Painful bone metastases are common in advanced prostate cancer. Physicians often order treatment with strontium- 89 ( $^{89} Sr$ ), a radionuclide with a strong affinity for bone tissue. A patient is given a sample containing $4 mCi$ of $^{89} Sr$ .
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(a) If $25 %$ of the $^{89} Sr$ remains in the body after 100 days, write a function of the form $Q (t) = Q_{0} e^{- k t}$ to model the amount $Q (t)$ of radioactivity in the body $t$ days after the initial dose. Round the value of $k$ to five decimal places. Do not round intermediate calculations.
$Q (t) =$

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Answer

$Q (t) = 4 e^{- 0.01386 t}$ is the function that models the amount of radioactivity in the body $t$ days after the initial dose.

Steps

Step 1 ：We are given that the initial dose of radioactivity, $Q_{0}$ , is 4 mCi.

Step 2 ：We are also given that 25% of the $^{89} S r$ remains in the body after 100 days. This means that $Q (100) = 0.25 * Q_{0} = 1$ mCi.

Step 3 ：We can use these two points to solve for the decay constant, $k$ , in the equation $1 = 4 e^{- 100 k}$ .

Step 4 ：Solving for $k$ gives us approximately 0.01386.

Step 5 ：Using this value of $k$ , we can write the function $Q (t)$ as $Q (t) = 4 e^{- 0.01386 t}$ .

Step 6 ： $Q (t) = 4 e^{- 0.01386 t}$ is the function that models the amount of radioactivity in the body $t$ days after the initial dose.