You own an antique that is currently worth 1500, and whose value increases linearly at a rate of 175 per year. If the prevailing interest rate remains constant at 5%, per year compounded continuously, when will it be most advantageous for you to sell the antique and invest the proceeds?
Is there some rule of thumb for setting up an equation for word problems or is there an actual formula to this?
On
Edit: I am expanding on the exposition, the math is unchanged.
As a standard economic principle, assets are valued on the basis of their marginal returns. Here we have a choice of two assets: holding the antique, and selling it for cash. How should we compare the marginal returns? Well, we know that the value of the antique grows linearly, it is $1500+175t$. Hence the marginal return, the derivative in $t$, is 175. Cash compounds continuously at $5 \% $ so the marginal return on \$ $x$ is the derivative of $e^{.05t}x$ (evaluated at $t=0$) or $.05x$.
Now, in this case, the marginal return on the antique is constant over time, while the marginal return on the cash grows to $\infty$. Initially, the antique grossly outperforms cash (as $ 5 \%$ of $1500$ is a lot less than \$ $175$) but over time this changes. Clearly you should switch the instant the marginal return on the cash value exceeds the marginal return on the antique itself.
Therefore, all you need to do here is to figure out when $5\%$ of the value = \$ $175$. But the value after $n$ years is just $1500+175n$ so all we want is $$.05(1500+175n)=175$$ this is easily solved to yield about $11.43$ years.
Suppose you sell the antique at the unknown time $t_s$. Then it is worth $1500+175t_s$. Then at the given time $t_f \geq t_s$, you will have $(1500+175t_s)e^{0.05(t_f-t_s)}$ in investment value. So you want to maximize $(1500+175t_s)e^{0.05(t_f-t_s)}$ over $t_s \in [0,t_f]$. This can be done with calculus. Notably, if the optimal choice of $t_s$ is $t_f$, then you don't sell the antique at all.
As an interesting aside, we can recast this as a (simple) control theory problem, by posing it as:
Maximize $V(t_f)$ over choices of $t_s$, where $V'(t)=0.05V(t)u(t-t_s)+175(1-u(t-t_s))$. Here $u(t)=\begin{cases} 1 & t \geq 0 \\ 0 & t < 0 \end{cases}$.