Question

The amount of outstanding consumer debt? (in trillions of? dollars) is approximated by g(t)= 0.365362 * 1.068022^t, where t=0

corresponds to 1980. Find the year in which consumer debt is

(a) $2 trillion

(b) $9 trillion

(a) The consumer debt will reach???$2 trillion in the latepart of year:

?(b) The consumer debt will reach???$9 trillion in late part of the year:

Answer 1

The amount of outstanding consumer debt (in trillions of dollars) is approximated by g(t)= 0.365362 * 1.068022^t, where t = 0 corresponds to 1980.

(a). Let the consumer debt reach $2 trillion t years after 1980. Then we have 2 = 0.365362 * 1.068022^t or, 1.068022^t = 2/0.365362 = 5.474023024. Now, on taking logarithm of both the sides, we get t log 1.068022 = log 5.474023024 ( as log mⁿ = n log m) or, t = log 5.474023024 /log1.068022 = 0.738306619/0.028580198 = 25.83 (on rounding off to 2 decimal places). Now, 1980+25.83 = 2005.83, hence the consumer debt will reach $ 2 trillion in the late part of year 2006.

(b). Let the consumer debt reach $3 trillion t years after 1980. Then we have 3 = 0.365362 * 1.068022^t or, 1.068022^t = 3/0.365362 =8.211034536. Now, on taking logarithm of both the sides, we get t log 1.068022 = log 8.211034536 or, t = log 8.211034536 /log1.068022 = 0.914397878/0.028580198 = 31.99 (on rounding off to 2 decimal places). Now, 1980+31.99 = 2011.99, hence the consumer debt will reach$ 3 trillion in the late part of year 2012.