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Q4. a. The amount, A, micrograms, of a certain radioactive substance remaining after t years decreases according to the formula: A_t = A₀ e^{- 0.001t}; where A₀ is the amount present initially.

i. Determine the amount of this substance present initially if 500 micrograms remain after 1000 years. (4)

ii. The half - life of a substance is the time taken for the amount to decrease to half its initial amount. Determine the half - life of this substance. (4)

b. Solve for x, x > 0, in the following equation: 5x² = 25^{5 - 3x} (8)