11 A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t .$$
- Solve the differential equation to find the general solution for \(x\) in terms of \(t\).
Initially there was 10 g of the substance.
- Find the particular solution of the differential equation.
- Find to 6 significant figures the amount of substance that would be predicted by the model at
(a) 6 hours,
(b) 6.25 hours. - Comment on the appropriateness of the model for predicting the amount of substance over time.
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