Exponential and Logarithmic Functions - Applications of Exponential and Logarithmic Functions
Exponential and Logarithmic Functions - Applications of Exponential and Logarithmic Functions
Secondary 4
8 marks
The mass, \(M\) grams, of a radioactive substance, present at time \(t\) years after first being observed, is given by the formula \(M = 200e^{-kt}\), where \(k\) is a constant. The mass of the substance was 123.7 g after being observed for 2 years.
- State the initial mass of the substance.[1]
- Show that \(k\) is approximately 0.240, correct to 3 significant figures.[1]
- Find the mass of the substance when \(t = 5\),[1]
- Find the value of \(t\) when the mass of the substance is 15% of its initial mass. Give your answers correct to three significant figures.[2]
- Explain, with clear working, why the mass of the substance can never be more than 200 grams.[1]
- Sketch the graph of \(M\) against \(t\).[2]
Solution:
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Answer:(a)(i) \(200\text{ g}\) (ii) \(k=0.240\) (iii) \(60.2\text{ g}\) (iv) \(7.90\text{ years}\) (b) \(M\leq200\text{ g}\) for \(t\geq0\) (c) Decreasing exponential through \((0,200)\), asymptotic to \(M=0\).
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