Lateral surface area

The ratio of the area of the base of the rotary cone to its lateral surface area is 3: 5. Calculate the surface and volume of the cone, if its height v = 4 cm.

Correct result:

S =  75.3982 cm²
V =  37.6991 cm³

Solution:

$$\begin{gathered} v=4\text{ cm } \\ {S}_1:S_2=3:5 \\ {S}_1=\pi r^2 \\ {S}_2=\pi rs \\ 3\mathrm{/}5={\displaystyle \frac{\pi r^2}{\pi rs}} \\ 3\mathrm{/}5=r\mathrm{/}s \\ {s}^2=r^2+v^2 \\ {s}^2=(3\mathrm{/}5s{)}^2+v^2 \\ s=v\mathrm{/}\sqrt{1-(3\mathrm{/}5{)}^2}=4\mathrm{/}\sqrt{1-(3\mathrm{/}5{)}^2}=5\text{ cm } \\ r=3\mathrm{/}5\cdot s=3\mathrm{/}5\cdot 5=3\text{ cm } \\ {S}_1=\pi \cdot r^2=3.1416\cdot 3^2\doteq 28.2743{\text{cm}}^2 \\ {S}_2=\pi \cdot r\cdot s=3.1416\cdot 3\cdot 5\doteq 47.1239{\text{cm}}^2 \\ k=S_1\mathrm{/}{S}_2=28.2743\mathrm{/}47.1239={\displaystyle \frac{3}{5}}=0.6 \\ S=S_1+S_2=28.2743+47.1239=75.3982{\text{cm}}^2 \end{gathered}$$

$V={\displaystyle \frac{1}{3}}\cdot S_1\cdot v={\displaystyle \frac{1}{3}}\cdot 28.2743\cdot 4=37.6991{\text{cm}}^3$

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