Max volume of a rectangular box inscribed in a sphere (KristaKingMath)

Learn how to find the largest possible volume of a rectangular box inscribed in a sphere of radius r. Write down the equation of a sphere in standard form and then write an equation for the volume of the rectangular box. Since the equation for volume is the equation that needs to be optimized, solve the other equation for one of the variables and substitute the value into the volume equation so that the volume equation is reduced to two variables instead of three. Then take first-order partial derivatives of the volume equation, set them equal to 0, and use the resulting equations as a system of simultaneous equations to find critical points. Since we're dealing with a 3D figure in real space, ignore any critical numbers that have negative values for x or y. Plug the resulting critical point into the volume equation in two variables to find a value that represents the maximum volume of the rectangular box.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[MauricedelPrado]

Fantastic presentation - very clear audio, very good graphics, very good walk-through - and the information I needed to answer a fun question at work!

[UAPfun]

You explanation is answering the propulsion mechanisms of currently observed UAP’s … thanks for the video

[prateekchaudhary]

Your voice is so sweet
thank you for helping me in understanding the concept mam
🙏🙏🙏🙏🙏

[danielalamparelli]

If I could give your video and explanation more than one 'thumbs up' I would. Thank you! Really clear and concise explanation.

[suithonn]

Gracias!! Estuve buscando esto un dia entero. Saludos desde Argentina

[artemisfaux]

Thank you very much, your vid helped me with my Lagrange multiplier problem, I was having a hard time seeing why it was 2x and 2y and 2z. Also, if you use LGM it would be a quicker calculation I think.

[forevereveryours]

This is a problem that i was stuck on the Stewart and here it is.. Thanks alot, your videos are always very helpful!!!

[mausefue]

Thank you, your videos are always so helpful!!

[boyang]

2 ways: 1st one is your idea that is use partial derivative to find critical point that can help us find maximum or minimum. 2nd way is using Lagrange Multiplier to find extrema. Are they right?

[laylazadina]

Thank you! that was helpful! I liked the voice too, it s so pleasant to listen

[poprostuja]

thank you very much for helping me to understand this topic!!!

[onionlayers]

Okay, you have a beautiful voice... I love it... I just subscribed to your channel, and I really love this... Thank you so much

[notofthisworld]

I was looking for this. I have a test soon, and this helped. Ty very much.

[ahlinad]

Thank you soo much, I really appreciate your effort into visualising a problem and to make seem easy and simple :)

[Lamo]

honestly I can't thank you enough!!!

[mauricioaguilar]

Super increíble, gracias por tu ayuda ahora comprendo mejor, que alivio.

[MoizAMarvi]

Hi! how is there only one '8y' after subtracting two terms with common denominator equation at 7:40? Your cooperating will be highly appreciated.

[bystekaminasgerais]

Can I use Lagrange multiplier? We have the sphere equation and volume equation, so I can use Lagrange multiplier, right?

[dolbydigitalzone]

thank u so much mam
it really helped me

[jigarkumargohil]

but you have not verified for maximum or minimum value of that X, Y and Z. we can verify by using hasian matrix with that second order derivation formula for jacobian