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Math  /  Algebra

QuestionYou go to the doctor and he gives you 15 milligrams of radioactive dye. After 12 minutes, 8 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.
You will spend □\square minutes at the doctor's office.

Studdy Solution

STEP 1

1. The decay of the radioactive dye follows an exponential decay model.
2. The initial amount of dye is 15 milligrams.
3. After 12 minutes, 8 milligrams remain.
4. The detector sounds an alarm if more than 2 milligrams are present.

STEP 2

1. Establish the exponential decay model.
2. Calculate the decay constant using the given information.
3. Determine the time when the dye amount is 2 milligrams.

STEP 3

The exponential decay model is given by the equation:
A(t)=A0e−kt A(t) = A_0 e^{-kt}
where A(t) A(t) is the amount of dye at time t t , A0 A_0 is the initial amount, and k k is the decay constant.

STEP 4

Given A0=15 A_0 = 15 milligrams and A(12)=8 A(12) = 8 milligrams, substitute these values into the decay model:
8=15e−12k 8 = 15 e^{-12k}
Solve for k k :
815=e−12k \frac{8}{15} = e^{-12k}
Take the natural logarithm of both sides:
ln⁡(815)=−12k \ln\left(\frac{8}{15}\right) = -12k
k=−112ln⁡(815) k = -\frac{1}{12} \ln\left(\frac{8}{15}\right)

STEP 5

We need to find the time t t when A(t)=2 A(t) = 2 milligrams. Substitute into the decay model:
2=15e−kt 2 = 15 e^{-kt}
Solve for t t :
215=e−kt \frac{2}{15} = e^{-kt}
Take the natural logarithm of both sides:
ln⁡(215)=−kt \ln\left(\frac{2}{15}\right) = -kt
Substitute the value of k k from Step 2:
t=−1kln⁡(215) t = -\frac{1}{k} \ln\left(\frac{2}{15}\right)
t=12ln⁡(815)ln⁡(215) t = \frac{12}{\ln\left(\frac{8}{15}\right)} \ln\left(\frac{2}{15}\right)
Calculate t t using a calculator:
t≈28.4 t \approx 28.4
Round to the nearest minute:
t≈28 t \approx 28
You will spend 28\boxed{28} minutes at the doctor's office.

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