The maximum volume of this cylinder = 256π cubic inches
What is volume?
In three-dimensional space, volume is the area occupied by an object within its borders. Another name for it is an object's capacity.
Let h be the height and r be the radius.
Then, h+r = 12
=> h=12-r
Volume of the cylinder (V):
V=πr²h
=> h= πr² ( 12-r )
=> h= π ( 12r² - r³ )
Differentiate with respect to r , we get
dv/dr = 24 πr - 3πr²
Find r by taking dv/dr=0
24 πr - 3πr² = 0
3π ( 8r - r² ) = 0
8r = r²
r=8
Again differentiating, we get
d²V/dr² = 24π - 6 π r
When r=8, 24π - 48 π <0
So, V is maximum at r= 8 inches
And h= 12-8= 4 inches
Hence, the required volume of the cylinder = π8²*4 = 256π cubic inches
To learn more about the volume from the given link
https://brainly.com/question/463363
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