The radioactive substance uranium- 240 has a half-life of 14 hours. The amount $A(t)$ of a sample of uranum-240 remaining (in grams) after $t$ hours is given by the following exponential function.
\[
A(t)=5600\left(\frac{1}{2}\right)^{\frac{t}{14}}
\]
Find the initial amount in the sample and the amount remaining after 60 hours.
Round your answers to the nearest gram as necessary.
\begin{tabular}{|c|c|}
\hline Initial amount: & Dgrams \\
\hline Amount after 60 hours: & Dgrams \\
\hline & $x$ \\
\hline
\end{tabular}
So, the amount remaining after 60 hours is \(\boxed{287}\) grams.
Step 1 :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: \[A(t)=5600\left(\frac{1}{2}\right)^{\frac{t}{14}}\]
Step 2 :The initial amount of the sample can be found by evaluating the function at \(t=0\).
Step 3 :Substitute \(t=0\) into the function: \[A(0)=5600\left(\frac{1}{2}\right)^{\frac{0}{14}} = 5600\]
Step 4 :So, the initial amount of the sample is \(\boxed{5600}\) grams.
Step 5 :The amount remaining after 60 hours can be found by evaluating the function at \(t=60\).
Step 6 :Substitute \(t=60\) into the function: \[A(60)=5600\left(\frac{1}{2}\right)^{\frac{60}{14}} = 287\]
Step 7 :So, the amount remaining after 60 hours is \(\boxed{287}\) grams.