1984 AIME Problems/Problem 9

Problem

In tetrahedron , edge has length 3 cm. The area of face is and the area of face is . These two faces meet each other at a angle. Find the volume of the tetrahedron in .

Solution

For non-asymptote version of image, see Image:1984_AIME-9.png.

/* modified version of olympiad modules */
real markscalefactor = 0.03;
path3 rightanglemark(triple A, triple B, triple C, real s=8)
{
	triple P,Q,R;
	P=s*markscalefactor*unit(A-B)+B;
	R=s*markscalefactor*unit(C-B)+B;
	Q=P+R-B;
	return P--Q--R;
}
path3 anglemark(triple A, triple B, triple C, real t=8 ... real[] s)
{ 
	triple M,N,P[],Q[]; 
	path3 mark; 
	int n=s.length;
	M=t*markscalefactor*unit(A-B)+B; 
	N=t*markscalefactor*unit(C-B)+B; 
	for (int i=0; i<n; ++i)  
	{  
		P[i]=s[i]*markscalefactor*unit(A-B)+B;  
		Q[i]=s[i]*markscalefactor*unit(C-B)+B; 
	} 
	mark=arc(B,M,N); 
	for (int i=0; i<n; ++i) 
	{  
		if (i%2==0)  
		{   
			mark=mark--reverse(arc(B,P[i],Q[i]));  
		}  
		else  
		{   
			mark=mark--arc(B,P[i],Q[i]);   
		} 
	} 
	if (n%2==0 && n!=0) 
	mark=(mark--B--P[n-1]); 
	else if (n!=0) 
	mark=(mark--B--Q[n-1]); 
	else mark=(mark--B--cycle); 
	return mark;
}

size(200);
import three; defaultpen(black+linewidth(0.7)); pen small = fontsize(10);
triple A=(0,0,0),B=(3,0,0),C=(1.8,10,0),D=(1.5,4,4),Da=(D.x,D.y,0),Db=(D.x,0,0); 
currentprojection=perspective(16,-10,8);

draw(surface(A--B--C--cycle),rgb(0.6,0.7,0.6),nolight);
draw(surface(A--B--D--cycle),rgb(0.7,0.6,0.6),nolight);

/* draw pyramid - other lines + angles */
draw(A--B--C--A--D--B--D--C); 
draw(D--Da--Db--cycle);
draw(rightanglemark(D,Da,Db));draw(rightanglemark(A,Db,D));draw(anglemark(Da,Db,D,15));

/* labeling points */
label("$A$",A,SW);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$30^{\circ}$",Db+(0,.35,0.08),(1.5,1.2),small);
label("$3$",(A+B)/2,S); label("$15\mathrm{cm}^2$",(Db+C)/2+(0,-0.5,-0.1),NE,small); label("$12\mathrm{cm}^2$",(A+D)/2,NW,small);
 (Error making remote request. Unknown error_msg)

Position face on the bottom. Since , we find that . The height of forms a with the height of the tetrahedron, so . The volume of the tetrahedron is thus .