2024 AMC 12A Problems/Problem 20
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[hide]Problem
Points and are chosen uniformly and independently at random on sides and respectively, of equilateral triangle Which of the following intervals contains the probability that the area of is less than half the area of
Solution 1
Let and . Applying the sine formula for a triangle's area, we see that
Without loss of generality, we let , and thus ; we therefore require for . (Note: You can skip most of this by using triangle area ratios ~A_MatheMagician.)
A quick rough sketch of on the square given by reveals that the curve intersects the boundaries at and , and it is actually quite (very) obvious that the area bounded by the inequality and the aforementioned unit square is more than but less than (cf. the diagram below). Thus, our answer is .
~Technodoggo
Solution 2
WLOG let Which we can express as .
We follow by graphing the equation on , and we follow the same way solution 1 does.
We see that the probability is slightly less than but definitely greater than
Thus answer choice
Solution 3
The actual probability can be found by using calculus, and taking the integral.
.
Thus, since , we get ~lptoggled
Note that if you do not know an approximation for , you can use the following power series formula: Using this formula to four terms gives the apprximation , which is good enough to solve the problem.
See also
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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 | |
All AMC 12 Problems and Solutions |
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