Question

1. (10 points) the half life of cesium - 137 is 30 years. suppose we have 10 gram sample

(a) find a function ( m(t) ) that models the mass remaining after ( t ) years.
( m(t)=a 2^{-t / t_{h}} )
( a)=10 ) grams
( t_{n}=30 ) years
write your answer in the box below:
( m(t)=10 cdot 2^{-t / 30} )
(b) after how many years will only 2.5 grams of the sample remain?

write your answer in the box below:
60 years
page 3

Explanation:

Step1: Substitute values into the formula

The formula for radioactive decay is \(m(t)=a\cdot2^{-\frac{t}{t_{h}}}\), where \(a = 10\) (initial mass) and \(t_{h}=30\) (half - life). So \(m(t)=10\cdot2^{-\frac{t}{30}}\).

Step2: Solve for \(t\) when \(m(t) = 2.5\)

Set up the equation \(2.5=10\cdot2^{-\frac{t}{30}}\). Divide both sides by \(10\) to get \(\frac{2.5}{10}=2^{-\frac{t}{30}}\), which simplifies to \(0.25 = 2^{-\frac{t}{30}}\). Since \(0.25=2^{- 2}\), then \(-\frac{t}{30}=-2\).

Step3: Solve for \(t\)

Multiply both sides of \(-\frac{t}{30}=-2\) by \(-30\). We get \(t = 60\).

Answer:

(a) \(m(t)=10\cdot2^{-\frac{t}{30}}\)
(b) \(60\) years