We have just finished step (b) — finding the decay constant — and now we are asking you to do step (c) — to find the time on the clock when the decaying temperature function `T=21+9 e^(-kt)` had the value `37`. The procedure is similar to step (b), except now we know `k` and want to solve for `t`. We substitute `37` in our formula, and we find `16=9 e^(-kt)`. Taking logarithms again, and substituting our known value for `k`, we find
`ln 16=ln 9-kt=ln 9-t text[(]ln 9-ln 7text[)]`.
We solve for `t` and find the time of death to be
`t=(ln 9-ln 16)/(ln 9-ln 7)~~-2.29`.
This still needs interpretation. The negative sign means the time of death was before the time of the first measurement at 8:50 A.M. — could it be otherwise? In fact, the time of death was 2 hours, 17 minutes before 8:50, or 6:33 A.M.
How does this help our detective sort out the suspects? Let's recall who they were: The industrialist's ex-wife, who reported for work at 9 A.M. Monday in Cleveland. His nephew, whose wife claims he was in bed with her from midnight Sunday to about 9 A.M. Monday. The vice-president who was at work two hours before the secretary discovered the body. We have found the time of death to be close enough to the time of discovery to make the rest of the details irrelevant.
Conclusion (of the calculation and of the TV show): The ex-wife didn't have time to commit the murder and drive to Cleveland, nor could she have gotten a plane that early. The nephew was probably in bed. So that leaves the vice-president, who was probably in the building at the right time. Or did the janitor do it?