Question 1023682
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Julia's Boston Whaler cruised 45 miles upstream and 45 miles back in a total of 8 hours. The speed of the river is 3 mph. Find the speed of the boat in still water
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Let &nbsp;{{{u}}}&nbsp; be the unknown speed of the boat in still water (in  miles per hour).

When the boat moves upstream, &nbsp;its speed relative to the bank of the river is &nbsp;{{{u-3}}}&nbsp; miles per hour, 
and the time spent moving upstream is &nbsp;{{{45/(u-3)}}} &nbsp;hours. 

When the boat moves downstream, &nbsp;its speed relative to the bank of the river is {{{u+3}}} &nbsp;miles per hour, 
and the time spent moving downstream is &nbsp;{{{45/(u+3)}}}&nbsp; hours. 

So, &nbsp;the total time upstream and downstream is &nbsp;{{{45/(u-3) + 45/(u+3)}}}, &nbsp;and it is equal to &nbsp;8&nbsp; hours, &nbsp;according to the problem's input.

This gives an equation &nbsp;{{{45/(u-3) + 45/(u+3) = 8}}}. 

To simplify the equation, &nbsp;multiply both sides by &nbsp;{{{(u-3)*(u+3)}}} &nbsp;and collect like terms. &nbsp;Step by step, &nbsp;you will get 

{{{45(u+3) + 45(u-3)}}} = {{{8*(u+3)(u-3)}}},
{{{90u}}} = {{{8*(u^2-9)}}},
{{{90u = 8u^2 - 72}}},
{{{8u^2 - 90u - 72}}} = {{{0}}}.

Now cancel both sides by the factor 2. You will get

{{{4u^2 - 45u - 36}}} = {{{0}}}.

Apply the quadratic formula

{{{u[1,2]}}} = {{{(45 +- sqrt(45^2 + 4*4*36))/8}}} = {{{(45 +- sqrt(2601))/8}}} = {{{(45 +- 51)/8}}}.

You need only positive root u = {{{96/8}}} = 12.

The boat speed in still water is 12 mph.

<B>Answer</B>. &nbsp;The boat speed in still water is &nbsp;12&nbsp; mile per hour.
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