Free Practice Question
PS · Problem Solving
705-805
Functions
In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), and C(0,12). If a point (x,y) is in the triangle ABC, what is the probability that y is smaller than x?
Answer Choices
Correct answer marked belowA
1/2
1/3
Correct
C
1/4
D
2/3
E
3/4
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Explanation
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Official Explanation
Official
The triangle with vertices A(0,0), B(6,0), and C(0,12) is a right triangle in the first quadrant.
Area of triangle ABC = .
We need the probability that a randomly chosen point inside the triangle satisfies . This is the region inside the triangle below the line .
Find intersection of line with line BC.
Equation of line BC: using points B(6,0) and C(0,12), slope = , so equation: .
Set : .
Thus intersection point is (4,4).
The region where inside the triangle is the quadrilateral from (0,0) to (4,4) along , then to (6,0) along BC, and back to (0,0).
Easier: find area where inside triangle (the smaller triangle from (0,0) to (4,4) to (0,12)? Wait, careful).
Actually, inside triangle ABC, the line cuts it into two regions. The region where is triangle with vertices (0,0), (4,4), and (6,0)? Not exactly: (6,0) is on x-axis, but line from (4,4) to (6,0) is along BC. So the region is triangle (0,0), (4,4), (6,0). Check: (6,0) satisfies , yes.…
Area of triangle ABC = .
We need the probability that a randomly chosen point inside the triangle satisfies . This is the region inside the triangle below the line .
Find intersection of line with line BC.
Equation of line BC: using points B(6,0) and C(0,12), slope = , so equation: .
Set : .
Thus intersection point is (4,4).
The region where inside the triangle is the quadrilateral from (0,0) to (4,4) along , then to (6,0) along BC, and back to (0,0).
Easier: find area where inside triangle (the smaller triangle from (0,0) to (4,4) to (0,12)? Wait, careful).
Actually, inside triangle ABC, the line cuts it into two regions. The region where is triangle with vertices (0,0), (4,4), and (6,0)? Not exactly: (6,0) is on x-axis, but line from (4,4) to (6,0) is along BC. So the region is triangle (0,0), (4,4), (6,0). Check: (6,0) satisfies , yes.…
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What it tests
Your ability to evaluate and compose functions, including unusual defined operations.
Common trap
Applying function operations in the wrong order or ignoring domain restrictions.
Details
Difficulty
705-805
Type
PS
Category