4 A college loan of 16000 is made at 25 interest compounded

4) A college loan of $16,000 is made at 2.5% interest, compounded annually. After t years, the amount due, A, is given by the function A( t) =16,000(1.025)^t . a) After what amount of time will the amount due reach $22,000? b) Find the doubling time.

Solution

Given that

p = $16,000 , r = 2.5%= 0.025

  A( t) =16,000(1.025)^t

a ) A = $22,000

A( t) =16,000(1.025)^t

22,000 = 16,000 ( 1.025 )^t

22,000 / 16,000 = ( 1.025 )^t

1.375 = ( 1.025 )^t

By applying log on both sides

log ( 1.375) = log (1.025)^t

  log ( 1.375) = t . log (1.025)

t =    log ( 1.375) /  log (1.025)

t = 0.138 / 0.011

t = 0.127

Therefore,

Time t = 0.127

b ) Amount = 2 x 16000 = 32,000

   A =16,000 (1.025)^t

32000 = 16000 (1.025)^t

2 = ( 1.025 )^t

take log on both sides

log 2 = log ( 1.025 )^t

log 2 = t . log (1.025)

t =  log 2 /  log (1.025)

t = 0.290

Therefore,

Doubling time = 0.290