Difference between revisions of "1987 AHSME Problems/Problem 13"
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\textbf{(D)}\ 72\pi \qquad | \textbf{(D)}\ 72\pi \qquad | ||
\textbf{(E)}\ 90\pi </math> | \textbf{(E)}\ 90\pi </math> | ||
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+ | == Solution == | ||
+ | Notice (by imagining unfolding the roll), that the length of the paper is equal to the sum of the circumferences of the concentric circles, which is <math>\pi</math> times the sum of the diameters. Now the, the diameters form an arithmetic series with first term <math>2</math>, last term <math>10</math>, and <math>600</math> terms in total, so using the formula <math>\frac{1}{2}n(a+l)</math>, the sum is <math>300 \times 12 = 3600</math>, so the length is <math>3600\pi</math> centimetres, or <math>36\pi</math> metres, which is answer <math>\boxed{\text{A}}</math>. | ||
== See also == | == See also == |
Latest revision as of 14:24, 1 March 2018
Problem
A long piece of paper cm wide is made into a roll for cash registers by wrapping it times around a cardboard tube of diameter cm, forming a roll cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms concentric circles with diameters evenly spaced from cm to cm.)
Solution
Notice (by imagining unfolding the roll), that the length of the paper is equal to the sum of the circumferences of the concentric circles, which is times the sum of the diameters. Now the, the diameters form an arithmetic series with first term , last term , and terms in total, so using the formula , the sum is , so the length is centimetres, or metres, which is answer .
See also
1987 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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