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A man stands at a point A on the bank of a straight river, 3 mi wide. To reach point B, L = 9 mi downstream on the opposite bank, he first rows his boat to point P on the opposite bank and then walks the remaining distance x to B, as shown in the figure. He can row at a speed of 2 mi/h and walk at a speed of 5 mi/h.

(a) Find a function that models the time T needed for the trip.
(b) Where should he land so that he reaches B as soon as possible? (Round your answer to two decimal places.)

A man stands at a point A on the bank of a straight river 3 mi wide To reach point B L 9 mi downstream on the opposite bank he first rows his boat to point P on class=

Respuesta :

so...hmm notice the picture

we get distance AP by using the pythagorean theorem

now, AP is just rowing, and PB is walking
their rates are 2mph and 5mph respectively
so... let us apply those rates to those distances of AP and PB

[tex]\bf \textit{distance}=AP+PB\to \sqrt{x^2-18x+90}+x\quad \begin{cases} AP\ rate=2mph\\ PB\ rate=5mph \end{cases} \\\\ thus \\\\ \textit{time it took in hours}=t=\cfrac{\sqrt{x^2-18x+90}\ mi}{\frac{2mi}{h}}+\cfrac{x\ mi}{\frac{5mi}{h}} \\\\\\ t=\cfrac{\sqrt{x^2-18x+90}}{2}+\cfrac{x}{5}[/tex]
Ver imagen jdoe0001
okpalawalter8

The functions that models the time T and position it lands is mathematically given as

[tex]T=\frac{\sqrt{x^2-18x+90}+x}{2ni/h}+x/5mi/ hr[/tex]

[tex]x=\sqrt{x^2-18x+90+x}[/tex]

What is a function that models the time T and position it lands?

a)

Generally, the equation for the distance  is mathematically given as

d=AP+PB

[tex]d=\sqrt{x^2-18x+90}+x[/tex]

Therefore

Time taken is

[tex]T=\frac{\sqrt{x^2-18x+90}+x}{2ni/h}+x/5mi/ hr[/tex]

b)

Hence The  position ist lands are given by

[tex]x=\sqrt{9+81-18x+x^2+x}[/tex]

[tex]x=\sqrt{x^2-18x+90+x}[/tex]

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