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Numerade Educator

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Problem 53 Hard Difficulty

Find the volume of the described solid $ S $.
A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.

Answer

10 $\mathrm{cm}^{3}$

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Video Transcript

were given a solid S and rest defying the volume of this solid fine. We're told the solid as a tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths three cm 4 cm story and five centimeters boyfriend. And you should know about my real quick said Anthony to lino. It's it's actually clouds long. Stern was he is from Al He volunteered his hometown. Yeah. We kind of took the initiative camp. It's kind of a field of dreams situation where it's like you build. This is a pretty complicated exercise. Um It really helps if you have some experience drawing three dimensional figures. Go to my days in the city. Yeah. So I'll draw a sketch. It's just different angular Greg. He's just from a different time. We'll have three axes which are mutually perpendicular. So we have an edge with a length of three. Make the parents. Yeah, that sounds that's pretty. Where are some other things? We also have an edge of length four. Where's the funny joke? The reveal was that his name was gay. Name's gay fucking. It is. And then we also have an edge of length five recipes that you know how I know you're gay or something. That was that was the improv. See No, that was 40 year old Virgin took his first. Sure. Did he apologize apology for him? And and paul right now to draw the tetrahedron, we have these three mutually perpendicular edges with these lengths and three mutually perpendicular faces which are like this. I know we I'll call this the X, Y and Z axes rescue like now, oh, want to find the equation of the line it passes through the edge that's sort of right in front of us. Mm The edge through I guess you would say from 0032 Uh 500. Do you read? We'll treat it however, as if it's the same as from zero. We're just looking at it essentially the Xz plane. So 03 250 in the Xz plane. So this is going to be New one switch. I played pressure. It's well we have why minus why? But z minus three Over X -0 0 is equal to uh let's see 3 -0-0 -5. So we have Z is equal to -3/5 x plus three. Now I want to find the equation of the line that passes through. Yes. The line from I guess you would say 040 To the .500. This is the same as the line from 04 250 in the X. Why plane? So to find the equation of this line we have Y -4 over X zero equals 4 0: 0 0 -5. Which on the right hand side this is -4/5. So we have that y equals negative 4/5 X plus four. Listening to reg. Mhm. That would liberate all of these oppressed. Mhm. We multiply these two together. This gives us a base times the height. You can use us to get the area of a cross section. So the area of the cross section shall call a it's gonna be a function of X is one half times. This is our area of the base one half times isn't it? Negative 3/5 X plus three times negative 4/5 X plus four. I always assumed it was. I think he's getting head off screen. This eventually simplifies to six 25th times X squared minus 10 X plus 25. Can hand every feel like a plumber. Yes. And therefore we have the volume of our tetrahedron. This is going to be the integral from X equals 02 X equals five. I know that it was red of our area A of X. D X. I was just studied, which is six 25th integral from 0 to 5 of x squared minus 10 X plus 25 D. X. And if you evaluate this integral, you should get eventually an answer of 10 yes that he got from the gay section of target and the answer is in cubic centimeters.