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Q4. (a) The amount, A_t, micrograms, of a certain radioactive substance remaining after t years decreases according to the formula:

A_t = A₀ e^{- 1.2} × 10^{- 4} t

Where A₀ is the amount present initially.

(i) Determine the amount of this substance present initially if 400 micrograms remain after 1000 years.          (3)

(ii) The half - life of a substance is the time taken for the amount to decrease to half

Its original amount.

Determine the half - life of this substance.          (5)

(b) Transpose the following formula to make t the subject:

i = Ie^{- 1/CR}

(c) Simplify the following as fully as possible:

(8a⁹ b³ c⁶)^2/3/(4(a³ b² c)²)