A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 30 grams.

Question

1.) Write an exponential equation f(t) representing this situation. (Let t be the amount of radioactive substance in grams and t be the time in minutes.)

2.) To the nearest minute, what is the half-life of this substance?

Answer 1

1. Let g be the initial mass, let t be the elapsed time in minutes, and let h be the half-life of this substance.

f(t) = g × (1/2)^t/h

2.

30 = 250 × (0.5)^250/h (divide both sides by 250)

30/250 = (0.5)^250/h

log _0.5 (0.12) = 250/h

h = 250/log _0.5 (0.12) = 81.73 minutes, rounded to two decimal places.

h = 82 minutes, rounded to the nearest minute.

Answer 2

One form of an exponential function that can be used to model either growth or decay is

f(t) = A₀·b^t where A₀ represents the initial amount and b is the base of the exponential function. When b > 1, this models exponential growth, when 0 < b < 1 it will model decay. For growth, the greater b is, the faster the growth; for decay, the closer b is to 0, the faster the decay.

Using this form of the exponential function, we have f(t) = 250·b^t and we can use the given point (250, 30) to solve for b:

30 = 250·b²⁵⁰

.12 = b²⁵⁰

b = ²⁵⁰√.12 ≈ .99155 so f(t) = 250·.99155^t

We can find the half-life by finding the time it takes until 125g of the substance remains:

125 = 250·.99155^t

.5 = .99155^t

ln(.5) = t·ln(.99155)

t ≈ 82 minutes (We can ballpark to see that this answer makes good sense: 250 minutes would represent just a little over 3 1/2 lives elapsed, which would mean the amount has halved 3 times, and about 1/8 of the initial amount would remain. 30 g is a little less than 1/8 of the original 250g.)

Btw, since we now have the length of the 1/2 life, an alternate form of the exponential function would be

f(t) = 250 · (1/2)^t/82

You can check to see that plugging in 250 for t in this equation also gives f(250) ≈ 30.