Answer: 9 miles per hour
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Explanation:
x = boat's speed on the lake (aka speed in still water)
Going with the river current, the boat will go x+6 mph, because it gets a 6 mph boost in speed. It travels for some amount of time, call it A. It travels 22.5 miles
So,
distance = rate*time
22.5 = (x+6)*A
A = 22.5/(x+6)
We'll use this later.
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Going against the current, the boat's initial speed x drops to x-6, because the current slows down the boat now.
It travels for some time B and goes a distance of 22.5 miles
So,
distance = rate*time
22.5 = (x-6)*B
B = 22.5/(x-6)
We'll use this later as well.
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The time values A and B must add to 9 hours
A + B = 9
Plug in A = 22.5/(x+6) and B = 22.5/(x-6)
22.5/(x+6) + 22.5/(x-6) = 9
At this point we have a single equation in terms of one variable. Solve for x.
To make things easier, it helps to get rid of the fractions. Multiply both sides by (x-6)(x+6) to clear out the denominators.
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[tex]\frac{22.5}{x+6} + \frac{22.5}{x-6} = 9\\\\(x-6)(x+6)\left(\frac{22.5}{x+6} + \frac{22.5}{x-6}\right) = 9(x-6)(x+6)\\\\22.5(x-6)+22.5(x+6) = 9(x-6)(x+6)\\\\22.5x-135+22.5x+135 = 9(x^2-36)\\\\45x = 9x^2-324\\\\0 = 9x^2-324-45x\\\\0 = 9x^2-45x-324\\\\9x^2-45x-324 = 0\\\\9(x^2-5x-36) = 0\\\\x^2-5x-36 = 0\\\\(x-9)(x+4) = 0\\\\x-9=0 \ \text{or} \ x+4 = 0\\\\x=9 \ \text{or} \ x = -4\\\\[/tex]
Ignore the solution x = -4 as it doesn't make sense to have a negative speed.
Therefore, the boat's speed in still water must be 9 miles per hour.
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If the boat's speed in still water is x = 9 mph, then,
speed with current = x+6 = 9+6 = 15 mph
speed against current = x-6 = 9-6 = 3 mph
Going with the current, the boat spends d/(x+6) = 22.5/15 = 1.5 hrs traveling
Going against the current, the boat spends d/(x-6) = 22.5/3 = 7.5 hrs traveling
total travel time = 1.5 hrs + 7.5 hrs = 9 hrs
This confirms we have the right answer.
