Questions: Radium-226, in grams, decays in such a way that after t years, the amount left over can be modeled by the equation A(t)=450 e^(-0.0004 t). How many grams of Radium-226 will remain after seven years? Round your answer to the nearest tenth.

Solution Steps

We are given a decay model for Radium-226, represented by the equation $A(t) = 450 e^{-0.0004 t}$, where $A(t)$ is the amount of Radium-226 remaining after $t$ years. We need to find the amount remaining after 7 years.

Substitute $t = 7$ into the equation to find $A(7)$: $$A(7) = 450 e^{-0.0004 \times 7}$$

Calculate the exponent: $$-0.0004 \times 7 = -0.0028$$ Now, calculate the exponential term: $$e^{-0.0028}$$

Substitute the calculated exponential term back into the equation: $$A(7) = 450 \times e^{-0.0028}$$ Using a calculator, compute $e^{-0.0028} \approx 0.9972$.

Now, calculate: $$A(7) = 450 \times 0.9972 \approx 448.74$$

Round the result to the nearest tenth: $$A(7) \approx 448.7$$

Final Answer