the radioactive substance uranium - 240 has a half - life of 14 hours. the amount a(t) of a sample of…

the radioactive substance uranium - 240 has a half - life of 14 hours. the amount a(t) of a sample of uranium - 240 remaining (in grams) after t hours is given by the following exponential function. a(t)=2800(1/2)^(t/14) find the initial amount in the sample and the amount remaining after 50 hours. round your answers to the nearest gram as necessary. initial amount: grams amount after 50 hours: grams

the radioactive substance uranium - 240 has a half - life of 14 hours. the amount a(t) of a sample of uranium - 240 remaining (in grams) after t hours is given by the following exponential function. a(t)=2800(1/2)^(t/14) find the initial amount in the sample and the amount remaining after 50 hours. round your answers to the nearest gram as necessary. initial amount: grams amount after 50 hours: grams

Answer

Explanation:

Step1: Find the initial amount

The initial amount is when $t = 0$. Substitute $t=0$ into $A(t)=2800(\frac{1}{2})^{\frac{t}{14}}$. When $t = 0$, $A(0)=2800(\frac{1}{2})^{\frac{0}{14}}=2800\times1 = 2800$.

Step2: Find the amount after 50 hours

Substitute $t = 50$ into $A(t)=2800(\frac{1}{2})^{\frac{t}{14}}$. $A(50)=2800(\frac{1}{2})^{\frac{50}{14}}=2800(\frac{1}{2})^{\frac{25}{7}}$. First, calculate $(\frac{1}{2})^{\frac{25}{7}}=2^{-\frac{25}{7}}$. $A(50)=2800\times2^{-\frac{25}{7}}\approx2800\times0.0595\approx167$.

Answer:

Initial amount: 2800 grams Amount after 50 hours: 167 grams