Discussion
A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is:
Answer: A
Write \( b \) for the boat's speed in still water and \( r \) for the stream's. Then
\[ \text{downstream} = b + r, \qquad \text{upstream} = b - r \]
\[ 90 \text{ min} = 1.5 \text{ h} \]
\[ \frac{36}{10 - r} - \frac{36}{10 + r} = 1.5 \]
\[ \frac{36(10 + r) - 36(10 - r)}{100 - r^{2}} = \frac{72r}{100 - r^{2}} = 1.5 \]
\[ 72r = 150 - 1.5r^{2} \quad\Longrightarrow\quad r^{2} + 48r - 100 = 0 \]
\[ r = \frac{-48 + \sqrt{2304 + 400}}{2} = \frac{-48 + 52}{2} = 2 \text{ mph} \]
The negative root is discarded as a speed. Checking: 36 miles at 8 mph takes 4.5 hours and at 12 mph takes 3 - exactly 90 minutes apart.
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