Problem 6 · Factor a difference of squares · 4 points
Made by AI and not reviewed by a real person.
A circular track has two concentric edges. Their circumferences add to 34π metres, and the track’s area is 51π square metres. What is the inner edge’s circumference?
- 14π metres
- 17π metres
- 6π metres
- 28π metres
- 20π metres
Show a hint
- The two radii appear in both pieces of information. Can you connect what you know without finding either radius first?
- Write the track’s area as the difference of two circle areas, then factor.
Show the answer and solution
Answer: A — 14π metres
Let the outer and inner radii be R and r. The combined circumference gives R + r = 17, while the area gives (R − r)(R + r) = 51. Thus R − r = 3 and r = 7, so the inner circumference is 14π metres. The tempting 6π metres is the difference between the two circumferences, not the inner circumference.