# calculus

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM dt equals negative 1 times k times M, where k is a positive constant. If the initial mass was 100g, then find the expression for the mass, M, at any time t.

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1. dM/dt = -k M
separate variables, M left, t right
dM/M = -k dt
integrate
ln M = -k t + c

note e^log a = a
ao
e^ln M = M = e^(-kt+c) = e^-kt e^c
or since e^c is some constant call it C
M = C e^-kt
note that when t = 0
e^-kt = 1
so
C = initial amount
so here
M = 100 e^-kt

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2. Damon is right

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