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

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 $0$ .

@@ 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>
+==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==
 {{AHSME box|year=1996|num-b=27|num-a=29}}

1996 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
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Difference between revisions of "1996 AHSME Problems/Problem 28"

Revision as of 17:54, 20 August 2011

Problem

Solution

See also