Question

How to Become a Millionaire Upon graduating from col- lege, Donna has no initial capital. However, during each year she makes deposits amounting to d = $1000 in a bank that pays interest at an annual rate of 8%, compounded continuously. (a) Find the future value, S(t), of Donna's account at any time. (b) What should be the value of the annual deposit d in order that the balance of Donna's account will be 1 million dollars when she retires in 40 years? (c) If d = $2500, what should be the value of r in order that Donna's account will have a balance of 1 million dollars in 40 years?

Answer 1

amount is 1000 ×
`e^(0.08t)`

$40762.20 balance of Donna's account will be 1 million dollars when she retires in 40 years

rate 14.97 % when Donna's account will have a balance of 1 million dollars in 40 years when principal is $2500

Explanation:

principal = $1000

rate = 8 % = 0.08

to find out

the future value, S(t)

principal when Donna's account will be 1 million dollars when she retires in 40 year

at what rate Donna's account will have a balance of 1 million dollars in 40 years

solution

we know compounded continuously formula i.e.

amount = principal ×
`e^(rt)` ..................1

put the value principal and rate in equation 1 to find amount any time

amount = principal ×
`e^(rt)`

amount = 1000 ×
`e^(0.08t)`

in 2nd part we have time 40 year and amount 1 million so put rate amount and time in equation 1 to find principal

rt = 0.08 × 40 = 3.2

amount = principal ×
`e^(rt)`

1000000 = principal ×
`e^(3.2)`

principal = 1000000 /
`e^(3.2)`

principal = 1000000 / 24.5325302

principal = 40762.20397

so $40762.20 balance of Donna's account will be 1 million dollars when she retires in 40 years

in 3rd part we have amount 1 million and principal $2500 and time 40 year put all these in equation 1 to find rate

amount = principal ×
`e^(rt)`

1000000 = 2500 ×
`e^(40r)`

take ln both side

ln
`e^(40r)` = ln (1000000 / 2500 )

40 r = ln 400

r = ln (400) / 40

r = 0.149787

so rate 14.97 % when Donna's account will have a balance of 1 million dollars in 40 years when principal is $2500