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A boat traveled upstream \(90\) miles at an average speed of \((\mathrm{v}-3)\) miles per hour and then traveled the same distance downstream at an average speed of \((\mathrm{v}+3)\) miles per hour. If the trip upstream took a half hour longer than the trip downstream, then how many hours did it take the boat to travel downstream?
Let's break down what's happening in this problem using everyday language first.
Imagine you're in a boat on a river. When you go upstream (against the current), the water pushes back against you, making you slower. When you go downstream (with the current), the water helps push you forward, making you faster.
In our problem:
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Now let's think about the relationship between distance, speed, and time in plain English.
When you travel a certain distance at a certain speed, the time it takes equals the distance divided by the speed. It's like asking: "If I need to go 60 miles at 30 mph, how long will it take?" The answer is \(60 ÷ 30 = 2\) hours.
For our boat problem:
Notice how the upstream time will be larger because \((\mathrm{v}-3)\) is smaller than \((\mathrm{v}+3)\), which makes sense since going against the current takes longer.
The problem tells us that the upstream trip takes half an hour longer than the downstream trip. In mathematical terms:
Time upstream - Time downstream = 0.5 hours
Substituting our expressions:
\(\frac{90}{\mathrm{v}-3} - \frac{90}{\mathrm{v}+3} = 0.5\)
This equation captures the key relationship in the problem. The left side represents the difference in travel times, and the right side is the given half-hour difference.
Now we need to solve this equation step by step. Let's work with the equation:
\(\frac{90}{\mathrm{v}-3} - \frac{90}{\mathrm{v}+3} = 0.5\)
To solve this, we'll find a common denominator for the fractions on the left side:
\(\frac{90(\mathrm{v}+3) - 90(\mathrm{v}-3)}{(\mathrm{v}-3)(\mathrm{v}+3)} = 0.5\)
Simplifying the numerator:
\(90(\mathrm{v}+3) - 90(\mathrm{v}-3) = 90\mathrm{v} + 270 - 90\mathrm{v} + 270 = 540\)
Simplifying the denominator:
\((\mathrm{v}-3)(\mathrm{v}+3) = \mathrm{v}² - 9\)
So our equation becomes:
\(\frac{540}{\mathrm{v}² - 9} = 0.5\)
Cross multiply:
\(540 = 0.5(\mathrm{v}² - 9)\)
\(540 = 0.5\mathrm{v}² - 4.5\)
\(544.5 = 0.5\mathrm{v}²\)
\(\mathrm{v}² = 1089\)
\(\mathrm{v} = 33\)
Process Skill: MANIPULATE - Algebraic manipulation to isolate the variable
Now that we know \(\mathrm{v} = 33\), we can find the downstream travel time.
Downstream speed = \(\mathrm{v} + 3 = 33 + 3 = 36\) mph
Downstream time = Distance ÷ Speed = \(90 ÷ 36 = 2.5\) hours
Let's verify this makes sense:
The boat traveled downstream for 2.5 hours.
This matches answer choice A. 2.5.
Students often confuse which direction gets which speed. They might think upstream speed is \((\mathrm{v}+3)\) and downstream speed is \((\mathrm{v}-3)\), reversing the logic. Remember: upstream means going AGAINST the current (so the current slows you down, hence \(\mathrm{v}-3\)), while downstream means going WITH the current (so the current speeds you up, hence \(\mathrm{v}+3\)).
Students frequently write the time difference equation backwards as: Time downstream - Time upstream = 0.5, instead of the correct Time upstream - Time downstream = 0.5. Since upstream takes longer, it should be the larger value being subtracted from.
The question asks for downstream travel time, but students might set up their entire approach to find upstream time, the speed v, or the time difference. Always double-check what the question is specifically asking for before you start solving.
When simplifying \(90(\mathrm{v}+3) - 90(\mathrm{v}-3)\), students often make sign errors and get \(90\mathrm{v} + 270 - 90\mathrm{v} - 270 = 0\) instead of the correct \(90\mathrm{v} + 270 - 90\mathrm{v} + 270 = 540\). Pay careful attention to the negative sign in front of \(90(\mathrm{v}-3)\) and how it affects the signs when distributing.
When cross multiplying \(\frac{540}{\mathrm{v}² - 9} = 0.5\), students might incorrectly calculate \(540 = 0.5(\mathrm{v}² - 9)\) as \(540 = 0.5\mathrm{v}² - 9\) instead of \(540 = 0.5\mathrm{v}² - 4.5\). Remember that \(0.5 × (-9) = -4.5\), not -9.
When solving \(\mathrm{v}² = 1089\), students might calculate the square root incorrectly or take the negative value. Since v represents speed, it must be positive, so \(\mathrm{v} = 33\), not \(\mathrm{v} = -33\).
After calculating both upstream time (3 hours) and downstream time (2.5 hours), students might select 3 hours because they spent more time calculating it or because it seems like the 'main' answer. Always refer back to what the question specifically asks for.
Since we calculated \(\mathrm{v} = 33\) mph during the solution process, students might mistakenly think this is the final answer. Remember that the question asks for time in hours, not speed in mph.