Question

A mass of 100 grams of a particular radioactive substance decays according to the function m(t)=100e−t850

, where t>0 measures time in years.

When does the mass reach 30 grams?

Answer 1

After 1023.4 years the mass of the substance will be 30 g.

Step-by-step explanation:

Hi there!

Let´s write the function (according to what I found on the web):

`m(t) = 100e^(-t/850)`

We have to find the time "t1" at which the mass of the substance is 30 g. Mathematically:

m(t1) = 30

Then:

`30 = 100e^(-t1/850)`

Let´s solve the equation for t1. First, divide by 100 both sides of the equation:

`0.3 = e^(-t1/850)`[/tex]

Apply ln to both sides of the equation:

`ln(0.3) = ln(e^(-t/850))`

Use the logarithm property: ln (aᵇ) = b ln(a)

ln(0.3) = -t/850 · ln (e) (ln (e) = 1)

ln(0.3) = -t/850

850 ln(0.3) = -t

t = -850 ln(0.3)

t = 1023.4

After 1023.4 years the mass of the substance will be 30 g.