Suppose a different benk has a rewards program for customers with large account balances. An account with $0$ dollas starts at an interest rate of $r.$ The interest rate is increased by one percentage point per million dollars in the account. So$,$ for example, an account with $$500,000$ earns an isterest rate of $r+\frac{1}{2}(\frac{1}{100})$

Let $B(t)$ be the balase after $t$ years, given in dollars, and let $B_0=B(0).$

$($a$)***$ th t t Write an autonomous differential equation describing $\frac{dB}{dt}.$

$($b$)$ It's tough to solve for $B$ explicitly in the form you found above. A substitution makes it easier. Let $M={10}^{-8}B.$

$2$

ii$.***$r $$r ${?}^2$ tr Write an autonomous differential equation describing $\frac{dM}{dt}.$ Your asswer may only depend on $M$ and $r.$

$($There are no polats for this question, but it will belp you later on$.)$

iv$.******$ Use your answers from parts if and iii to solve for $M(t)$ explicitly.

i$.***$ th $$r

ii$.***$ t$$ t$$ tor Write an autonomous differential equstion describing $\frac{dM}{dt}.$

Your answer may only depend on $M$ and $r.$

iii. $***$ th th th Verify algebraichlly that $\frac{1}{M}-\frac{1}{r+M}=\frac{r}{(r+M)M}.$

$($There are no points for this question, but it will belp you later on$.)$

iv$.******$ &r &t Use your answers from parts ii and iil to solve for $M(t)$ explicitly.

Your answer should only depend on $M,r,t,$ and the namber $M_0=M(0).$ You may assume $M$ and $r$ are positive.

$($c$)***$ fr tr b$$ Use your answer from part iv to find $B(t)$ explicitly. Your answer should only depend on $B,t,r,$ and $B_0=B(0).$

$($d$)$ The formala for $B(t)$ is not continusous for all $t0.$ Let $t=a$ be the point at which $B(t)$ is discontinuous.

i$.***$r $$ th Find $a.$

Your answer may depend on $r$ and $B_0.$

ii$.***$ th t$$ Evaluate $\underset{t{0}^{+}}{lim}B(t),$ and interpret it in terms of the model. acounts with larger balances. Suggest a better system than the one described in this question.