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Problem 65

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 $z=f(x, y)=6-2 x-2 y$ .

Answer

9 $\mathrm{in}^{3}$

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## Discussion

## Video Transcript

All right, so we have a huge where problem on, uh, guys are probably thinking, What the heck? Impossible. However, I do this hideaway even approach this one of my even Silvestri is solving for and it's it's actually quite easy just breaking it down. All we're looking for is the volume of a container, and instantly you should be like, Oh, double integral. I should be using a doubling your goal here. Um and yes, we're gonna be using a double integral to solve for the volume of this new fruit juice drink. Ah, packaging, which is kind of weird, but whatever anyways, so essentially what they're saying, this problem is the volume bounded by this container is actually described by the function Z equals six miles to X minus two y. And if we just rolled out out, draw that out in the x y Z cornet plane. Just strong straight line. Those yucky squiggly lines are yucky. X y z um, so let's just solve for r X intercepts of this plane. So this is actually a plane in this case and it should sweep out a tetrahedron volume. Um, so the X intercept is described when both y in ZR zero. So when there's zero, you have zero equals six months to X minus zero. So two X is equal to six and extra B three. So we have one intercept right here. You have her? Why? Intercepts right here. And the Y intercept is described when xn 00 should be along this line right here. So when those air 06 minus zero minus two y, why should just also be three? Okay, so 123 You have another, uh, nurse up there and then finally R Z intercept. That's when X and wire zero. When There. It's basically along this, uh, according to access. So when x of wire zero g of z is equal to six. So our last intercept is Z equals six right here. And our plane should kind of look like this. And it does expand out in this direction, this direction at this direction. Um, but I'm just drawing this in the first quadrant or the first often of our, uh, x Y Z space and the tetra asian I'm talking about is this little triangular or actually has four sides. It looks like a triangle, but has four sides. 123 and then four along the actual plane. That eyes downing the first often. So now we basically have to just find the volume of this. And it's pretty simple. You just have to find the volume between the plane and the Z equals zero plane. So the Z equals zero plane is right here. The evil zero. Um, it's also bounded by the Y equals zero. Or enough of the X equals zero plane. It was just getting messy. The X equals zero plane. The y equals zero plane. Uh, and so on. Okay. Okay. So now what we have to do is basically say, where are we captaining a double integral. Over. Okay. And it's basically justice region in the X y plane, but we kind of need to figure out what that is. Uh, and the most important thing is, we have to find this intercept right here. And it's basically just the intercept between the Z equals zero plane on the, uh, this plane right here. So Z equals six minus two x. Okay. On my screen. Six miles to X minus, two y. In order to find the intercept, we just said them equal to each other. We get that Why is equal Teoh Three minus X And if we draw this out, they say that this is our region are right here. So we draw out region are 123123 Why equals three minus x x y It's basically this triangular region right here and now we can properly describe a double integral so area of the container equal to the double integral of our function six miles, two x As to why, uh, remember, DeLorean was always copy for the volume between the, uh, surface and this surface and the Z equals zero plane. Always remember that. So we would be sweeping out all of this volume right here. So and then we're gonna just do it D y d accidental. Just because I feel like it does really matter. You could do a dxy wide angle and you could verify for yourself that it works as well. Um, and you'll get the same answer. But I was going to do I d. X. So for doing d y dx, we think about how why ranges why ranges from basically Ziegel y equals zero to y equals streamlined sex. So sure to three mine sex. And then basically, why ranges from those two limits for only when acts strangers from zero to right here. And this would just be X equals three because, uh, Y is equal to zero if why is zero on X is equal to three. So this intercept right here is 30 So a ranges from 0 to 3. Okay. And then we could just solve for this long TV ists in a girl. Okay, something for this. We have six. Why my two x y or our regard every other available that you're not integrating respect to as a constant. So this would be just two x y. It's not x squared because we're not integrating respect to X minus y squared problems. Your three, my sex. Okay, And then there should be equal to six times we hit when you plug in three. My sex. So remember to plug into why not X? Because we're in a great insight to why so Six miles six times through my sex minus two x times three My sex minus y squared So three mice x 20 squared, then we subtract from what we get when we plug in zero. And it should just be zero. Because all these have factors apply and zero times any number is zero. So let's just certify this. 18 minus six X minus six X plus two X squared, I believe, Um yep. And then minus nine minus six X plus X squared. This should simplify too. Uh, a team my 12 x plus two X squared, minus nine plus six x minus X squared. And this should be equal Teoh X squared by the six X plus nine. And I'm just gonna factor this X minus 3 20 squared because then when we played into the outer angle concerted three of X minus three quantity squared DX We can just use the fact that this is a linear function inside of a, um, square function. And we can just, uh, anti drive it in this fashion. So we think of the X my story as an ex insight just for now, 3/3, using the reverse parral. And then we just divide by p linear coefficient inside, which is just one. So we have a division by one here. It doesn't really matter. But don't forget to divide by this linear coefficient because that is very important. Or else you're anti derivative will be wrong. And that's a big no, no. Um, plugging in three, we get 0/0 to third over. Three minus will be getting plug in zero. Get negative. Three to third over three. This is equal to zero plus 27/3. And our final result would be nine inches cute. Don't forget this, uh, unit or else your teacher's gonna market wrong. And that's to be That's gonna be a big goof. Okay? And that is our final answer.

## Recommended Questions

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 shape of tetrahedra in which three edges are perpendicular, as shown in the figure on the next page. 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 $z=f(x, y)=6-2 x-2 y . )$

Find the volume of the following solids.

The solid beneath the plane $f(x, y)=6-x-2 y$ and above the region $R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 1\}$

(FIGURE CAN'T COPY)

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $ x + 2y + 3z = 6 $.

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The solid in the first octant bounded by the plane $2 x+3 y+6 z=12$ and the coordinate planes.

A rectangular box is inscribed in the region in the first octant

bounded above by the plane with $x$ -intercept $6, y$ -intercept $6,$ and

$z$ -intercept $6 .$

a. Find an equation for the plane.

b. Find the dimensions of the box of maximum volume.

Find the volume of the following solids using triple integrals.

The region in the first octant bounded by the plane $2 x+3 y+6 z=12$ and the coordinate planes

(FIGURE CAN'T COPY)

(a) Sketch the solid in the first octant that is enclosed

by the planes $x=0, z=0, x=5, z-y=0,$ and $z=-2 y+6 .$

(b) Find the volume of the solid by breaking it into two parts.

Find the volume of each composite figure.

This problem will prepare you for the Concept Connection on page 724

A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw can be inserted.

a. Find the height $h$ of the container to the nearest tenth.

b. Find the volume of the container to the nearest tenth.

c. How many ounces of juice does the container hold? (Hint: 1 in $^{3} \approx 0.55$ oz)

(IMAGE CAN'T COPY)

Find the volume of the wedge cut from the first octant by the cylinder $z=12-3 y^{2}$ and the plane $x+y=2$

Find the volume of the wedge cut from the first octant by the cylinder $z=12-3 y^{2}$ and the plane $x+y=2$