Mass of Decay question see below for question

Question

. The general decay of mass equation is 𝑀(𝑡) = 𝑀0 𝑒^kt, where M(t) is the mass of

substance at time t, in days, and M0 is the original amount. A chemist uses this equation to model

the mass of radioactive substance remaining when 100g decays exponentially to 25g in 10 days.

(a) Calculate the decay constant k and write the exponential decay function.

(b) Determine the time, in days & hours, for the substance to decay to 10g.

(c) Determine how fast the substance is decaying at two weeks. (2 decimal accuracy)

Answer 1

Part a)

M(t) = M₀e^kt

Plug in M = 25, M₀ = 100, t = 10

25 = 100e^10k

0.25 = e^10k

ln(0.25) = ln(e^10k)

ln(0.25) = (10k)ln(e)

-1.386294 = 10k

k = -0.1386294

So M(t) = 100e^-0.1386294t

Part b)

M(t) = 100e^-0.1386294t

10 = 100e^-0.1386294t

0.1 = e^-0.1386294t

ln(0.1) = ln(e^-0.1386294t)

ln(0.1) = (-0.1386294t)ln(e)

-2.302585 = -0.1386294t

t = 16.6096 days

t = 16 days + 0.6096 days

t = 16 days + 0.6096(24) hours

t = 16 days + 14.63 hours

t = 16 days, and 15 hours (rounding to the nearest hour)

Part c)

"How fast the substance is decaying" infers a rate. To get the rate, take the derivative:

M(t) = 100e^-0.1386294t

M' = 100(-0.1386294)e^-0.1386294t

M' = -13.86294e^-0.1386294t

at 2 weeks, t = 14 days

M' = -13.86294e^{-0.1386294•14}

M' = -13.86294e^-1.940812

M' = -13.86294(0.1435873)

M' = -1.99 grams/day