following exponential function. \[ A(t)=266\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining afte Round your answers to the nearest gram as necessary. Initial amount: grams Amount after 100 years: grams
Solution
Step 1 ：The given function is \(A(t)=266\left(\frac{1}{2}\right)^{\frac{t}{30}}\). This function represents the amount of a certain substance remaining after a certain time 't'.
Step 2 ：The initial amount is the value of the function at t=0. Substituting t=0 into the function, we get \(A(0)=266\left(\frac{1}{2}\right)^{\frac{0}{30}} = 266\).
Step 3 ：To find the amount remaining after 100 years, we need to substitute t=100 into the function and calculate the result. \(A(100)=266\left(\frac{1}{2}\right)^{\frac{100}{30}}\).
Step 4 ：After calculating, we find that the amount remaining after 100 years is approximately 26.39 grams.
Step 5 ：Rounding to the nearest gram, we get 26 grams.
Step 6 ：Final Answer: The initial amount is \(\boxed{266}\) grams and the amount remaining after 100 years is approximately \(\boxed{26}\) grams.
