Cone

Into rotating cone with dimensions r = 8 cm and h = 8 cm is an inscribed cylinder with maximum volume so that the cylinder axis is perpendicular to the cone's axis. Determine the dimensions of the cylinder.

Correct answer:

r =  2.67 cm
h =  5.33 cm

Step-by-step explanation:

$$\begin{gathered} V=\frac{1}{3}\pi r^{2}h \\ x/R=2r/H \\ r=\frac{H}{2R}x=\frac{8}{2\cdot 8}x \\ h=2(R-x)=2(8-x) \\ V=\frac{1}{3}\pi (\frac{8}{2\cdot 8}x{)}^2\cdot 2(8-x) \\ V=\frac{\pi 8^2}{6\cdot 8^2}(8x^2-x^3) \\ V‘=\frac{\pi 8^2}{6\cdot 8^2}(2\cdot 8x-3x^2) \\ V‘=0 \\ 2\cdot 8x-3x^2=0 \\ {x}_1=0 \\ 2\cdot 8-3x=0 \\ 16=3x \\ x=5.333cm \\ r=\frac{8}{2\cdot 8}\cdot 5.333=2.67\text{cm} \end{gathered}$$

$h=2(8-5.333)=5.33\text{cm}$

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