Evaluating an exponential function that models a real-world situation
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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.
\[
A(t)=934\left(\frac{1}{2}\right)^{\frac{1}{30}}
\]

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:

Answer

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Answer

Approximate the value using a calculator: $A(80) \approx 214$ grams

Steps

Step 1 ：Evaluate the exponent: $\left(\frac{1}{2}\right)^{\frac{1}{30}}$

Step 2 ：Substitute the evaluated exponent back into the function: $A(0) = 934\sqrt[30]{\frac{1}{2}}$

Step 3 ：Approximate the value using a calculator: $A(0) \approx 903$ grams

Step 4 ：Evaluate the exponent: $\left(\frac{1}{2}\right)^{\frac{8}{3}}$

Step 5 ：Substitute the evaluated exponent back into the function: $A(80) = 934\left(\frac{1}{2}\right)^{\frac{8}{3}}$

Step 6 ：Approximate the value using a calculator: $A(80) \approx 214$ grams