Rotary bodies

The rotating cone and the rotary cylinder have the same volume of 180 cm³ and the same height, v = 15 cm. Which of these two bodies has a larger surface area?

Correct answer:

S₁ =  199.5 cm²
S₂ =  208.2 cm²

Step-by-step explanation:

$$\begin{gathered} V=180{\text{cm}}^3 \\ h=15\text{cm} \\ V=\pi r^{2}h/3 \\ {r}_1=\sqrt{3\cdot V/\pi /h}=\sqrt{3\cdot 180/3.1416/15}\doteq 3.3851\text{cm} \\ s=\sqrt{h^2+r_1^2}=\sqrt{15^2+3.3851^2}\doteq 15.3772\text{cm} \\ {S}_1=\pi \cdot r_1^2+\pi \cdot r_1\cdot s=3.1416\cdot 3.3851^2+3.1416\cdot 3.3851\cdot 15.3772=199.5{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} V=\pi r^{2}h \\ {r}_2=\sqrt{V/\pi /h}=\sqrt{180/3.1416/15}\doteq 1.9544\text{cm} \\ {S}_2=2\pi \cdot r_2^2+2\pi \cdot r_2\cdot h=2\cdot 3.1416\cdot 1.9544^2+2\cdot 3.1416\cdot 1.9544\cdot 15\doteq 208.1988{\text{cm}}^2 \\ {S}_2>S_1 \end{gathered}$$

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