Rotary cylinder

In the rotary cylinder it is given: surface S = 96 cm² and volume V = 192 cm cubic. Calculate its radius and height.

Correct result:

r =  -5.1994 cm
h =  2.2607 cm

Solution:

$$\begin{gathered} S=96{\text{cm}}^2 \\ V=192{\text{cm}}^3 \\ S=2\pi \cdot r^2+2\pi \cdot r\cdot h \\ V=\pi r^{2}h \\ S\mathrm{/}(2\pi )=r^2+r\cdot h \\ V\mathrm{/}\pi =r^{2}h \\ rh=V\mathrm{/}(\pi \cdot r) \\ S\mathrm{/}(2\pi )=r^2+V\mathrm{/}(\pi \cdot r) \\ S\mathrm{/}2=\pi \cdot r^2+V\mathrm{/}r \\ r\cdot S\mathrm{/}2=\pi \cdot r^3+V \\ r=-5.1994=-5.1994\text{ cm } \\ h={\displaystyle \frac{V}{\pi \cdot r^2}}={\displaystyle \frac{192}{3.1416\cdot (-5.1994{)}^2}}\doteq 2.2607\text{ cm } \\ \text{ Verifying Solution: } \\ {S}_1=2\pi \cdot r^2+2\pi \cdot r\cdot h=2\cdot 3.1416\cdot (-5.1994{)}^2+2\cdot 3.1416\cdot (-5.1994)\cdot 2.2607\doteq 96.0035 \\ {V}_1=\pi \cdot r^2\cdot h=3.1416\cdot (-5.1994{)}^2\cdot 2.2607=192 \\ {S}_1=S,V_1=V \end{gathered}$$

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