an investor estimates his fortune in "t" years' time will be: (in thousands of dollars)
$ y(t) = 12[(t/10) + 1]^{3/2}$
a) what is the initial growth rate of his fortune? (give as percent per year)
$ y'/y $ (at t=0)
b) determine the time when the growth rate is equal to $0.1$ (10% per year)
I cannot for the life of me differentiate this and get to a proper conclusion. for part a, i keep getting values that do not make sense. I know the initial growth rate should logically be be > 10%, as part b asks you when it will go down to 10%.
please show a step by step solution, I would like to use this question as a reference to plug in other values to study from.
You don't need to take the log, just differentiate $y(t)$ to get $$y'(t)=12 \frac 32[(t/10)+1]^{1/2}\frac 1{10}$$ Now divide this by $y(t)$ to get the ratio you desire. A lot of factors cancel.