- Translate the problem requirements: Both rims rotate at the same linear speed (inches per second), but we need to find how many complete rotations the larger rim makes per minute given that the smaller rim makes x rotations per second
- Establish the equal linear speed relationship: Since both rims travel the same distance per second, set up the fundamental relationship between their circumferences and rotation rates
- Connect rotation rates using circumference ratio: Use the fact that when linear speeds are equal, rotation rates are inversely proportional to circumferences
- Convert time units and finalize: Transform the result from revolutions per second to revolutions per minute
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what's happening here in everyday terms:
- We have two circular rims - think of them like bicycle wheels of different sizes
- The smaller rim has a diameter of 28 inches
- The larger rim has a diameter of 35 inches
- Both rims are moving at the same linear speed - this means if you put a mark on each rim's edge, both marks travel the same distance per second
- The smaller rim completes x full rotations every second
- We need to find how many full rotations the larger rim makes per minute (not second)
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
2. Establish the equal linear speed relationship
Here's the key insight: when both rims travel the same distance per second, we can set up a relationship.
Think about it this way: every time the smaller rim completes one full rotation, a point on its edge travels a distance equal to the rim's circumference. The same is true for the larger rim.
Since both rims have the same linear speed (inches per second):
- Smaller rim's linear speed = (circumference of smaller rim) × (rotations per second)
- Larger rim's linear speed = (circumference of larger rim) × (rotations per second)
- These two linear speeds are equal!
The circumferences are:
- Smaller rim: \(\pi \times 28\) inches
- Larger rim: \(\pi \times 35\) inches
So our equal speed relationship becomes:
\((\pi \times 28) \times \mathrm{x} = (\pi \times 35) \times \mathrm{(rotations\,per\,second\,of\,larger\,rim)}\)
3. Connect rotation rates using circumference ratio
Now let's solve for the larger rim's rotation rate per second.
From our equation: \((\pi \times 28) \times \mathrm{x} = (\pi \times 35) \times \mathrm{(rotations\,per\,second\,of\,larger\,rim)}\)
The \(\pi\) cancels out from both sides:
\(28\mathrm{x} = 35 \times \mathrm{(rotations\,per\,second\,of\,larger\,rim)}\)
Solving for the larger rim's rotations per second:
Rotations per second of larger rim = \(\frac{28\mathrm{x}}{35} = \frac{4\mathrm{x}}{5}\)
Process Skill: SIMPLIFY - Reducing fractions to make calculations manageable
4. Convert time units and finalize
We found that the larger rim makes \(\frac{4\mathrm{x}}{5}\) rotations per second, but the question asks for rotations per minute.
Since there are 60 seconds in a minute:
Rotations per minute = \(\frac{4\mathrm{x}}{5} \times 60\)
\(= \frac{4\mathrm{x} \times 60}{5}\)
\(= 4\mathrm{x} \times 12\)
\(= 48\mathrm{x}\)
Final Answer
The larger rim makes \(48\mathrm{x}\) revolutions per minute.
Looking at our answer choices, this matches choice C. \(48\mathrm{x}\)
This makes intuitive sense: the larger rim rotates slower than the smaller rim (since it has a larger circumference but the same linear speed), but when we convert from seconds to minutes by multiplying by 60, we get a result that's larger than x.
Common Faltering Points
Errors while devising the approach
- Confusing linear speed with rotational speed: Students often miss that "rotates the same number of inches per second" refers to linear speed (distance traveled by a point on the rim's edge), not rotational speed (revolutions per second). This leads them to incorrectly assume both rims have the same rotational speed, making x revolutions per second each.
- Misunderstanding the time unit conversion requirement: Students may overlook that the question asks for revolutions per minute for the larger rim while giving information about the smaller rim in revolutions per second. This causes them to work entirely in seconds and forget the final conversion step.
- Setting up incorrect relationships: Some students might try to directly relate the diameters to the rotation rates without understanding that equal linear speeds means the product of circumference and rotational speed must be equal for both rims.
Errors while executing the approach
- Arithmetic errors in fraction simplification: When simplifying \(\frac{28\mathrm{x}}{35}\), students commonly make mistakes like getting \(\frac{3\mathrm{x}}{4}\) instead of \(\frac{4\mathrm{x}}{5}\), or failing to reduce the fraction to its simplest form altogether.
- Incorrect cancellation of π: While \(\pi\) does cancel out correctly from both sides of the equation \((\pi \times 28) \times \mathrm{x} = (\pi \times 35) \times \mathrm{r}\), some students either forget to include \(\pi\) initially or make errors when canceling it out.
- Time conversion calculation errors: When converting from seconds to minutes by multiplying by 60, students often make computational mistakes like \(\frac{4\mathrm{x}}{5} \times 60 = \frac{240\mathrm{x}}{5} = 40\mathrm{x}\) instead of the correct \(48\mathrm{x}\).
Errors while selecting the answer
- Selecting the intermediate result: Students who correctly find that the larger rim makes \(\frac{4\mathrm{x}}{5}\) revolutions per second might accidentally select an answer choice that matches this value (if available) instead of completing the conversion to revolutions per minute.
- Choosing the reciprocal or inverse relationship: Students might incorrectly select choice E (\(\frac{\mathrm{x}}{75}\)) thinking that since the larger rim should rotate slower, the answer should be a fraction of x, not a multiple of x.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose convenient values for the problem
Let's assign a concrete value to x (revolutions per second for the smaller rim). Since we need to work with circumferences and the diameters are 28 and 35 inches, let's choose \(\mathrm{x} = 5\) revolutions per second for the smaller rim. This gives us clean numbers to work with.
Step 2: Calculate the linear speed using the smaller rim
• Smaller rim diameter = 28 inches, so circumference = \(28\pi\) inches
• Smaller rim makes 5 revolutions per second
• Linear speed = \(5 \times 28\pi = 140\pi\) inches per second
Step 3: Find the revolution rate of the larger rim
• Larger rim diameter = 35 inches, so circumference = \(35\pi\) inches
• Both rims have the same linear speed: \(140\pi\) inches per second
• Larger rim revolutions per second = \(\frac{140\pi}{35\pi} = 4\) revolutions per second
Step 4: Convert to revolutions per minute
• Larger rim revolutions per minute = \(4 \times 60 = 240\) revolutions per minute
Step 5: Express in terms of x
• We used \(\mathrm{x} = 5\), and got 240 revolutions per minute
• Notice that \(240 = 48 \times 5 = 48\mathrm{x}\)
• Therefore, the larger rim makes \(48\mathrm{x}\) revolutions per minute
Verification: This matches answer choice C: \(48\mathrm{x}\), confirming our approach is correct.
