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Giải bài 10 trang 23 sách bài tập toán 11 - Chân trời sáng tạo tập 2

Đồng vị phóng xạ Uranium-235 (thường được sử dụng trong điện hạt nhân) có chu kì bán rã là (T = 703;800;000) năm.

ꦺTổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Đồng vị phóng xạ Uranium-235 (thường được sử dụng trong điện hạt nhân) có chu kì bán rã là \(T = 703\;800\;000\) năm. Theo đó, nếu ban đầu có 100 gam Uranium-235 thì sau t năm, do bị phân rã, lượng Uranium-235 còn lại được tính bởi công thức \(M = 100{\left( {\frac{1}{2}} \right)^{\frac{t}{T}}}\) (g). Sau thời gian bao lâu thì lượng Uranium-235 còn lại bằng 90% so với ban đầu?

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về giải phương trình mũ cơ bản để giải phương trình: \({a^x} = b\left( {a > 0,a \ne 1} \right)\) + Nếu \(b \le 0\) thì phương trình vô nghiệm. + Nếu \(b > 0\) thì phương trình có nghiệm duy nhất \(x = {\log _a}b\)

Chú ý: Với \(a > 0,a \ne 1\) thì \({a^x} = {a^\alpha } \Leftrightarrow x = \alpha \), tổng quát hơn: \({a^{u\left( x \right)}} = {a^{v\left( x \right)}} \Leftrightarrow u\left( x \right) = v\left( x \right)\)

Lời giải chi tiết

Khi \(M = 100.90\%  = 90\left( g \right)\) thì ta có: \(90 = 100{\left( {\frac{1}{2}} \right)^{\frac{t}{T}}} \Leftrightarrow {\left( {\frac{1}{2}} \right)^{\frac{t}{T}}} = 0,9 \Leftrightarrow \frac{t}{T} = {\log _{\frac{1}{2}}}0,9\) \( \Leftrightarrow t = T.{\log _{\frac{1}{2}}}0,9 \approx 106\;979\;777\) (năm) Vậy sau khoảng \(106\;979\;777\) năm thì lượng Uranium-235 còn lại bằng 90% so với ban đầu.

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