The radius and height of the cylinder of maximum volume are approximately 4.041 and 5.715, respectively.
The volume of the hemisphere ([tex]V_{h}[/tex]) is described by the following expression:
[tex]V_{h} = \frac{2\pi}{3}\cdot R^{3}[/tex] (1)
Where:
- [tex]R[/tex] - Radius of the hemisphere.
The volume of the cylinder inscribed in the hemisphere ([tex]V_{c}[/tex]) is:
[tex]V_{c} = \pi\cdot r^{2}\cdot h[/tex] (2)
Where:
- [tex]r[/tex] - Radius of the cylinder.
- [tex]h[/tex] - Height of the cylinder.
By Pythagorean theorem we derive a relationship between the radius of the hemisphere and the radius of the cylinder:
[tex]R^{2} = r^{2}+h^{2}[/tex] (3)
By applying (3) in (1) and simplifying the resulting expression we have this outcome:
[tex]V_{c} = \pi\cdot R^{2}\cdot h - \pi\cdot h^{3}[/tex] (4)
Now we perform first and second derivative tests to determine the dimensions of the cylinder so that volume found is a maximum:
FDT
[tex]2\cdot \pi\cdot R^{2} - 3\cdot \pi\cdot h^{2} = 0[/tex]
[tex]h = \sqrt{\frac{2}{3} }\cdot R[/tex] (5)
SDT
[tex]V''_{c} = -6\pi\cdot h[/tex]
Since [tex]h > 0[/tex], [tex]V_{c}''[/tex] indicates that critical value found in the previous step is associated to a maximum.
By (3) we have an expression for the radius of the cylinder:
[tex]r = \sqrt{R^{2}-\frac{2}{3}\cdot R^{2} }[/tex]
[tex]r = \sqrt{\frac{1}{3} }\cdot R[/tex] (6)
If we know that [tex]R = 7[/tex], then the radius and the height of the cylinder of maximum volume are:
[tex]h\approx 5.715[/tex], [tex]r \approx 4.041[/tex]
The radius and height of the cylinder of maximum volume are approximately 4.041 and 5.715, respectively.
We kindly invite to check this question on optimization: https://brainly.com/question/14160739
