Cone - from volume surface area

The volume of the rotating cone is 1,018.87 dm³, and its height is 120 cm. What is the surface area of the cone?

Correct answer:

S =  679.1141 dm²

Step-by-step explanation:

$$\begin{gathered} V=1018.87{\text{dm}}^3 \\ h=120\text{ cm}\to \text{ dm}=120\mathrm{/}10\text{ dm}=12\text{ dm } \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h \\ {S}_1={\displaystyle \frac{3\cdot V}{h}}={\displaystyle \frac{3\cdot 1018.87}{12}}=254.7175{\text{dm}}^2 \\ {S}_1=\pi \cdot r^2 \\ r=\sqrt{S_1\mathrm{/}\pi }=\sqrt{254.7175\mathrm{/}3.1416}\doteq 9.0044\text{ dm } \\ {s}^2=h^2+r^2 \\ s=\sqrt{h^2+r^2}=\sqrt{12^2+9.0044^2}\doteq 15.0026\text{ dm } \\ {S}_2=\pi \cdot r\cdot s=3.1416\cdot 9.0044\cdot 15.0026\doteq 424.3966{\text{dm}}^2 \\ S=S_1+S_2=254.7175+424.3966=679.1141{\text{dm}}^2 \end{gathered}$$

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