Find the dimensions of the right circular cylinder of maximum volume that can be placed inside of a sphere of radius R.
1 answer:
This question is incomplete, the complete question is;
Find the dimensions of the right circular cylinder of maximum volume that can be placed inside of a sphere of radius R(10cm)
What is the maximum volume?
Answer:
a) Dimensions of the cylinder are; Radius = 8.1650 cm , Height = 11.547 cm
b) the maximum volume is 2418 cm³
Step-by-step explanation:
From the image
radius of the sphere is 10cm
radius of the cylinder is x and its height is 2y
so
The volume of cylinder is V = πr²h = πx²(2y)
Get V as function of just one variable x² + y² = 100
x² = 100 - y²
Therefore V = π( 100-y² )(2y) = 200πy-2πy³
V will be a maximum when V' = 0
V' = 200π - 6πy² =0
y² = 200π / 6π = 100/3
y = √(100/3) = 5.7735
x² = 100 - y²
x² =100 - (100/3)
x = √(200/3)
x = 8.1650
So The maximum volume will occur when the radius is 8.1650 cm
and the height 2y is 2(5.7735) = 11.547 cm
The maximum volume is
πr²h = π(8.1650)² (11.547 )
= 2418 cm³
Therefore the maximum volume is 2418 cm³
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