Question

Find the base radius of a cylindrical hole with maximum volume which can be drilled into a cone of height 16 cm and radius 12 cm as shown below. 

Answer 1

^r/₁₂ = ^16-h/₁₆
16r = 12(16 -h)
⁴/₃r 16 -h
h = 16 - ⁴/₃ r
V = πr²h
=πr²( 16 - ⁴/₃ r)
= (16πr² - ⁴/₃πr³)
^dv/_dr = 32πr - 4πr²
at max, ^dv/_dr = 0
4πr(8 -r) = 0
r=0 or 8
r=8cm