the radioactive substance cesium - 137 has a ...
the radioactive substance cesium - 137 has a half - life of 30 years. the amount a(t) (in grams) of a sample of cesium - 137 remaining after t years is given by the following exponential function. a(t)=381(1/2)^(t/30) find the initial amount in the sample and the amount remaining after 80 years. round your answers to the nearest gram as necessary. initial amount: grams amount after 80 years: grams
Answer
# Explanation:
## Step1: Find initial amount
Set $t = 0$ in $A(t)=381\left(\frac{1}{2}\right)^{\frac{t}{30}}$. Since any non - zero number to the power of 0 is 1, $A(0)=381\left(\frac{1}{2}\right)^0 = 381$ grams.
## Step2: Find amount after 80 years
Substitute $t = 80$ into $A(t)=381\left(\frac{1}{2}\right)^{\frac{t}{30}}$. So $A(80)=381\left(\frac{1}{2}\right)^{\frac{80}{30}}=381\left(\frac{1}{2}\right)^{\frac{8}{3}}$. Calculate $\left(\frac{1}{2}\right)^{\frac{8}{3}}=\frac{1}{2^{\frac{8}{3}}}=\frac{1}{\sqrt[3]{256}}\approx\frac{1}{6.3496}$. Then $A(80)=381\times\frac{1}{6.3496}\approx60$ grams.
# Answer:
Initial amount: 381 grams
Amount after 80 years: 60 grams