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.

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.

Transcript text: 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. (1 point)

Solution

Solution Steps

Step 1: Understand the Problem

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.

Step 2: Substitute the Given Time into the Equation

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

Step 3: Calculate the Exponential Term

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

Step 4: Compute the Amount of Radium-226 Remaining

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: \[ A(7) = 450 \times 0.9972 \approx 448.74 \]

Step 5: Round the Result

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