My Math Forum HELP! ASAP!! A rally car in cross country race is travelling at night.

 Calculus Calculus Math Forum

October 25th, 2018, 11:38 AM   #11
Member

Joined: Oct 2018
From: Netherlands

Posts: 39
Thanks: 3

Quote:
 Originally Posted by Arisktotle I guess the only escape from that is to assume that the meters do not correspond to the coordinates.
That was a confusing sentence. It should have read:

I guess the only escape from that is to assume that the coordinate values are not the same as the meters.

Last edited by Arisktotle; October 25th, 2018 at 11:44 AM. October 25th, 2018, 01:03 PM #12 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 The problem is complete as given. y=ax^2 is the shape of the track. The parabola has to be shifted along x axis so that x=-100m (W) and y=100m (N) is a point on the track. Then ck the sign is outside the track and find the line from sign tangent to track (equate slope of line through sign to slope of a point on parabola. EDIT: To translate, y=a(x-k)^2 and -100,100 is a point on the curve gives k. Last edited by zylo; October 25th, 2018 at 01:38 PM. October 25th, 2018, 01:58 PM   #13
Member

Joined: Oct 2018
From: Netherlands

Posts: 39
Thanks: 3

Quote:
 Originally Posted by zylo The problem is complete as given.
Agree the problem is complete but have issues with the wind directions specified. Do not appear to be consistent with common conventions. October 25th, 2018, 02:18 PM   #14
Global Moderator

Joined: Dec 2006

Posts: 21,105
Thanks: 2324

Quote:
 Originally Posted by zylo To translate, y=a(x-k)^2 and -100,100 is a point on the curve gives k.
That doesn't work, as the given distances are supposed to be from the apex (i.e. the vertex) of the parabola. October 26th, 2018, 06:08 AM   #15
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 126

Quote:
 Originally Posted by skipjack That doesn't work, as the given distances are supposed to be from the apex (i.e. the vertex) of the parabola.
With vertex at the origin, rotate parabola about origin/vertex till it touches -100,100 and solve for angle of rotation. Exercise in analytic geometry. October 26th, 2018, 08:50 AM   #16
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 126

Quote:
 Originally Posted by zylo With vertex at the origin, rotate parabola about origin/vertex till it touches -100,100 and solve for angle of rotation. Exercise in analytic geometry.
ccw rotation t
x'=xcost+ysint
y'=-xsint+ycost
y'=ax'^2
Find t for case y=-x and then let x=-100.
-x(sint+cost)=ax^2(cost-sint)^2
(sint+cost)/(cost-sint)^2=-ax=18
Couldn't solve it. October 26th, 2018, 10:10 AM #17 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 There is a way to find rotation angle. Let R,$\displaystyle \phi$ = (sqrt2)100,135 be the location of -100.100 in polar coordinates. In polar coordinates, x=rcos$\displaystyle \theta$, y=rsin$\displaystyle \theta$ and y=ax^2 becomes sin$\displaystyle \theta$=(ar)cos^2$\displaystyle \theta$=ar(1-sin^2$\displaystyle \theta$) substitute a=.18, R=(sqrt2)100 and solve for $\displaystyle \theta$. Then the rotation angle is 135-$\displaystyle \theta$ Now you can find equation of rotated track in x,y coordinates, ck sign is outside track, and find tangent to parabola from line through sign. EDIT Better yet. Solve the problem in the rotated coordinate system. Then all you need is the coordinates of the sign in the rotated coordinate system and y'=ax'^2. Last edited by zylo; October 26th, 2018 at 10:33 AM. October 26th, 2018, 12:34 PM #18 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Solving for $\displaystyle \theta$ gives that a radius of (sqrt2)100 intersects y=ax^2 at 78 and 102deg. To -100,100 the parabola has to be rotated 135-102 =33deg ccw. In a coordinate system rotated $\displaystyle \alpha$ = 33deg ccw the coordinates of the sign become: x'=xcos$\displaystyle \alpha$+ysin$\displaystyle \alpha$=5cos33+4sin33=6.37 y'=ycos$\displaystyle \alpha$-xsin$\displaystyle \alpha$=4cos33-5sin33=.631 In the new coordinate system, dropping primes, the problem becomes y=ax^2 with sign at 6.37,.631 At x=6.37 distance to track is .18(6.37)^2= 7.30 and sign is outside track. Final step is to find line through sign tangent to track, and give result in original coordinate system. OP can do that. October 30th, 2018, 04:22 PM #19 Newbie   Joined: Oct 2018 From: Deep Outta Space Posts: 4 Thanks: 0 The equation is broken because the sign is inside the parabola therefore it's never going to touch. Last edited by skipjack; October 30th, 2018 at 11:23 PM. November 2nd, 2018, 06:08 PM #20 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Equation of curve in rotated coordinate system: y=ax$\displaystyle ^{2}$ Coordinates of signal in rotated coordinate system: (r,s)=(6.37,.631) See my previous post. Slope of curve = dy/dx = 2ax Equation of straight line through signal: y=mx+b s=mr+b $\displaystyle \rightarrow$ b=s-mr y=mx+s-mr This line is also tangent to curve at x$\displaystyle _{0}$, y$\displaystyle _{0}$ where m=2ax$\displaystyle _{0}$ and y$\displaystyle _{0}$=ax$\displaystyle _{0}^{2}$ $\displaystyle ax_{0}^{2}= (2ax_{0})x_{0}+s-2ax_{0}r$ This gives two tangent points from signal as expected. (solve for x$\displaystyle _{0}$ then y$\displaystyle _{0}$=ax$\displaystyle _{0}^{2}$) Tags asap, calculus, car, country, cross, modelling, night, problem solving, race, rally, travelling Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post MMath Art 17 October 24th, 2016 08:43 PM MMath Art 3 June 3rd, 2016 06:58 AM shunya Probability and Statistics 1 January 22nd, 2016 09:28 AM jillsandr New Users 7 December 15th, 2006 12:43 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      