Question

A sample of an unknown chemical element naturally loses its mass over time. The relationship between the elapsed time ttt, in months, since the mass of the sample was initially measured, and its mass, M(t)M(t)M, left parenthesis, t, right parenthesis, in grams, is modeled by the following function: M(t)=120⋅(81625)t Complete the following sentence about the rate of change in the mass of the sample. Round your answer to two decimal places. The mass of the sample decays by a factor of \dfrac35 5 3 ​ start fraction, 3, divided by, 5, end fraction every months.

Answer 1

Answer:

Step-by-step explanation:

Given a sample M(t)

M(t) = 120 • ( 81 / 625)^t

When is the fraction of the mass decay to 3/5 of it's mass

Generally

M(t) = Mo•(k^t)

The original mass is 120

Mo = 120

So, we want to find time when it decay to 3/5 of it's original mas

M = 3/5 × 120

M = 72

Then,

M(t) = 120 • ( 81 / 625)^t

72 = 120 • ( 81 / 625)^t

72 / 120 = ( 81 / 625)^t

0.6 = ( 81 / 625)^t

Take natural logarithmic of both sides

In(0.6) = In(81/625)^t

In(0.6) = t•In(81/625)

t = In(0.6) / In(81/625)

t = In(0.6) / In(0.1296)

t = 0.25 monthly

t = ¼ monthly