Người ta xây dựng một chân tháp bằng bê tông có dạng khối chóp cụt tứ giác đều (Hình bên dưới). Cạnh đáy dưới dài 5m, cạnh đáy trên dài 2m, cạnh bên dài 3m. Biết rằng chân tháp được làm bằng bê tông tươi với giá tiền là 1470000 đồng/m3. Tính số tiền để mua bê tông tươi làm chân tháp theo đơn vị đồng.
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Mô hình hoá chân tháp bằng cụt chóp tứ giác đều ABCD.A′B′C′D′ với O, O′ là tâm của hai đáy.
Vậy \(A B=5, A^{\prime} B^{\prime}=2, C C^{\prime}=3\).
\(ABCD\) là hình vuông \(\Rightarrow A C=\sqrt{A B^2+B C^2}=5 \sqrt{2} \Rightarrow C O=\frac{1}{2} A C=\frac{5 \sqrt{2}}{2}\)
\(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) là hình vuông \(\Rightarrow {{A}^{\prime }}{{C}^{\prime }}=\sqrt{{{A}^{\prime }}{{B}^{'}}^{2}+{{B}^{\prime }}{{C}^{'}}^{2}}=2\sqrt{2}\Rightarrow {{C}^{\prime }}{{O}^{\prime }}=\frac{1}{2}{{A}^{\prime }}{{C}^{\prime }}=\sqrt{2}\)
Kẻ \({{C}^{\prime }}H\bot OC\,\,(H\in OC)\)
\(O H C^{\prime} O^{\prime}\) là hình chữ nhật
\(\Rightarrow O H=O^{\prime} C^{\prime}=\sqrt{2}, O O^{\prime}=C^{\prime} H \Rightarrow C H=O C-O H=\frac{3 \sqrt{2}}{2}\)
\(\Delta C C^{\prime} H\) vuông tại \(H \Rightarrow C^{\prime} H=\sqrt{C C^{\prime 2}-C H^2}=\frac{3 \sqrt{2}}{2} \Rightarrow O O^{\prime}=C^{\prime} H=\frac{3 \sqrt{2}}{2}\)
Diện tích đáy lớn là: \(S=A B^2=5^2=25\left(m^2\right)\)
Diện tích đáy bé là: \(S^{\prime}=A^{\prime} B^{\prime 2}=2^2=4\left(m^2\right)\)
Thể tích hình chóp cụt là:
\(V=\frac{1}{3}h\left( S+\sqrt{S{{S}^{\prime }}}+{{S}^{\prime }} \right)=\frac{1}{3}.\frac{3\sqrt{2}}{2}(25+\sqrt{25.4}+4)=\frac{39\sqrt{2}}{2}\left( {{m}^{3}} \right)\)
Số tiền để mua bê tông tươi làm chân tháp là: \(\frac{39\sqrt{2}}{2}.1470000\approx 40538432\) (đồng).
Với cấu trúc mới lạ và đa dạng, Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 4 giúp thí sinh phát huy tối đa khả năng Giải Quyết Vấn Đề, Sáng Tạo và Tư Duy Định Lượng. Bài thi bao gồm ba phần chính: Toán Học Và Xử Lí Số Liệu, Văn Học - Ngôn Ngữ, Khoa Học/Tiếng Anh. Mỗi phần thi đều được thiết kế với các dạng câu hỏi phong phú, không chỉ đánh giá kiến thức lý thuyết mà còn kiểm tra khả năng phân tích dữ liệu, xử lý thông tin và vận dụng thực tế. Đây là lựa chọn hoàn hảo cho học sinh muốn tự tin chinh phục kỳ thi đánh giá năng lực.
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