Wooden bowls

20 wooden bowls in the shape of a truncated cone should be painted on the outside and inside with wood varnish. We need 0.1 l of paint to paint 200 cm². How many liters of paint do we have to buy if the bowls are 25 cm high, the bottom of the bowl has a diameter of 20 cm and the upper base has a diameter of 30 cm?

Correct result:

x =  46.3308 l

Solution:

$$\begin{gathered} n=20 \\ \rho =0.1\mathrm{/}200={\displaystyle \frac{1}{2000}}=0.0005{\text{l/cm}}^2 \\ h=25\text{ cm } \\ d=20\text{ cm } \\ D=30\text{ cm } \\ r=d\mathrm{/}2=20\mathrm{/}2=10\text{ cm } \\ R=D\mathrm{/}2=30\mathrm{/}2=15\text{ cm } \\ {S}_1=\pi \cdot r^2=3.1416\cdot 10^2\doteq 314.1593{\text{cm}}^2 \\ {S}_2=\pi \cdot (r+R)\cdot \sqrt{h^2+(R-r{)}^2}=3.1416\cdot (10+15)\cdot \sqrt{25^2+(15-10{)}^2}\doteq 2002.3803{\text{cm}}^2 \\ S=S_1+S_2=314.1593+2002.3803\doteq 2316.5395 \\ x=2\cdot n\cdot \rho \cdot S=2\cdot 20\cdot 0.0005\cdot 2316.5395=46.3308\text{ l} \end{gathered}$$

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