What is the biggest cylinder that you can put in a sphere? Calculus 1 optimization tutorial

We will find the largest possible volume of a cylinder inscribed in a sphere. Calculus 1 optimization tutorial.
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#calculus #bprpcalculus #apcalculus #tutorial #math

[joshuafreeze]

I made the derivative of the volume somewhat easier why substituting in for x rather than y:
V = pi * r^2 * h = pi * x^2 * 2y
x^2 + y^2 = 16 –> x^2 = 16 - y^2
V = pi * (16 - y^2) * 2y = 32pi * y - 2pi * y^3
V' = 32pi - 6pi * y^2

[rob]

let radius of cylinder = r
height of cylinder = h
radius of sphere = R
r^2 = R^2 - h^2/4
Vol of cylinder, V = πr^2h = π(R^2-h^2/4)h
dV/dh = πR^2 - 3πh^2/4
for maximum volume, dV/dh = 0
R^2 - 3h^2/4 = 0
h^2 = 4/3R^2
h = 2R/√3, r = √(2/3)R

[crafters]

Without watching it yet i would have tried to simplify into a rectangle within a circle and gone from there

[selxo]

just want to mention that there was no need to make the question this long by using the product rule, as we already know that the x and y must turn out to be postive values (they are lengths) you could have changed x^2 into root(x^4) and then multiplied the two function leaving you with 2pi *root(16X^4-x^6) deriving this is easy you can just derive what's inside the root and equal that to zero, so 64x^3 -6x^5 = zero (we don't need the denominator) x^3(64-6x^2)=zero which leave you with x= zero or x= +- 4root(6)/3 (root(32/3))! hope you ever see this !!

[ratandmonkey]

nice ! I did it differently by varying the angle between (the axis of rotation) and (the line from the center of the sphere to where the cylinder touches the sphere). Lots of sines and cosines !

[abhishekbhattach]

6:13 how can I know when I should use the product rule?