(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a long straight river flowing north with parallel banks 40m apart. Let us use

the function u(x) = 3 sin([tex]\pi[/tex]x/40) to model the rate of water flow x units from the west bank.A boat proceeds at a constant speed of 5m/s from a point A on the west bank

while maintaining a heading perpendicular to the bank. How far down the river

on the opposite bank will the boat touch shore?

3. The attempt at a solution

The velocity vector is V(x) = <5, u(x)>.

As far as my understanding of the problem goes I should use V(x) to develop a vector function for dispacement D(x), parameterize it to D(t), and then find the arc length by integrating

[tex]\int ||D'(t)||dt[/tex].

My initial attempt was to multiply V(x) by a variable for time (T) and parameterize with x = t, however the bounds I have are for x (0 and 40) not time so that didn't get me anywhere. My next thought was that I remembered that displacement was the integral of velocity and I should just integrate V(x) with x = t, lower bound t =0, and upper bound t = 40.

[tex]\int V(t)dt[/tex]

This gives me <200,240/pi> which means the answer is 214 (applying the typical norm) which seems pretty high

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# Vector Calculus Boat Traversing a river problem

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