rate

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bianca bianca
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rate

A small motorboat travels 10 mph in still water.  It takes 3 hours longer to travel 70 miles going upstream than it does going downstream.  Find the rate of the current.
AlgebraHouse AlgebraHouse
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Re: rate

r = rate of current
10 + r = speed of boat going downstream   {with the current}
10 - r = speed of boat going upstream  {against the current}

t = time in hours going downstream
t + 3 = time in hours going upstream

distance = rate x time

Upstream
d = 70
rate = 10 - r
time = t + 3
70 = 10t + 30 - rt - 3r   {distance = rate x time}
70 = 10t - rt + 30 - 3r   {re-arranged}
70 = t(10 - r) + 30 - 3r   {factored t out of first two terms}
t(10 - r) = 3r + 40   {subtracted 30 and added 3r}

 3r + 40
---------- = t   {divided each side by (10 - r)
  10 - r

Downstream
d = 70
r = 10 + r
t = t
70 = (10 + r)(t)   {distance = rate x time}

   70
-------- = t   {solved for t}
10 + r

3r + 40         70
---------- = --------   {both were equal to t, so set them equal to each other}
  10 - r        10 + r

30r + 3r² + 400 + 40r = 700 - 70r   {cross-multiplied}
3r² + 70r + 400 = 700 - 70r   {combined like terms}
3r² + 140r - 300 = 0   {subtracted 700 and added 70r}
r ≈ 2.05  {used quadratic formula and rounded}

rate of current is approximately 2.05 mph

- Algebra House