- Translate the problem requirements: We need to find the maximum possible difference between the actual volume of a cube and the volume calculated using a potentially incorrect measurement of 100 cm (with up to 1 cm error).
- Establish the range of actual measurements: Determine the minimum and maximum possible actual edge lengths given the measurement error.
- Calculate volume extremes: Find the actual volume at both extreme edge lengths and compare with the computed volume using the 100 cm measurement.
- Identify maximum difference: Compare the differences to determine which gives the larger deviation from the computed volume.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're looking for in plain English. We have a cube whose edge is measured as 100 centimeters, but this measurement might be wrong by up to 1 centimeter in either direction. We need to find the maximum possible difference between:
- The actual volume of the cube (whatever its true edge length is)
- The volume we calculate using our potentially incorrect measurement of 100 cm
Think of it this way: if someone gives you a measurement and says "it's 100 cm, give or take 1 cm," how wrong could your volume calculation be at worst?
The volume we'll compute using the measurement is: \(100^3 = 1,000,000\) cubic centimeters
Process Skill: TRANSLATE - Converting the error scenario into a clear mathematical comparison
2. Establish the range of actual measurements
Since the measurement error is "no more than 1 centimeter," this means the actual edge length could be anywhere in this range:
- Minimum actual edge length: \(100 - 1 = 99\) centimeters
- Maximum actual edge length: \(100 + 1 = 101\) centimeters
So while we think the edge is 100 cm, it could actually be as small as 99 cm or as large as 101 cm.
3. Calculate volume extremes
Now let's find the actual volumes at both extremes and see how they differ from our computed volume of 1,000,000 cubic centimeters.
Case 1: Actual edge length is 99 cm
Actual volume = \(99^3\)
Let's calculate this step by step:
- \(99^2 = (100-1)^2 = 100^2 - 2(100)(1) + 1^2 = 10,000 - 200 + 1 = 9,801\)
- \(99^3 = 99 \times 9,801 = (100-1) \times 9,801 = 100 \times 9,801 - 9,801 = 980,100 - 9,801 = 970,299\)
Difference from computed volume = \(1,000,000 - 970,299 = 29,701\) cubic centimeters
Case 2: Actual edge length is 101 cm
Actual volume = \(101^3\)
Let's calculate this step by step:
- \(101^2 = (100+1)^2 = 100^2 + 2(100)(1) + 1^2 = 10,000 + 200 + 1 = 10,201\)
- \(101^3 = 101 \times 10,201 = (100+1) \times 10,201 = 100 \times 10,201 + 10,201 = 1,020,100 + 10,201 = 1,030,301\)
Difference from computed volume = \(1,030,301 - 1,000,000 = 30,301\) cubic centimeters
4. Identify maximum difference
Comparing our two cases:
- When actual edge is 99 cm: difference = 29,701 cubic centimeters
- When actual edge is 101 cm: difference = 30,301 cubic centimeters
The maximum possible difference is 30,301 cubic centimeters.
Looking at our answer choices, 30,301 is closest to 30,000.
Process Skill: CONSIDER ALL CASES - We examined both directions of error to find the maximum
Final Answer
The maximum possible difference between the actual volume and the computed volume is approximately 30,301 cubic centimeters, which is closest to choice B. 30,000.
This makes intuitive sense: when dealing with cubes, small errors in edge measurement get amplified significantly because we're cubing the measurement. A 1% error in edge length leads to about a 3% error in volume.
Common Faltering Points
Errors while devising the approach
- Misunderstanding what "maximum possible difference" means: Students might think they need to find the difference between the maximum and minimum possible volumes (\(101^3 - 99^3\)), rather than finding which of these volumes differs most from the computed volume of \(100^3\). This leads to a completely different calculation approach.
- Incorrectly interpreting the error constraint: Students might misread "error is no more than 1 centimeter" as meaning the error is exactly 1 centimeter, or they might think the actual measurement must be either exactly 99 cm or exactly 101 cm, missing that any value in the range [99, 101] is possible.
- Forgetting to consider both directions of error: Students might only calculate the difference for one case (say, when actual edge is 101 cm) and assume this gives the maximum difference, without checking the other extreme (when actual edge is 99 cm).
Errors while executing the approach
- Arithmetic errors in cubing calculations: When calculating \(99^3\) or \(101^3\), students often make mistakes in the step-by-step multiplication, especially when expanding \((100±1)^3\) or when multiplying large numbers like \(99 \times 9,801\).
- Sign errors when calculating differences: Students might incorrectly subtract in the wrong direction, calculating \(970,299 - 1,000,000\) instead of \(1,000,000 - 970,299\), leading to negative differences that they then incorrectly handle.
Errors while selecting the answer
- Choosing the wrong case as the maximum: After calculating both differences (29,701 and 30,301), students might accidentally select the answer choice closest to the smaller difference (29,701) rather than the larger one (30,301), leading them to choose an incorrect answer.
