Expert: Paul Klarreich - 6/25/2006

Question
My name is Chris and I am studying Business Calculus.  The
problem is as follows: Your family owns a rare book whose value
t years from now will be V(i)=7e^(4t)^1/2 dollars.  If the
prevailing interest rate remains constant at 6% per year
compounded continuously, when will it be most advantageous
for your family to sell the book and invest the proceeds.

I started by equating the formula for the book value to the
compound interest formula V=Pe^rt.  I then solved for t using a
natural log and got 1111.11 years.  I'm pretty sure I've missed
something.

Thanks for your help.

Answer
Hi, Chris,

Subject:  Continuously Compounded Interest
Question:  My name is Chris and I am studying Business Calculus.  The problem is as follows: Your family owns a rare book whose value t years from now will be V(i)=7e^(4t)^1/2 dollars.  If the prevailing interest rate remains constant at 6% per year compounded continuously, when will it be most advantageous for your family to sell the book and invest the proceeds.

I started by equating the formula for the book value to the compound interest formula V=Pe^rt.  I then solved for t using a natural log and got 1111.11 years.  I'm pretty sure I've missed something.

Thanks for your help.
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I will assume that:

1. The book is worth $7 today.  That is your V(0)
2. You invest $7 today at 6%, compounded continuously, which does fit your formula  I(t) = 7 exp(0.06t).

So you want to know whether the graphs of  V(t) and I(t) ever intersect.  Easy:

Set  exp(2 sqrt(t)) = exp(0.06t)

Then   2 sqrt(t) = 0.06t   --- don't need any natural logs here.

sqrt(t) = 2/0.06 = 33.33
t = 33.33^2 = 1111.11 , just as you got.

Good news -- the book will be worth a colossal amount at the crossover time.
Bad news -- well, I think you know what that is.