Question

The total amount of money in circulation for the years 1990-2012 can be closely approximated by M(t) = 0.05022t^3 - 1.125t^2 + 37.511t + 247.4. where t represents the number of years since 1990 and M(t) is in billions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years. a. 1991 b. 1998 c. 2005 d. 2011 e. What do your answers in parts a-d tell you about the amount of money in circulation in those years?

Answer 1

Answer:

(a) 35.41166

(b) 29.15324

(c) 37.6595

(d) 56.70206

(e) From 1991 to 1998 the change in amount of money circulation decreases and from 1998 to 2011 the change in amount of money circulation increases.

Step-by-step explanation:

The given function is

$M(t)= 0.05022t^3 - 1.125t^2 + 37.511t + 247.4$

where t represents the number of years since 1990 and M(t) is in billions of dollars.

Differentiate with respect to t.

$M'(t)= 0.05022(3t^2) - 1.125(2t) + 37.511(1) + (0)$

$M'(t)=0.15066t^2 - 2.25t + 37.511$            .... (1)

M'(t) represents the rate of change of money in circulation, where t represents the number of years since 1990.

(a) For 1991, substitute t=1 in equation (1).

$M'(1)=0.15066(1)^2 - 2.25(1) + 37.511=35.41166$

(b) For 1998, substitute t=8 in equation (1).

$M'(8)=0.15066(8)^2 - 2.25(8) + 37.511=29.15324$

(c) For 2005, substitute t=15 in equation (1).

$M'(15)=0.15066(15)^2 - 2.25(15) + 37.511=37.6595$

(d) For 2011, substitute t=21 in equation (1).

$M'(21)=0.15066(21)^2 - 2.25(21) + 37.511=56.70206$

(e) From 1991 to 1998 the change in amount of money circulation decreases and from 1998 to 2011 the change in amount of money circulation increases.