Question 172553
Let {{{s}}}= her speed in still water
Let {{{c}}}= the speed of the current
Let {{{d}}}= the distance she has to go
Let {{{t[u]}}}= time going upstream in hrs
Let {{{t[d]}}}= time going downstream in hrs
Then
{{{s - c}}}= speed of boat going upstream
{{{s + c}}}= speed of boat going downstream
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The problem says
{{{c = 4}}} mi/hr
{{{t[u] = t[d] + (1/3)}}}
{{{d = 5}}} mi
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{{{d = (s - c)*t[u]}}} going upstream
{{{5 = (s - 4)*(t[d] + (1/3))}}}
{{{5 = t[d]*s - 4t[d] + (1/3)*s - (4/3)}}}
and, going downstream
{{{5 = (s + 4)*t[d]}}}
{{{t[d] = 5 / (s + 4)}}}
Substitute this into the upstream equation
{{{5 = (5/(s + 4))*s - 4*5 / (s + 4) + (1/3)*s - 4/3}}}
Multiply both sides by {{{3*(s + 4)}}}
{{{15*(s + 4) = 15s - 60 + s*(s + 4) - 4*(s + 4)}}}
{{{15s + 60 = 15s - 60 + s^2 + 4s - 4s - 16}}}
{{{s^2 - 120 - 16 = 0}}}
{{{s^2 - 136 = 0}}}
{{{s^2 = 136}}}
{{{s = 11.662}}} mi/hr
The speed of the boat in still water is 11.662 mi/hr
check answer:
{{{s - c = s - 4}}}
{{{s - 4 = 7.662}}}
{{{s + c = s + 4}}}
{{{s + 4 = 15.662}}}
{{{t[d] = 5 / (s + 4)}}}
{{{t[d] = 5 / 15.662}}}
{{{t[d] = .319}}}hrs
{{{t[u] = t[d] + (1/3)}}}
{{{t[u] = .6525}}}hrs
{{{d = (s - c)*t[u]}}} going upstream
{{{5 = 7.662*.6525}}}
{{{5 = 4.9998}}} close enough
also
{{{d = (s + c)*t[d]}}}
{{{5 = 15.662*.319}}}
{{{5 = 4.996}}} close enough