The maximum volume of cylinder is 39808.9
Let us consider that, radius of cylinder is r and height is h .
Volume of cylinder is,
V = πr²h
Since, the sum of its height and its circumference is 150 cm.
So,
h + 2πr = 150
h = 150 - 2πr
Substituting value of h in volume equation of cylinder.
V = πr²(150 - 2πr)
V = 150πr² - 2π²r³
For maximum volume, differentiate above expression with respect to r and equate with zero.
dv/dr = 300πr - 6π²r²
300πr - 6π²r² = 0
6πr(50 - πr) = 0
6πr = 0
r = 0
50 - πr = 0
r = 50/π
So, radius of cylinder is, 50/π
Now,
h = 150 - 2πr
h = 150 - 2π(50/π)
h = 50
Substituting value of r and h in volume formula of cylinder.
V = πr²h
V = 3.14(50/3.14)(50/3.14)50
V = 39808.9
Thus, the maximum volume of cylinder is 39808.9
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