The speed of the current in a river is 6 mph. A ferry operator who works that part of the river is looking to buy a new boat for his business. Every day, his route takes him 22.5 miles each way against the current and back to his dock, and he needs to make this trip in a total of 9 hours. He has a boat in mind, but he can only test it on a lake where there is no current. How fast must the boat go on the lake in order for it to serve the ferry operator's needs?
PLEASE SHOW YOUR WORK

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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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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.

25/6 hope it : )

step-by-step explanation: