Business and Economics

Packaging The manufacturer of a fruit juice drink has decided to try innovative packaging in order to revitalize sagging sales. The fruit juice drink is to be packaged in containers in the shapeof tetrahedra in which three edges are perpendicular, as shown in the figure below. Two of the perpendicular edges will be 3 in. long, and the third edge will be 6 in. long. Find the volume of the container. (Hint: The equation of the plane shown in the figure is $z=f(x, y)=6-2 x-2 y$ .

## Discussion

## Video Transcript

Okay, so you find the average value of this function choose X squared plus y squared on this rectangle zero to cross Syria three. And so the average value of dysfunction on this rectangle what we want. Teo waits with the area which will just be the area of a rectangle, that area that we're, ah, integrating the function over. And then we would just wantto multiply that by the actual and everyone. Dean, why the x o gets a first of all, let's just note that what is the area of this rectangle? Well, it has based two in height three. So that ships two times three two six said this is one sixth. And now let's take anti derivative with respect to why first. So that's going to be X squared. Why? Plus one third why, kid evaluated terms here. Two, three. Okay, so that's one sixth cleared to X. This is this going to be three x squared and then plus well, we're just gonna keep twenty seven to about three, seven nine. Okay, now, just an X and a derivative someone Sixth, this is just going to be x cubed plus nine x five age from zero to two. Okay. So justly applying into she we have eight plus eighteen, which is twenty six, So it looks like we get twenty six over six.

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