PET studies require fluorine-18, which is produced in a cyclotron and decays with a half-life of 1.83 hours. Assuming that the F-18 can be transported at 60.0 miles/hour, how close must the hospital be to the cyclotron if 65% of the F-18 produced makes it to the hospital?

Verified step by step guidance

1. <insert step 1: Understand the problem by identifying the key information: the half-life of fluorine-18 (1.83 hours), the transportation speed (60.0 miles/hour), and the requirement that 65% of the F-18 must reach the hospital.>
2. <insert step 2: Use the concept of half-life to determine the decay constant (k) using the formula: k = \frac{0.693}{t_{1/2}}, where t_{1/2} is the half-life.>
3. <insert step 3: Calculate the time (t) it takes for the F-18 to decay to 65% of its original amount using the first-order decay equation: N_t = N_0 e^{-kt}, where N_t/N_0 = 0.65.>
4. <insert step 4: Solve for t in the equation from step 3 to find the time it takes for the F-18 to decay to 65% of its original amount.>
5. <insert step 5: Calculate the maximum distance the F-18 can be transported by multiplying the time (t) from step 4 by the transportation speed (60.0 miles/hour).>