Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AB}$, $\overline{PA}\perp\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$?
[asy]
import three;
triple A = (4,8,0);
triple B= (4,0,0);
triple C = (0,0,0);
triple D = (0,8,0);
triple P = (4,8,6);
draw(B--P--D--A--B);
draw(A--P);
draw(C--P, dashed);
draw(B--C--D,dashed);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$P$",P,N);
[/asy]

Since $\overline{PA}$ is perpendicular to both $\overline{AB}$ and $\overline{AD}$, the segment $\overline{PA}$ is the altitude from the apex to the base of the pyramid.   The area of the base is $[ABCD] = (AB)(BC) = 32$, and the height from the apex to the base is 6, so the volume of the pyramid is $\frac13(32)(6) = \boxed{64}$ cubic units.
64
Level 4
Geometry
Problem: Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AB}$, $\overline{PA}\perp\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the volume of pyramid $PABCD$, we use the formula for the volume of a pyramid, which is given by:

Volume = (1/3) * Base Area * Height.

First, we need to calculate the area of the base rectangle $ABCD$. The base is defined by the lengths of sides $AB$ and $BC$. 

Given:
AB = 8 (length)
BC = 4 (width)

The area of rectangle $ABCD$ can be calculated as:
Base Area = AB * BC = 8 * 4 = 32.

Next, we identify the height of the pyramid. The height is the perpendicular distance from point $P$ to the base $ABCD$. It is given that $PA = 6$, and since $PA$ is perpendicular to the base, this value represents the height of the pyramid.

Now we can substitute the values into the volume formula:
Volume = (1/3) * Base Area * Height
Volume = (1/3) * 32 * 6.

Calculating this step-by-step:
1. Multiply the base area by the height: 32 * 6 = 192.
2. Divide by 3 to find the volume: 192 / 3 = 64.

Thus, the volume of pyramid $PABCD$ is 64.<|END_OF_LATENT|> The final answer is $64$.