anonymous
  • anonymous
A car is traveling east at 100 mph and a truck is traveling north at 80 mph. when the car is 3 miles east of an intersection and the truck is 4 miles north of the same intersection, how fast is the distance between them changing?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
@DanJS
DanJS
  • DanJS
o, man , i havent been getting notification i dont think, sorry
DanJS
  • DanJS
did you realize that the intersection , and north and east, form a right-triangle?

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anonymous
  • anonymous
Yes, I am aware of that
anonymous
  • anonymous
that's alright lol
DanJS
  • DanJS
let the east direction for the car be called X and the north direction for the truck be called y, to make it look familiar
DanJS
  • DanJS
|dw:1449364749078:dw|
DanJS
  • DanJS
|dw:1449364891337:dw|
DanJS
  • DanJS
nothing is stated about where the car and truck came from, all they give you is a position and velocity in relation to this intersection...
DanJS
  • DanJS
car is at x = 3, and moving 100 mi/h east truck is at y = 4 and moving at 80 mi /h north
DanJS
  • DanJS
the direct distance between them , hypotenuse of right triangle, at this instant , is z = 5
DanJS
  • DanJS
so you have all three sides of a right triangle, and also the rates that two of the three sides are changing
DanJS
  • DanJS
the actual time here at this instant is irrelevant
DanJS
  • DanJS
if you differentiate the x^2 + y^2 = z^2 w.r.t time you get the rates of changes of distance, the velocities
DanJS
  • DanJS
\[\frac{ d }{ dt }[x^2 + y^2] = \frac{ d }{ dt }[z^2]\] to get derivative of x, w.r.t t, use chain rule, \[\large \frac{ d }{ dt }[x^2] = \frac{ d }{ dx }[x^2] * \frac{ dx }{ dt }\] same for the Y and Z
DanJS
  • DanJS
\[2x \frac{ dx }{ dt } + 2y \frac{ dy }{ dt } = 2z \frac{ dz }{ dt }\] you know all those terms except dz/dt
DanJS
  • DanJS
2 * 3 * 100 + 2*4 * 80 = 2*5 * (dz/dt) dz/dt = ( 600 + 160 ) / 10 = 76 mi/hr
DanJS
  • DanJS
brb
DanJS
  • DanJS
if you have some other practice probs on related rates or optimization , ill do another one
DanJS
  • DanJS
if i remember, most are the same process, for the calculus, you take a derivative of a property usually to relate what you know to what you dont just other scenarios , like water tanks draining and the situation at a instant for height and stuff, different shapes..
anonymous
  • anonymous
Sorry got cut of but I am lookin gover ur work.
DanJS
  • DanJS
k,

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