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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

recall. The formula for point slope form is wise and times x minus X juan plus y one, which means plugging end through a 0.32500 minus three. Divided by five minutes zero times X minus zero plus three Simplifies to negative three. Divide by five X plus three. Okay, now we have this. We know we're doing the same thing with the 0.4250 times X minus zero plus four. This gives us why is negative four divided by five acts plus four. Okay, now that we have this, we need to some the two parts that we have. So am Max is gonna be 1/2 times negative. 3/5 X plus three comes negative. 4/5 I suppose for. Okay, now that we have this simplified as much as we can, we end up with 6 25th times X squared, minus 10 acts plus 25. Okay, now, this is going to be are integral, Like I specified earlier. We're gonna be integrating it from 0 to 5 when we integrate. We know the first thing we can do is we can pull out a constant which is six over 25. We can use the power rule. Increase the explain it by one divide by the new exponents. I didn't Exxon to any constant. The value becomes the new coefficient. Now we can simply pull again. And this is all just zero. Only putting zero from the twenties just get zero. This gives us our final solution of two times five, which is 10 units centimeters cute.