The cylinder

The cylinder has a surface area of 300 square meters, while the height of the cylinder is 12 m. Calculate the volume of this cylinder.

Correct result:

V =  374.3807 m³

Solution:

$$\begin{gathered} S=300{\text{m}}^2 \\ h=12\text{ cm } \\ S=2\ast pi\ast r^2+2\ast pi\ast r\ast h \\ 300=2\cdot 3.1415926\cdot r^2+2\cdot 3.1415926\cdot r\cdot 12 \\ -6.2831852r^2-75.398r+300=0 \\ 6.2831852r^2+75.398r-300=0 \\ a=6.2831852;b=75.398;c=-300 \\ D=b^2-4ac=75.398^2-4\cdot 6.2831852\cdot (-300)=13224.7141811 \\ D>0 \\ {r}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{-75.4\pm \sqrt{13224.71}}{12.5663704}} \\ {r}_{1,2}=-6\pm 9.15131049315 \\ {r}_1=3.15131049315 \\ {r}_2=-15.1513104931 \\ \text{ Factored form of the equation: } \\ 6.2831852(r-3.15131049315)(r+15.1513104931)=0 \\ r=r_1=3.1513\doteq 3.1513 \\ {S}_1=\pi \cdot r^2=3.1416\cdot 3.1513^2\doteq 31.1984{\text{m}}^2 \\ V=S_1\cdot h=31.1984\cdot 12\doteq 374.3807=374.3807{\text{m}}^3 \\ \text{ Verifying Solution: } \\ {S}_2=2\pi \cdot r^2+2\pi \cdot r\cdot h=2\cdot 3.1416\cdot 3.1513^2+2\cdot 3.1416\cdot 3.1513\cdot 12=300 \\ S=S_2 \end{gathered}$$

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