Difference between revisions of "1996 AHSME Problems/Problem 28"

Line 21: Line 21:
  
 
<math> \text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9 </math>
 
<math> \text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9 </math>
 +
 +
==Solution==
 +
 +
By placing the cube in a coordinate system such that <math>D</math> is at the origin, <math>A(0,0,3)</math>, <math>B(4,0,0)</math>, and <math>C(0,4,0)</math>, we find that the equation of plane <math>ABC</math> is:
 +
 +
<cmath>\frac{x}{4} + \frac{y}{4} + \frac{z}{3} = 1</cmath>
 +
 +
<cmath>3x + 3y + 4z = 12</cmath>
 +
 +
<cmath>3x + 3y + 4z - 12 = 0</cmath>
 +
 +
The equation for the distance of a point <math>(a,b,c)</math> to a plane <math>Ax + By + Cz + D = 0</math> is given by:
 +
 +
<cmath>\frac{Aa + Bb + Cc + D}{\sqrt{A^2 + B^2 + C^2}}</cmath>
 +
 +
Note that the capital letters are coefficients, while the lower case is the point itself.
 +
 +
Thus, the distance from the origin (where <math>a=b=c=0</math>) to the plane is given by:
 +
 +
<cmath>\frac{D}{\sqrt{A^2 + B^2 + C^2}}</cmath>
 +
 +
<cmath>\frac{12}{\sqrt{9 + 9 + 16}}</cmath>
 +
 +
<cmath>\frac{12}{\sqrt{34}}</cmath>
 +
 +
Since <math>\sqrt{34} < 6</math>, this number should be just a little over <math>2</math>, and the correct answer is <math>\boxed{C}</math>.
 +
 +
Note that the equation above for the distance from a point to a plane is a 3D analogue of the 2D case of the [[distance formula]], where you take the distance from a point to a plane.  In the 2D case, both <math>c</math> and <math>C</math> are set equal to <math>0</math>. 
  
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=27|num-a=29}}
 
{{AHSME box|year=1996|num-b=27|num-a=29}}

Revision as of 17:54, 20 August 2011

Problem

On a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to

[asy] size(120); import three; currentprojection=orthographic(1, 4/5, 1/3); draw(box(O, (4,4,3))); triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); draw(A--B--C--cycle, linewidth(0.9)); label("$A$", A, NE); label("$B$", B, NW); label("$C$", C, S); label("$D$", D, E); label("$4$", (4,2,0), SW); label("$4$", (2,4,0), SE); label("$3$", (0, 4, 1.5), E); [/asy]

$\text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9$

Solution

By placing the cube in a coordinate system such that $D$ is at the origin, $A(0,0,3)$, $B(4,0,0)$, and $C(0,4,0)$, we find that the equation of plane $ABC$ is:

\[\frac{x}{4} + \frac{y}{4} + \frac{z}{3} = 1\]

\[3x + 3y + 4z = 12\]

\[3x + 3y + 4z - 12 = 0\]

The equation for the distance of a point $(a,b,c)$ to a plane $Ax + By + Cz + D = 0$ is given by:

\[\frac{Aa + Bb + Cc + D}{\sqrt{A^2 + B^2 + C^2}}\]

Note that the capital letters are coefficients, while the lower case is the point itself.

Thus, the distance from the origin (where $a=b=c=0$) to the plane is given by:

\[\frac{D}{\sqrt{A^2 + B^2 + C^2}}\]

\[\frac{12}{\sqrt{9 + 9 + 16}}\]

\[\frac{12}{\sqrt{34}}\]

Since $\sqrt{34} < 6$, this number should be just a little over $2$, and the correct answer is $\boxed{C}$.

Note that the equation above for the distance from a point to a plane is a 3D analogue of the 2D case of the distance formula, where you take the distance from a point to a plane. In the 2D case, both $c$ and $C$ are set equal to $0$.

See also

1996 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 27
Followed by
Problem 29
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 26 27 28 29 30
All AHSME Problems and Solutions
Invalid username
Login to AoPS