Two bodies

The rectangle with dimensions 8 cm and 4 cm is rotated 360º first around the longer side to form the first body. Then, we similarly rotate the rectangle around the shorter side b to form a second body. Determine the ratio of surfaces of the first and second bodies.

Correct result:

r =  1:2

Solution:

$$\begin{gathered} a=8\text{ cm } \\ b=4\text{ cm } \\ {r}_1=b=4=4\text{ cm } \\ {h}_1=a=8=8\text{ cm } \\ {S}_1=2\pi \cdot r_1^2+2\pi \cdot r_1\cdot h_1=2\cdot 3.1416\cdot 4^2+2\cdot 3.1416\cdot 4\cdot 8\doteq 301.5929{\text{cm}}^2 \\ {r}_2=a=8=8\text{ cm } \\ {h}_2=b=4=4\text{ cm } \\ {S}_2=2\pi \cdot r_2^2+2\pi \cdot r_2\cdot h_2=2\cdot 3.1416\cdot 8^2+2\cdot 3.1416\cdot 8\cdot 4\doteq 603.1858{\text{cm}}^2 \\ r=b:a \\ r=S_1\mathrm{/}{S}_2=301.5929\mathrm{/}603.1858={\displaystyle \frac{1}{2}}=0.5=1:2 \end{gathered}$$

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