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.

Result

r =  1:2

Solution:

a=8 cm b=4 cm  r1=b=4 cm h1=a=8 cm  S1=2π⋅ r12+2π⋅ r1⋅ h1=2⋅ 3.1416⋅ 42+2⋅ 3.1416⋅ 4⋅ 8≐301.5929 cm2  r2=a=8 cm h2=b=4 cm  S2=2π⋅ r22+2π⋅ r2⋅ h2=2⋅ 3.1416⋅ 82+2⋅ 3.1416⋅ 8⋅ 4≐603.1858 cm2  r=b:a r=S1/S2=301.5929/603.1858=12=0.5=1:2a=8 \ \text{cm} \ \\ b=4 \ \text{cm} \ \\ \ \\ r_{1}=b=4 \ \text{cm} \ \\ h_{1}=a=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 \ \text{cm} \ \\ h_{2}=b=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}/S_{2}=301.5929/603.1858=\dfrac{ 1 }{ 2 }=0.5=1:2

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