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Difference between revisions of "2016 AMC 10B Problems/Problem 9"

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(→‎Solution)
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==Solution==
 
==Solution==
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Let the point in the first quadrant be <math>(a, a^2)</math>.
+
<asy>import graph;size(7cm,IgnoreAspect);
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Then, the area of the triangle is <math>\frac{2a\cdot a^2}{2}=a^3</math>.
+
real f(real x) {return x*x;}
−
Solving the equation <math>a^3=64</math> for <math>a</math>, we get <math>a=4</math>, so <math>BC=2a=8</math>.
+
draw((0,0)--(4,16)--(-4,16)--cycle,blue);
−
So the answer is <math>\boxed{\textbf{(C)}\ 8}</math>.
+
draw(graph(f,-5,5,operator ..),gray);
 +
xaxis("$x$");yaxis("$y$",-1);
 +
label("$y=x^2$",(4.5,20.25),E);
 +
draw((4.2,0)--(4.2,16),Arrows);
 +
label("$r^2$",(4.2,0)--(4.2,16),E);
 +
draw((0,17)--(4,17),Arrows);
 +
label("$r$",(0,17)--(4,17),N);
 +
</asy>
 +
The area of the triangle is <math>r^3</math>, so <math>r^3=64\implies r=4</math>, giving a total distance across the top of <math>8</math>, which is answer <math>\textbf{(C)}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2016|ab=B|num-b=8|num-a=10}}
 
{{AMC10 box|year=2016|ab=B|num-b=8|num-a=10}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:05, 21 February 2016

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

[asy]import graph;size(7cm,IgnoreAspect); real f(real x) {return x*x;} draw((0,0)--(4,16)--(-4,16)--cycle,blue); draw(graph(f,-5,5,operator ..),gray); xaxis("$x$");yaxis("$y$",-1); label("$y=x^2$",(4.5,20.25),E); draw((4.2,0)--(4.2,16),Arrows); label("$r^2$",(4.2,0)--(4.2,16),E); draw((0,17)--(4,17),Arrows); label("$r$",(0,17)--(4,17),N); [/asy] The area of the triangle is $r^3$, so $r^3=64\implies r=4$, giving a total distance across the top of $8$, which is answer $\textbf{(C)}$.

See Also

2016 AMC 10B (Problems • Answer Key • Resources)
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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