Question

The equation of a plane in space is x + 3y + 2z = 6.

1.) The three planes of the axes and the plane you have sketched create a triangular pyramid. Find the volume of this pyramid, showing the formula you are using and all
calculations below.

For my intercepts I got values of, (6, 2, 3), with x = 6, y = 2, and z = 3. I plotted these intercepts on the graph.

How would I do part B of this problem (finding the volume)?

Answer 1

are you sure you are on track? maybe you are but thought i would ask

write the plane \( x + 3y + 2z = 6 \) as

\(z = 3 - \frac{x}{2} - \frac{3y}{2}\)

gives

Answer 2

OK

i now see that you have just presented your intercepts differently/ unconventionally [as if they were one]

the next step is to integrate

\(\large \int \int z(x,y) \ dA = \int \int 3 - \frac{x}{2} - \frac{3y}{2} \ dx \ dy\)

which is either:

\(\large \int_{y=0}^{2} \int_{x = 0}^{6 - 3y} 3 - \frac{x}{2} - \frac{3y}{2} \ dx \ dy\)

or

\(\large \int_{x=0}^{6} \int_{y = 0}^{2 -\frac{x}{3}} 3 - \frac{x}{2} - \frac{3y}{2} \ dy \ dx\)


i get 6.

Answer 3

@Irishboy123 If I'm going to be quite honest with you... I have no idea what most of that means. I'm taking a geometry course as a sophomore in high school. Could you explain that set of equations to me in a simpler way?

Answer 4

\[volume=\frac{ 1 }{ 3 }\left( area ~of~base \right)\times height\]

Answer 5

Would this be the correct equation, then? \[\frac{ 1 }{ 3 } (6) (3) = 6\]

Answer 6

@surjithayer

Answer 7

i think so.

Answer 8

Assuming that my calculation of 6 equaling the base area is correct.