Answer :
The point where the boat should be landed is the point 3.4 miles from point P towards the town.
The point where the boat should be landed can be found by expressing
the distance traveled on the boat and walking as a function of time.
Given that x represents the distance from point P to the boat landing point.
Therefore, distance of rowing the boat = √((12 - x)² + 3²)
The total time, t, is, therefore;
[tex]t=\frac{12-x}{4} +\frac{\sqrt{x^2+3^2} }{3}-----------(1)[/tex]
differentiate the above equation we get:
[tex]\frac{dt}{dx} =\frac{d}{dx} (\frac{12-x}{4} +\frac{\sqrt{x^2+3^2} }{3} )\\\\\\=\frac{12.(-3+4*\frac{2x}{2*\sqrt{x^2+9} } )}{144} \\\\\\=\frac{12(-3+4*\frac{2x}{2*\sqrt{x^2+9} }) }{144}\\\\=-\frac{1}{4} +\frac{x}{3*\sqrt{x^2+9} } \\\\\frac{x}{3*\sqrt{x^2+9} }=\frac{1}{4} \\\\cross multiplying both terms we get:\\\\4x=3*\sqrt{x^2+9}[/tex]
squaring on both sides we get:
16x^2=9(X^2+9)
16x^2=9x^2+81
7x^2=81
[tex]x=\frac{9*\sqrt{x} }{7}[/tex]
Approximately,x=3.4 miles.
To know more about the Distance of points:
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