can you please check over why my method is incorrect for this question. Topic differentiation A Cylinder of height h and radius r is inside a sphere of of radius 10. find the value of r that will maximise volume of cylinder

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can you please check over why my method is incorrect for this question. Topic differentiation A Cylinder of height h and radius r is inside a sphere of of radius 10. find the value of r that will maximise volume of cylinder

Mathematics
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Volume of a cylinder is v=pi r^2*h so i went about getting an expression or h to substitute in by using similar triangles
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so i used similar triangles and obtained h=20-2r .. the model answers showed that they used pythagoras.. why is my methods incorrect?

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height is based off a curve, not a line
height = sqrt(10^2 - x^2)
oh, and even tho thats pythagorean, its really based off of the equation for the circle of the sphere
but if u take the into consideration the height and radius measurements from the origin of x and y axis - wont it be a straight line?
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all points equidistant from the center ... in this case the origin ... is the equation x^2 + y^2 = r^2
height is measured as 2y in this case. radius is x
but if u take the into consideration the height and radius measurements from the origin of x and y axis - wont it be a straight line?
repeating faulty arguments do not make them any more correct ...
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where r is the radius of the cylinder, h is the height of it
i am not trying sound smart .. i was just generally thinking about these problems... i did a problem before with a cylinder inside a triangle..
a sphere is not a triangle :)
a cone would have a linear setup ... since its profile is a triangle
in that case- similar triangle works.. but u cant apply the height is based off a curve...I am sorry i didnt mean to sound arrogant.. im just thinking
thinking is good ... just some thoughts are worse than others
I appreciate your answer. thank you
x^2 + y^2 = 10^2 is the equation of your circle, which your cylindar sets in y = sqrt(100-x^2) is the top of the sphere, which is 1/2 the height of the cylindar from the x axis
yes i get how that formula is used here. so i guess it really comes down to what the cylinder is inscribed inside...if it is a sphere then its properties would be different to a cone. Thank you
V = pi r^2 h, r=x, and h=2sqrt(100-x^2) V' = pi (2r r' + r^2 h') V' = 0 when, 2r r' + r^2 h' = 0 just some of my thoughts is all
correct, good luck :)
thank you!

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