Difference between revisions of "1986 AHSME Problems/Problem 18"
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A plane intersects a right circular cylinder of radius <math>1</math> forming an ellipse. | A plane intersects a right circular cylinder of radius <math>1</math> forming an ellipse. | ||
| − | If the major axis of the ellipse | + | If the major axis of the ellipse is <math>50\%</math> longer than the minor axis, the length of the major axis is |
<math>\textbf{(A)}\ 1\qquad | <math>\textbf{(A)}\ 1\qquad | ||
| Line 8: | Line 8: | ||
\textbf{(C)}\ 2\qquad | \textbf{(C)}\ 2\qquad | ||
\textbf{(D)}\ \frac{9}{4}\qquad | \textbf{(D)}\ \frac{9}{4}\qquad | ||
| − | \textbf{(E)}\ 3 </math> | + | \textbf{(E)}\ 3 </math> |
==Solution== | ==Solution== | ||
| + | ===Solution 1=== | ||
| + | The length of the minor axis is the distance from the opposite points of the cylinder, which is simply <math>2 \cdot 1 =2</math>, and according to the question, the length of the major axis is simply <math>2 \cdot 150\% = \boxed{(E) 3}</math> | ||
| + | |||
| + | ~ <math>shalomkeshet</math> | ||
| + | |||
| + | ===Solution 2=== | ||
We note that we can draw the minor axis to see that because the minor axis is the minimum distance between two opposite points on the ellipse, we can draw a line through two opposite points of the cylinder, and so the minor axis is <math>2(1) = 2</math>. Therefore, our answer is <math>2(1.5) = 3</math>, and so our answer is <math>\boxed{E}</math>. | We note that we can draw the minor axis to see that because the minor axis is the minimum distance between two opposite points on the ellipse, we can draw a line through two opposite points of the cylinder, and so the minor axis is <math>2(1) = 2</math>. Therefore, our answer is <math>2(1.5) = 3</math>, and so our answer is <math>\boxed{E}</math>. | ||
Latest revision as of 02:26, 31 October 2024
Problem
A plane intersects a right circular cylinder of radius 
forming an ellipse.
If the major axis of the ellipse is 
longer than the minor axis, the length of the major axis is
Solution
Solution 1
The length of the minor axis is the distance from the opposite points of the cylinder, which is simply 
, and according to the question, the length of the major axis is simply 
~ 
Solution 2
We note that we can draw the minor axis to see that because the minor axis is the minimum distance between two opposite points on the ellipse, we can draw a line through two opposite points of the cylinder, and so the minor axis is 
. Therefore, our answer is 
, and so our answer is 
.
See also
| 1986 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
