$\mathrm{}$You are $40$ years old and wish to retire at age $65.$ You expect to be able to average a $6\%$ annual rate on savings over your lifetime. You would like to save enough money to provide $$8,000$ per year beginning at age $66$ in retirement income to supplement other sources $($social security, pension plans, etc.$).$ Suppose you decide that the extra income need be provided for only $15$ years $($up to age $80).$  

 

a$)$ How much will you have saved when you are $65$ to make this plan possible?

b$)$ How much must you save each year between now and retirement to achieve your goal? Assume that your first contribution to the savings plan will take place one year from now.