# Math Help - two optimum stopping time questions

1. ## two optimum stopping time questions

Hello,
I have to solve two optimum stopping time questions. I have no idea where to start them. Or what the right answers are. Here they are

Your father gives you a famous painting by an artist. Its current value is estimated at $2 000 000. If the value of the painting grows according to the formula $V=1000000(2t+1/3)^{1/2}$, when is it optimal to sell it. Assume that the interst rate is 12% per year continuously compounded. t stands for time and is measured and years. Time is set to zero at the present time. I assume in this question I am supposed to use a continuously compounding formula, that is $e^{-0.12t}$ and just tack it onto the end and then take the derivative. Is this correct? I got 4 years for an answer. I included my worksheet as a pdf and would like to know if my methodology is correct. I am not 100% sure of the removal of the $e^{-0.12t}$ factor by dividing the left side 0 by it. It makes sense in my head and 0 divided by a number is 0 but is it right? The second question goes like so. You bought a regular house with a current value of$500 000. If the value of hte house grows according to the formula $V=1000000(t+0.25)^{1/2}$, when is it optimal to sell the house? t is time and measured in years. Interest is 5% per year. What is it's optimal selling value?

The second part is easy if you have the first part right, since you just plug in the numbers into the original equation. This one is not continuously compounded (assuming yearly since it doesn't specify). Would I simply remove the constant, tack on a $1.5^{-t}$, to the end and take the derivative then set it to 0 to find the critical point? If that is correct, I got an optimal selling time of 9.95 years with an optimal selling value of \$3 193 743.88

Any help/verification/instruction is greatly appreciated. Thank you,
Chris