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# Changing dimensions in a box The lengths $a, b,$ and $c$ of theedges of a rectangular box are changing with time. At the instantin question, $a=1 \mathrm{m}, b=2 \mathrm{m}, c=3 \mathrm{m}, d a / d t=d b / d t=$$1 \mathrm{m} / \mathrm{sec},$ and $d c / d t=-3 \mathrm{m} / \mathrm{sec} .$ At what rates are the box's volume $V$ and surface area $S$ changing at that instant? Are the box'sinterior diagonals increasing in length or decreasing?

## $V_{t}=3 \frac{m^{3}}{s}, S_{t}=0 \frac{m^{2}}{s}, D_{t}=\frac{-6}{\sqrt{14}} \frac{m}{s}(\text {decreasing})$

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##### Lily A.

Johns Hopkins University

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

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### Video Transcript

public 48 before this Probably have a rectangle box, which is changing with time. We have already knowing the changing rate here. So the first problem. Hey, that's us. Tow. Checkmate. The ray off the box Valium changes, right? So that is asking the b he he And we know that we is ABC, right? So that is a function about ABC, so that is just equal toe partial b o a times e p thus partial b r b times, baby. Yeah, And plus certainly so that has, uh, uh, DVDs the time. See, sometimes one. Right. And that a time. See, that is also one. Uh, here's a minus three on That's a B bye. So when a swan piece to a three that the sex last three minors takes right that the tube So history Don't forget the unit here for a second. I hear this. So it's the ass as this, uh, two times a B. That's a C spc right. The stakes. Six service. So we have the US key Seiko suit with partial on a tries. Be fussy times. What right. And the patrol B, That's a pussy on minus three times, they must be right. So he's too. So this is five before and the other three and minus ni so that the zero actually, the service doesn't change, right? That zero. And for this area, you have the d uh, be here the equal Stuart squirt a square us. He's where us Cease Where? Right that is equals to one this for first night. So it's scored 40. At this moment. We want to Cabaret, the d dd Did he? So that's also equals two to a Here. Is that the right And to a times what Thus will be times Why by us to see times three. Right? So that's all scored 14 here on how for, uh for us to that the stakes minus six time three. 18 right? 123 Yeah, but it's a team so that my style ready by 14 that is Lauren and zero. So it's de crazy, right? I'm so sorry. The directive dd divided by partial D part away. This just people So to the on the factor. So there's two here, so there's two here. So theon says is finest sakes. Minus six, divided by scored 14. Here's to the other factor, too. Way out there to be up there and to see up there. So there's a tool here. I forgot that.

##### Lily A.

Johns Hopkins University

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

Lectures

Join Bootcamp