Let $ABCDEFGH$ be a rectangular prism, where $EF = 4$, $EH = 5$, and $EA = 6$. Find the volume of pyramid $CFAH$. https://latex.artofproblemsolving.com/0/a/3/0a3404805162de58ee7c86ac6b0e5d23327c0026.png

Guest Apr 13, 2023

#1**0 **

To find the volume of pyramid CFAH, we need to first find the height of the pyramid. Since CFAH is a pyramid with a rectangular base, the height of the pyramid is the perpendicular distance from point F to plane ABCD.

Let's first draw a diagram of the rectangular prism:

We know that EF = 4, EH = 5, and EA = 6. Let's use the Pythagorean theorem to find the length of segment FH:

FH² = EF² + EH² FH² = 4² + 5² FH² = 41 FH = sqrt(41)

Now, let's draw a line segment from point F perpendicular to plane ABCD:

Let's call the point where this line segment intersects plane ABCD point X. We want to find the length of segment FX, which is the height of the pyramid.

Since EF is perpendicular to plane ABCD, we know that segment FX is perpendicular to segment EF. Therefore, triangle FEX is a right triangle, with legs of length FX and EF, and hypotenuse of length FH.

Using the Pythagorean theorem again, we can solve for FX:

FH² = FX² + EF² 41 = FX² + 4² FX² = 41 - 16 FX = sqrt(25) FX = 5

So the height of the pyramid is 5.

Now, to find the volume of the pyramid, we need to multiply the area of the base by the height and divide by 3 (since it's a pyramid). The area of the base is the area of rectangle AFCH, which is the product of the length and width:

AF = CH = EA = 6 CF = AH = EH - EF = 5 Area of rectangle AFCH = 6 * 5 = 30

So the volume of pyramid CFAH is:

(1/3) * 30 * 5 = 50

Therefore, the volume of pyramid CFAH is 50 cubic units.

Guest Apr 13, 2023