Based on data from the US Treasury Department the federal de

Based on data from the U.S. Treasury Department, the federal debt (in trillions of dollars) for the years 1995 to 2004 was given approximately by the formula: D(x) = 4.95 + .402x - .1067x^2 + .0124x^3 - .00024x^4, where x is the number of years elapsed since the end of 1995. a. Estimate the federal debt at the end of 1999 and the rate at which it was increasing at that time. b. React to the statement \"It is true that by the end of 2003 the federal debt was twice as large as it was at the end of 2001\", using mathematical evidence to support or disprove it. c. React to the statement \"It is true that by the end of 2003 the federal debt was increasing at a rate that was more than twice the rate the end of 2001\", using mathematical evidence to support or disprove it.

Solution

D(x) = - 0.00024x^4 + 0.0124x^3 - 0.1067x^2 + 0.402x + 4.95

x no. of years since 1995

a) 1999 ; x = 4

D(4) = - 0.00024(4)^4 + 0.0124(4^3 - 0.1067(4^2 + 0.402*4 + 4.95

= 5.583 trillion dollars

Rate of increase : dD/dx = 0.0372x^2 - 0.00096x^3 - 0.2134x + 0.402

dD/dx at x = 4 ; Rate at which it was increasing =0.082 trillion/year

b) At the end of 2003 means x= 8

D(8) = - 0.00024(8)^4 + 0.0124(8)^3 - 0.1067(8)^2 + 0.402(8) + 4.95

= 6.703 trillion dollars

at the end of 2001 ; x= 6

D(6) = - 0.00024(6)^4 + 0.0124(6)^3 - 0.1067(6)^2 + 0.402(6) + 4.95

= 5.888 trillion dollars

It is not true the federal debt was not twice

c) 2003 : dD/dx at x =8 ; dD/dx = 0.0372(8)^2 - 0.00096(8)^3 - 0.2134*8 + 0.402 = 0.584 trillon/year

2001 ; dD/dx at x= 6 ; dD/dx = 0.0372(6)^2 - 0.00096(6)^3 - 0.2134(6) + 0.402 = 0.253

So, at the end of 2003 it is true federal budget was increasing at a rate more than twice than that at the end of 2001