2016 AMC 10B Problems/Problem 9

Revision as of 13:57, 21 February 2016 by Wholeworld (talk | contribs) (Solution)

Problem

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

Solution

Let the point in the first quadrant be $(a, a^2)$. Then, the area of the triangle is $\frac{2a\cdot a^2}{2}=a^3$. Solving the equation $a^3=64$ for $a$, we get $a=4$, so $BC=2a=8$. So the answer is $\boxed{(C) 8}$.

See Also

2016 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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 10 Problems and Solutions

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