Difference between revisions of "1976 AHSME Problems/Problem 8"
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| + | == Problem 8 == | ||
| + | |||
| + | A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, | ||
| + | is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance | ||
| + | from the point to the origin is at most two units? | ||
| + | |||
| + | <math>\textbf{(A) }\frac{13}{81}\qquad | ||
| + | \textbf{(B) }\frac{15}{81}\qquad | ||
| + | \textbf{(C) }\frac{13}{64}\qquad | ||
| + | \textbf{(D) }\frac{\pi}{16}\qquad | ||
| + | \textbf{(E) }\text{the square of a rational number}</math> | ||
| + | |||
| + | == Solution == | ||
| + | |||
In order for the coordinates to have absolute values less than <math>4</math>, the points must lie in the <math>8</math> by <math>8</math> square passing through the points <math>(-4,-4), (-4, 4), (4,4)</math> and <math>(4, -4)</math>. | In order for the coordinates to have absolute values less than <math>4</math>, the points must lie in the <math>8</math> by <math>8</math> square passing through the points <math>(-4,-4), (-4, 4), (4,4)</math> and <math>(4, -4)</math>. | ||
For the points to be at most <math>2</math> units from the origin, the points must lie in a circle of radius <math>2</math> centered at the origin. Thus, the probability is the area of the circle over the area of the square, or | For the points to be at most <math>2</math> units from the origin, the points must lie in a circle of radius <math>2</math> centered at the origin. Thus, the probability is the area of the circle over the area of the square, or | ||
Revision as of 12:06, 29 June 2022
Problem 8
A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?
Solution
In order for the coordinates to have absolute values less than 
, the points must lie in the 
by square passing through the points 
and 
.
For the points to be at most 
units from the origin, the points must lie in a circle of radius centered at the origin. Thus, the probability is the area of the circle over the area of the square, or
![\[\frac{2^2 \pi}{8^2} = \frac{4 \pi}{64} = \frac{\pi}{16}\]](http://latex.artofproblemsolving.com/2/4/e/24ee483faa9afa0af224b91a1268959d6c7a0717.png)
~JustinLee2017
