Answer :
The half-life of the radioisotope is approximately 14.72 hours, which is closest to option d) 1.44 hr (1 hour and 26.4 minutes).
Radioisotopes are radioactive isotopes of elements that are used in a variety of applications, including medical imaging and cancer treatment. They are also used in geology and archaeology to determine the age of rocks and artifacts. The activity of a radioisotope is the rate at which it decays, measured in counts per minute (CPM). The half-life of a radioisotope is the amount of time it takes for half of the atoms to decay.
Given that the activity of a radioisotope is 3000 counts per minute at one time and 2736 counts per minute 48 hours later, we can use the formula A = A₀ (1/2)^(t/T) to find the half-life of the radioisotope.
Where A is the activity after time t, A₀ is the initial activity, T is the half-life, and t is the time elapsed.
Substituting the values given in the problem, we get:
2736 = 3000 (1/2)^(48/T)
Dividing both sides by 3000, we get:
0.912 = (1/2)^(48/T)
Taking the natural logarithm of both sides, we get:
ln 0.912 = ln (1/2)^(48/T)
Using the rule that ln (a^b) = b ln a, we get:
ln 0.912 = (48/T) ln (1/2)
Dividing both sides by ln (1/2), we get:
ln 0.912 / ln (1/2) = 48/T
Using a calculator to evaluate the left-hand side, we get:
3.26 = 48/T
Multiplying both sides by T, we get:
3.26T = 48
Dividing both sides by 3.26, we get:
T ≈ 14.72 hours
Therefore, the half-life of the radioisotope is approximately 14.72 hours, which is closest to option d) 1.44 hr (1 hour and 26.4 minutes).
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