Rachel is $20$ years old. She plans on retiring in $40$ years when she will be $60$ years old. Rachel believes she will live until she is $105.$

In order to live comfortably, she needs a substantial retirement income. She wants to receive a weekly income of $$5,000$ during retirement. The payments will be made at the beginning of each week during her retirement.

Also, Rachel has pledged to make an annual donation to her favorite charity during her retirement. There will be a total of $45$ payments. The payments will be made at the end of each year. The first annual payment will be for $$20,000.$ Rachel wants the annual payments to increase by $3\%$ per year. The payments will end when she dies.

In addition, she would like to establish a scholarship at Toronto Metropolitan University. The first payment from scholarship would be $$50,000.$ The first scholarship payment would be made $7$ years after she retires. Thereafter scholarship payments will be made every year. She wants the payments to continue after her death, therefore the payments will go on forever. To keep pace with inflation, Rachel would like the amount of scholarship payments to increase by $4\%$ each year.

Rachel has a niece, Rebecca. Rebecca is $5$ years old. Rachel plans on giving Rebecca $$1,000,000$ when the Rebecca turns $50$ years old.

During retirement, Rachel expects to earn $6\mathrm{.}5\%$ per year compounded annually.

Rachel currently has $$100,000$ in investment account A$,$ that earns $6\%$ interest per year compounded monthly. Rachel currently contributes $$100$ every week to an investment account B$.$ These contributions are made at the end of each week and will continue until she retires at $60.$ Rachel expects to earn $7\%$ per year compounded annually on her weekly contributions to her retirement account.

a$)$ How much money does she need when she retires at the age of $60?$ Be certain to include all her financial goals.

b$)$ How much money will she have when she retires?

c$)$ How much is she short? $[$Hint: that would be the difference between the amounts in parts $($a$)$ and $($b$)]$

d$)$ In order to finance any shortfall, Rachel will make monthly contributions into a new retirement account. This new retirement account, investment account C$,$ will earn $7\mathrm{.}5\%$ per year compounded annually. The contributions will be made at the end of each month until she retires at $60.$ How much must she contribute each month to the new retirement account?