Help Maths Differential equations?

Question related to modelling with first-order differential equations. Q. A radioactive isotope has a half-life of 18 days. We have 50g at the end of 70 days. How much radioisotope was initially present.

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  • 50 = P * (1/2)^(70/18)

    50 / (1/2)^(70/18) = P

    50 * 2^(70/18) = P

    50 * 2^(35/9) = P

    P = ‭740.69976982983234336375276861581‬

    A(0) = P

    A(18) = P/2

    A(t) = P * e^(k * t)

    (P/2) = P * e^(k * 18)

    1/2 = e^(18k)

    ln(1/2) = 18k

    (1/18) * ln(1/2) = k

    A(t) = P * e^((1/18) * ln(1/2) * t)

    A(70) = 50

    50 = P * e^((1/18) * ln(1/2) * 70)

    Which will give us the same result we got before.

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