Question

a tumor is injected with 220 milligrams of iodine-125, which decays exponentially. after 64 weeks, there are 116 milligrams of iodine-125 remaining. assuming that iodine-125 decays at a continuous rate, write an exponential model for this situation in terms of t, the number of weeks passed. round to two decimal places, when necessary. show your work here hint: to add an exponent (x^n), type ^exponent or press “^”

Explanation:

Step1: Recall continuous decay model

The continuous exponential decay model is $A(t) = A_0 e^{kt}$, where $A_0$ is the initial amount, $k$ is the decay rate, and $t$ is time.
Here, $A_0 = 220$, $A(64) = 116$, $t=64$.

Step2: Substitute known values into model

Substitute into the formula:
$116 = 220 e^{k \times 64}$

Step3: Isolate the exponential term

Divide both sides by 220:
$\frac{116}{220} = e^{64k}$
Simplify $\frac{116}{220} \approx 0.5273$, so $0.5273 = e^{64k}$

Step4: Solve for k using natural log

Take natural log of both sides:
$\ln(0.5273) = 64k$
Calculate $\ln(0.5273) \approx -0.6401$, so $-0.6401 = 64k$
Solve for $k$: $k = \frac{-0.6401}{64} \approx -0.01$

Step5: Write the final model

Substitute $A_0=220$ and $k\approx-0.01$ back into the original model.

Answer:

$A(t) = 220e^{-0.01t}$