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Interactive video lesson plan for: Max volume of a rectangular box inscribed in a sphere (KristaKingMath)

Activity overview:

► My Partial Derivatives course:

► My Applications of Derivatives course:

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.

● ● ● GET EXTRA HELP ● ● ●

If you could use some extra help with your math class, then check out Krista’s website //


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?!”

So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here:


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