Difference between revisions of "2020 AMC 12B Problems/Problem 25"
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(Similar solution, but different approach to maximization) |
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Therefore, the maximum value of <math>P(a)</math> is <math>1-\min(Q(a))</math>, which is <math>1-(\sqrt{2}-1) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math> | Therefore, the maximum value of <math>P(a)</math> is <math>1-\min(Q(a))</math>, which is <math>1-(\sqrt{2}-1) =\boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math> | ||
+ | |||
+ | ==Solution 2 (Calculus finish)== | ||
+ | |||
+ | We find the same region as in the first solution, and again notice we must have <math>a \geq \frac{1}{2}</math>. | ||
+ | |||
+ | We now express <math>P</math> as a function of <math>b=(1-a)</math>. The triangle on the right of the line <math>x=b</math> is an isosceles right triangle with altitude <math>b</math>, so it has area <math>b^2</math>. The total area of the region to the left of <math>x=b</math> has area <math>1-b</math>. So | ||
+ | <cmath>P(b) = \frac{1/2 - b^2}{1-b}</cmath> | ||
+ | Differentiating using the quotient rule, we find <math>P</math> has local extrema at | ||
+ | <cmath>P'(b) = \frac{(1-h)(-2h)-(-1/2 - h^2)(-1)}{(1-h)^2} = 0</cmath> | ||
+ | Setting the numerator equal to <math>0</math> and solving the quadratic, we find <math>P</math> has extrema at <math>b = 1 \pm \frac{\sqrt{2}}{2}</math>. Only <math>b=1-\frac{\sqrt 2}{2}</math> is within the desired region. Plugging in <math>P(1-\frac{\sqrt 2}{2}) = \boxed{\textbf{(B)}\ 2 - \sqrt{2}}</math> as our solution. | ||
+ | |||
+ | We also need to check <math>P(b=0) = 1/2</math>. But <math>1/2 < 2 - \sqrt{2}</math>, and if this isn't immediately obvious, <math>1/2</math> isn't an answer choice anyways. | ||
+ | |||
+ | ~jd9 (AMGM scary) | ||
==Video Solution by On The Spot STEM== | ==Video Solution by On The Spot STEM== |
Revision as of 09:44, 27 October 2023
Contents
Problem
For each real number with
, let numbers
and
be chosen independently at random from the intervals
and
, respectively, and let
be the probability that
What is the maximum value of
Solution
Let's start first by manipulating the given inequality.
Let's consider the boundary cases: .
Solving the first case gives us Solving the second case gives us
If we graph these equations in
, we get a rhombus shape.
Testing points in each section tells us that the inside of the rhombus satisfies the inequality in the problem statement.
From the region graph, notice that in order to maximize ,
. We can solve the rest with geometric probability.
Instead of maximizing , we minimize
.
consists of two squares (each broken into two triangles), one of area
and another of area
. To calculate
, we divide this area by
, so
By AM-GM, , which we can achieve by setting
.
Therefore, the maximum value of is
, which is
Solution 2 (Calculus finish)
We find the same region as in the first solution, and again notice we must have .
We now express as a function of
. The triangle on the right of the line
is an isosceles right triangle with altitude
, so it has area
. The total area of the region to the left of
has area
. So
Differentiating using the quotient rule, we find
has local extrema at
Setting the numerator equal to
and solving the quadratic, we find
has extrema at
. Only
is within the desired region. Plugging in
as our solution.
We also need to check . But
, and if this isn't immediately obvious,
isn't an answer choice anyways.
~jd9 (AMGM scary)
Video Solution by On The Spot STEM
https://www.youtube.com/watch?v=5goLUdObBrY
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.