$($a$)$ Let time t be measured in years. Assume that the force of interest per unit time is given by $\backslash$delta $($t$)=(\backslash$alpha $+2$t$)(\backslash$alpha $2$t$)$ where $\backslash$alpha is a constant to be found.

If the accumulation of an investment of $1$ unit of currency from t $=0$ to t $=1$ is A$(0,1)=1,$ find the general formula A$(0,$ t$)$ for all t$.$

$($b$)$ Now let $\backslash$kappa $,\backslash$beta and $\backslash$gamma be constants that may be chosen to model a smoothly decreasing or increasing force of interest given by $\backslash$delta $($t$)=\backslash$kappa $+\backslash$beta $/(1+\backslash$gamma e$\backslash$beta t $)$

$$ Show that v$($t$)$ is given by

v$($t$)=\{1/(1+\backslash$gamma $)\}$e$(\backslash$kappa $+\backslash$beta $)$t $+\{\backslash$gamma $/(1+\backslash$gamma $)\}$e$\backslash$kappa t $(3)$

Let $\backslash$beta $=0\mathrm{.}12,\backslash$kappa $=0\mathrm{.}08,$ and $\backslash$gamma $=0\mathrm{.}5.$ Find the present value of $100$ due in $10$ years time.