You hire a printer to print concert tickets. He delivers them in circular rolls labeled as 1000 tickets each. You want to check
the number of tickets in each roll without counting thousands of tickets. You decide to do it by measuring the diameter of the rolls. The tickets are 3 in long and 0.28 mm thick and are rolled on a core 4 cm in diameter.
n-th wrap: diameter = D + (n-1)*(2t) length = π[D + (n-1)*(2t)] The length forms an arithmetic sequence, with a₁ = πD = 1256.6 mm (first term) d = 2πt = 1.7593 mm
The n-th term is 1256.6 + 1.7593(n-1) = 1254.8 + 1.7593n
The total length of n wraps is (n/2)*(1256.6 + 1254.8 + 1.7593n) = 1255.7n + 0.8797n²
The total length should be equal to 762,000. Therefore 0.8797n² + 1255.7n - 762000 = 0
Solve with the quadratic formula. n = (1/1.7594)*[-1255.7 +/- √(1255.7² + 2.6812 x 10⁶ )] = 459.14 or -1886.56 Reject negative length, so that n = 459.14, rroundto the larger value of 460.
The number of tickets will be 1000 if the 460-th wrap (outer wrap) has a diameter of 1254.8 + 1.7593*460 = 2064.1 mm, or 2064.1/254 = 8.13 in
Answer: The diameter of each roll should be about 8.1 in (or 2064 mm)