+1207 A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?
+4622 Label the square bases' area as \(x.\) The area of one of the triangular face is thus \(\frac{1}{2}x\), and we have \(4*(\frac{1}{2}x)+x=432\),, and we have \(2x+x=432\), and thus \(x=144\) square units. The side length of the square is the square root of 144 or twelve inches. And, the height of the triangle is twelve inches. The Pythagorean theorem gives us \(12^2-6^2=144-36=72\longrightarrow\sqrt{72}\). Thus, the answer is \(\frac{144*\sqrt{72}}{3}=\boxed{48\sqrt{72}}\) cubic units.
+4622 Label the square bases' area as \(x.\) The area of one of the triangular face is thus \(\frac{1}{2}x\), and we have \(4*(\frac{1}{2}x)+x=432\),, and we have \(2x+x=432\), and thus \(x=144\) square units. The side length of the square is the square root of 144 or twelve inches. And, the height of the triangle is twelve inches. The Pythagorean theorem gives us \(12^2-6^2=144-36=72\longrightarrow\sqrt{72}\). Thus, the answer is \(\frac{144*\sqrt{72}}{3}=\boxed{48\sqrt{72}}\) cubic units.
