Exponential question

Question

The half-life of Radium-225 is 14.9 years. What is the annual decay rate? Express the result to four decimal places.

Express the annual decay rate as a percentage to two decimal places.

Answer 1

Consider the standard exponential equation of:

A(t) = A₀(1 + r)^t

In this case If we start with an amount of material "A₀" then after 14.9 years, we will have 1/2 of that so A₀/2. Plugging those values into our equation we get:

A₀/2 = A₀(1 + r)^14.9 then dividing both sides by A₀ we get:

1/2 = (1 + r)^14.9 then taking the log of both sides we get:

log(1/2) = log(1 + r)^14.9

-0.30103 = 14.9•log(1 + r) then dividing both sides by 14.9 we get:

-0.0202 = log(1 + r) Now, converting to an exponential:

10^-0.0202 = (1 + r)

0.95455 = 1 + r

r = 0.95455 - 1

r = -0.0454 or the rate is -4.54%