the radioactive substance cesium - 137 has a half - life of 30 years. the amount a(t) (in grams) of a sample…

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 100 years. round your answers to the nearest gram as necessary. initial amount: grams amount after 100 years: grams

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 100 years. round your answers to the nearest gram as necessary. initial amount: grams amount after 100 years: grams

Answer

Explanation:

Step1: Find initial amount

Set $t = 0$ in $A(t)=381\left(\frac{1}{2}\right)^{\frac{t}{30}}$. Then $A(0)=381\left(\frac{1}{2}\right)^{\frac{0}{30}}=381\times1 = 381$.

Step2: Find amount after 100 years

Set $t = 100$ in $A(t)=381\left(\frac{1}{2}\right)^{\frac{t}{30}}$. So $A(100)=381\left(\frac{1}{2}\right)^{\frac{100}{30}}=381\left(\frac{1}{2}\right)^{\frac{10}{3}}$. Calculate $\left(\frac{1}{2}\right)^{\frac{10}{3}}=\frac{1}{2^{\frac{10}{3}}}=\frac{1}{\sqrt[3]{1024}}\approx\frac{1}{10.079}\approx0.0992$. Then $A(100)=381\times0.0992\approx38$.

Answer:

Initial amount: 381 grams Amount after 100 years: 38 grams