6 years ago Rosa had $2,000 in the bank. She now has $4,500. The amount of money at the end of each year increases exponentially. How many more years (to the nearest year) will pass before Rosa has $20,000 in the bank? (Assume that she doesn't deposit or withdraw any money.) a. 7 years
b. 11 years
c. 17years
d. 39 years

9514 1404 393

  b.  11 years

Step-by-step explanation:

Since we're interested in the time going forward, we can write the expnential function for Rosa's balance as ...

  y = 4500(4500/2000)^(x/6)

where x is the number of years from now, and y is the balance at that time.

  20,000 = 4,500(2.25^(x/6))

Dividing by 4500, we get ...

  40/9 = 2.25^(x/6)

  log(40/9) = (x/6)log(2.25) . . . . take logs

  x = 6·log(40/9)/log(2.25) ≈ 11.037 . . . . divide by the coefficient of x

It will take 11 more years for Rosa to have $20,000 in the bank.

Comment on the exponential function

You can generally write an exponential function for a question of this sort pretty easily. The form of it is ...

  f(t) = (initial value)×(growth factor)^(t/(growth period))

Here, we're given a growth factor of 4500/2000 = 2.25 corresponding to a period of 6 years. This tells us t is in years, and the exponential term will be ...

  2.25^(t/6)

The "initial value" we choose corresponds to t=0. Since we want 'years from now', the "initial value" will be the value in the account now, 4500.

true

explanation: