Pyramid cut

We cut the regular square pyramid with a parallel plane to the two parts (see figure). The volume of the smaller pyramid is 20% of the volume of the original one. The bottom of the base of the smaller pyramid has an area of 10 cm². Find the area of the original pyramid.

Final Answer:

S₁ =  29.2402 cm²

Step-by-step explanation:

$$\begin{gathered} S_2=10{\text{cm}}^2 \\ {S}_2=a_2^2 \\ {a}_2=\sqrt{S_2}=\sqrt{10}=\sqrt{10}\text{cm}\doteq 3.1623\text{cm} \\ {V}_2=\frac{20}{100}{V}_1 \\ \frac{1}{3}{S}_2{h}_2=\frac{20}{100}\frac{1}{3}{S}_1{h}_1 \\ {S}_2{h}_2=0.20S_1{h}_1 \\ {a}_1:h_1=a_2:h_2 \\ {a}_1:a_2=h_1:h_2 \\ {S}_2=a_1/a_2\cdot 0.20S_1 \\ {a}_2^3=a_1^3\cdot 0.20 \\ {a}_1=(a_2^3/0.20{)}^{1/3} \\ {a}_1=a_2\cdot \sqrt[3]{1/0.20}=3.1623\cdot \sqrt[3]{1/0.20}\doteq 5.4074\text{cm} \\ {S}_1=a_1^2=5.4074^2=29.2402{\text{cm}}^2 \end{gathered}$$

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