According to me everything you have done is correct
The number of observable facts that a witness to a motor vehicle accident can reliably recall can be modeled by the function R(t) = Ae^-kt , where A represents the number of observable facts a witness can recall immediately following an accident, t is the time in weeks after the accident, and k (where 0 < k <1) is an index related to the individual. For a particular witness to an accident, k =0.75, and A=140.
1. Determine the number of facts that she should be able to recall after 2 weeks.
31 facts, after 2 weeks
R(t) = Ae^-kt
R(2) = 140e^-0.75*2
140e^-1.5 = 31.238
2. How long to the nearest half-day would it take according to this model for the witness to forget half the facts (i.e. find the ‘recall half-life’)?
e^-0.75 = 0.5
ln e^-0.75 = ln 0.5
-0.75 = ln 0.5
t = ln 0.5/-075 = 0.92419624
0.92419624 * 7 (days in a week) = 6.5 (1 dp)
3. Find a value of k for another witness that would give a ‘recall half life’ of 5 days (to 3dp). [Care with units, the unit for time t used in this model is weeks]
k = 0.97
5 days, 5/7 = 0.714 weeks
e^-k*0.714 = 0.5
ln e^-k*0.714 = ln 0.5
-k(0.714) = ln 0.5
k = ln 0.5/-0.714 = 0.970179437
I'm unsure of the last one, can anyone please have a look at these?