Cuboid cone volume

We used the same amount of paint to paint a cuboid with dimensions of 10 cm, 15 cm, and 3 cm to paint the shell of a cone whose radius is 8 cm. How tall is this cone? Calculate its volume in liters.

Final Answer:

h =  4.9525 cm
V =  0.3319 L

Step-by-step explanation:

$$\begin{gathered} a=10\text{cm} \\ b=15\text{cm} \\ c=3\text{cm} \\ r=8\text{cm} \\ S=2\cdot (a\cdot b+b\cdot c+c\cdot a)=2\cdot (10\cdot 15+15\cdot 3+3\cdot 10)=450{\text{cm}}^2 \\ S=\pi \cdot r^2+2\pi \cdot r\cdot h \\ {S}_1=\pi \cdot r^2=3.1416\cdot 8^2\doteq 201.0619{\text{cm}}^2 \\ h=\frac{S-S_1}{2\pi \cdot r}=\frac{450-201.0619}{2\cdot 3.1416\cdot 8}=4.9525\text{cm} \end{gathered}$$

$$\begin{gathered} V_1=\frac{1}{3}\cdot S_1\cdot h=\frac{1}{3}\cdot 201.0619\cdot 4.9525\doteq 331.9174{\text{cm}}^3 \\ V=V_1\to \text{l}=V_1:1000\text{l}=331.9174:1000\text{l}=0.332\text{l}=0.3319\text{L} \end{gathered}$$

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