Question

Đã trả lời bởi chuyên gia

Answer 1

Đáp án D

Kẻ $A H ⊥ S B \Rightarrow d (A, (S B C)) = A H = \frac{a \sqrt{2}}{2} \Rightarrow Δ S A B$ vuông cân tại $A \Rightarrow S A = a$

$\Rightarrow V_{S . A B C D} = \frac{1}{3} . S A . S_{A B C D} = \frac{1}{3} . a . a^{2} = \frac{a^{3}}{3} .$

Kẻ $M N / / C D \Rightarrow \frac{S M}{S D} = \frac{S N}{S C} = \frac{3}{4}$

Ta có: $V_{S . A B D} = V_{S . B C D} = \frac{1}{2} V_{S . A B C D}$

$\begin{array}{l} \frac{V_{S . A M N B}}{V_{S . A B C D}} = \frac{V_{S . A B M} + V_{S . B M N}}{2 V_{S . A B D}} = \frac{1}{2} (\frac{V_{S . A B M}}{V_{S . A B D}} + \frac{V_{S . B M N}}{V_{S . A B D}}) = \frac{1}{2} (\frac{S M}{S D} + \frac{S M}{S D} . \frac{S N}{S C}) = \frac{1}{2} (\frac{3}{4} + \frac{3}{4} . \frac{3}{4}) = \frac{21}{32} \\ \Rightarrow \frac{V_{M N A B C D}}{V_{S . A B C D}} = \frac{11}{32} V_{S . A B C D} = \frac{11}{32} . \frac{a^{3}}{3} = \frac{11 a^{3}}{96} \end{array}$

Vậy

$V_{M N A B C D} = \frac{11}{32} V_{S . A B C D} = \frac{11}{32} . \frac{a^{3}}{3} = \frac{11 a^{3}}{96}$

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Answer 2