In hot pursuit, Agent Logan of the FBI

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Homework Statement

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In hot pursuit, Agent Logan of the FBI must get directly across a 1200-m-wide river in minimum time. The river's current is 0.80 m/s, he can row a boat at 1.60 m/s, and he can run 3.00 m/s. Describe the path he should take(rowing plus running along the shore) for the minimum crossing time, and determine the minimum time.

Homework Equations

x = v*t

The Attempt at a Solution

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1
2 <---vRiver= 0.80 m/s
0
0

m
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Let the angle between dashes and slash be x.
The agent must be at the dot when he's done with rowing and running.
The distance between the dot and the point he arrives after rowing(bold dashes) is y.
Rowing time is: t₁
Running time is: t₂So the equations are:

1200 = 1.60*(sinx)*t₁
y = (1.60*(cosx) - 0.80)*t₁
y = 3.00 * t₂

t₁ + t₂ = 750/sinx + y/3.00
t₁ + t₂ = (750/sinx) + (1.60*(cosx) - 0.80)*1200/(3*1.60*(sinx))

In short,

t₁ + t₂ = 550/(sinx) + 400cot(x)

After taking derivative,

-550(cosx)/(sin²x) - 400/(sin²x) = 0

The angle I got from this is 136.7 degrees.
180-136.7 = 43.3 degrees is the angle between the river current and the boat.

But according to my book, the correct answer is the following.
"Row at an angle of 24.9 degrees upstream and run 104m along the bank in a total time of 862 seconds."I guess my thinking on this question is wrong. Any ideas how to draw the vectors here?

 

[luisbv880000]

I solved the problem and I got the result from the book. I suggest keeping your variables as symbolic during the calculations, and using their numerical values just at the end. (In that way you get rid of all those numbers and you get pretty neat expressions). Give names to the quantities:
$d=1200$m
$v_{row}=1.60$m/s
$v_{riv}=0.80$m/s
$v_{run}=3.00$m/s
Apart from that, you should use the angle defined as you want it in the answer. With that, the velocity of the boat can be written as:
$\vec{v}=(v_{row}\cos\theta,v_{row}\sin\theta-v_{riv})$

I consider your approach is correct, just revise your calculations using what I just told you.

At the end of the calculation you will notice that the value of the angle does not depend on the width of the river. (If you replace the numbers from the beginning it is impossible to notice that)

 

[gerardooo]

you add 0.8 not subtract it xd

 

[haruspex]

you add 0.8 not subtract it xd

Not quite.
He will row partly against the current, so 1.60*(cosx) - 0.80 is correct. The error is that the distance to run is never negative, so it should read y = |(1.60*(cosx) - 0.80)|*t1.
And since there is no benefit in arriving upstream of the target, we can simplify that to y = -(1.60*(cosx) - 0.80)*t1.