Cone and the ratio

Rotational cone has a height 23 cm and the ratio of the base surface to lateral surface is 7: 9. Calculate a surface of the base and the lateral surface.

Correct result:

S₁ =  2544.79 cm²
S₂ =  3271.87 cm²

Solution:

$$\begin{gathered} v=23 \\ {S}_1=\pi r^2 \\ {S}_2=\pi rs \\ {S}_1:S_2=7\mathrm{/}9 \\ {\displaystyle \frac{\pi r^2}{\pi rs}}=7\mathrm{/}9 \\ r\mathrm{/}s=7\mathrm{/}9 \\ s=r\cdot 9\mathrm{/}7; \\ {s}^2=v^2+r^2 \\ {r}^2\cdot 9^2\mathrm{/}{7}^2=23^2+r^2 \\ {r}^2={\displaystyle \frac{529}{1.653-1}} \\ {r}^2=810.031 \\ r=\sqrt{810.031}=28.46cm \\ s=r\cdot 9\mathrm{/}7=36.59cm \\ {S}_1=\pi r^2=\pi 28.46^2=2544.79{\text{cm}}^2 \end{gathered}$$

$S_2=\pi rs=\pi \cdot 28.46\cdot 36.59=3271.87{\text{cm}}^2$

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