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 really compare a region they can't be expressed. I just kind of is the difference you are The region bounded between either two functions of X or pounded between two functions of why and one example that they give. So here's our first example. Is this annual ous So X squared? That's why squared We want to be inside a circle of radius two we're renting and then outside Circle Radius one. So here's circle raise to er to an outside circle. Raise one on the inside. Let me see that this is not bounded between two functions of ice and destruction of why Now it is bounded, huh? It is the region bounded between two functions of kind of the angle state. So if we look in terms of polar coordinates, this is R squared between two and one. So you know it's bounded between these two functions. Far. So another example. Well, I mean, we could do another Angelus that we could do something a little bit different, too. We could look at, say, the region, how founded between why equals X cute. Why Eagle's eggs and then say X nickel's closer, minus one. So this would be another example. So we plot this. We have excuse you. No, we have y equals X and Dan years X equals minus one. X equals one saw region like something like this. And you see that? No, this region is not the region bounded between two functions of I mean it is the region bounded between two functions of X, but the functions actually switched signs that we're gonna have to Just like you know, when we do this integral and Anya List will probably have to do like the top half. And in the bottom half here, we're gonna have to do the region between negative one to zero and in the region from zero one, because thes dysfunctions were actually switching, which is bigger rights over here, X cubed is bigger. So why's going from next X cube butt over here? Why is going from execute two x

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