Difference between revisions of "1985 AHSME Problems/Problem 10"
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==Problem== | ==Problem== | ||
− | An arbitrary | + | An arbitrary circle can intersect the graph of <math>y = \sin x</math> in |
− | <math> \mathrm{(A) | + | <math> \mathrm{(A) \ } \text{at most }2\text{ points} \qquad \mathrm{(B) \ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}</math> |
+ | <math>\mathrm{(E) \ }\text{more than }16\text{ points} </math> | ||
==Solution== | ==Solution== | ||
− | Consider a circle | + | Consider a circle whose center lies on the positive <math>y</math>-axis and which passes through the origin. As the radius of this circle becomes arbitrarily large, its curvature near the <math>x</math>-axis becomes almost flat, and so it can intersect the curve <math>y = \sin x</math> arbitrarily many times (since the <math>x</math>-axis itself intersects the curve infinitely many times). Hence, in particular, we can choose a radius sufficiently large that the circle intersects the curve at <math>\boxed{\text{(E)} \ \text{more than }16 \text{ points}}</math>. |
==See Also== | ==See Also== | ||
{{AHSME box|year=1985|num-b=9|num-a=11}} | {{AHSME box|year=1985|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:35, 19 March 2024
Problem
An arbitrary circle can intersect the graph of in
Solution
Consider a circle whose center lies on the positive -axis and which passes through the origin. As the radius of this circle becomes arbitrarily large, its curvature near the
-axis becomes almost flat, and so it can intersect the curve
arbitrarily many times (since the
-axis itself intersects the curve infinitely many times). Hence, in particular, we can choose a radius sufficiently large that the circle intersects the curve at
.
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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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