Area of a Walkway: Confusing Linear and Square Units
A geometry class is solving a landscaping problem: a circular fountain has a radius of 4.5 feet. A concrete walkway 3 feet wide is to be built around it. Students must estimate the area of the walkway to order concrete, which is sold by the cubic yard at a recommended depth of 2 inches. The teacher provides the formulas A = πr² and C = 2πr. Student A computes π(7.5)² − π(4.5)² = 36π ≈ 113.1 square feet. Student B computes 2π(7.5) − 2π(4.5) = 6π ≈ 18.8 and labels the result 'square feet.' Student C uses 7.5 as the diameter of the outer circle, computes π(3.75)² − π(4.5)², and gets a negative area, then says the problem is impossible. The teacher wants to use these responses to discuss units, radius versus diameter, and precision. The class has already learned area and circumference but has not yet converted cubic yards.
1. Diagnose Student B's error. What does the computation 2π(7.5) − 2π(4.5) actually represent, and why is the label 'square feet' a serious issue?
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Student B computed a difference of circumferences: 2π(7.5) - 2π(4.5) = 15π - 9π = 6π ≈ 18.8. That is a length in feet, not an area. It represents how much longer the outer boundary is than the inner boundary, not the amount of surface between them. Labeling it 'square feet' is a dimensional error: feet minus feet yields feet, and area requires square feet. The error suggests B is treating the walkway as a one-dimensional strip or confusing perimeter with area. The teacher should ask what units result from multiplying two lengths and have B sketch the ring and shade the region whose area is wanted.
2. Student C gets a negative area. Identify the likely error and explain how the teacher can respond without simply correcting the number.
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Student C used 7.5 as the outer diameter, then divided by 2 to get 3.75 as the outer radius. But 7.5 is already the outer radius: the fountain radius is 4.5 ft, and the walkway adds 3 ft, so 4.5 + 3 = 7.5 ft. Since 3.75 < 4.5, subtracting the inner area gives a negative result. The problem is not impossible; C's radius assignment is wrong. The teacher should have C draw a line from the center to the outer edge and label it 7.5, then compare that with the inner radius 4.5. Recomputing gives π(7.5² - 4.5²) = π(56.25 - 20.25) = 36π ft². Asking C to explain why a negative area is impossible can reinforce the meaning of radius.
3. Student A's answer 36π ft² is exact for the area. The concrete is sold by the cubic yard at 2 inches deep. What should the teacher emphasize about precision and unit conversion when the class orders concrete? Include a reasonable estimate.
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First convert depth: 2 in = 1/6 ft. Volume = 36π ft² × 1/6 ft = 6π ft³ ≈ 18.85 ft³. Since 1 yd³ = 27 ft³, this is about 0.70 yd³. A practical order might be 0.75 or 1 yd³ depending on the supplier's minimum and allowance for uneven ground. The teacher should emphasize that 36π is exact only for the idealized dimensions; the 2-inch depth and the 4.5-ft radius are measurements with uncertainty. Reporting 0.698... yd³ would be false precision. Students should round to a sensible order quantity, state the rounding decision, and distinguish square feet (area) from cubic feet and cubic yards (volume).