Problem:
A long piece of paper 5 cm wide is made into a roll for cash registers by wrapping it 600 times around a cardboard tube of diameter 2 cm, forming a roll 10 cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm.)
Answer Choices:
A. 36Ο
B. 45Ο
C. 60Ο
D. 72Ο
E. 90Ο
Solution:
Let d1β,d2β,β¦,d600β be the diameters of the concentric circles in the model. The d 's form an arithmetic sequence with d1β=2 cm and d600β=10 cm. If L is the total length, then
Lβ=Οd1β+Οd2β+β―+Οd600β=Ο(d1β+d2β+β―+d600β)=Ο600(2d1β+d600ββ)=Ο600(212β)cm=36Ο meters. β
OR
Let L be the length of the tape in cm. The thickness of paper on the roll is (10β2)/2=4 cm. Therefore, the thickness of the tape is 4/600=1/150 cm. We may assume that unfolding the paper and laying it out flat has negligible effect on its cross-sectional area. Therefore, we may equate the cross-sectional area of the laid out paper, which is L/150 cm2, to the cross-sectional area while it is on the roll, which is
Ο52βΟ12=24Ο cm2
Solving for L gives L=3600Ο cm=36Ο meters.