Question

A developing country’s gross domestic product (GDP) from 1998 to 2008 is approximated by: G(t) = -.02t 3 + 2.4t2 + 57 where G(t) is in billions of dollars and t is the number of years past 2000. a. Find G(0) and interpret. b. Find G’(3) and interpret. c. Approximately what will the difference be in GDP going from 2001 to 2003? d. In 2008 was the GDP increasing or decreasing? e. Find the marginal GDP in 2005. Interpret.

Answer 1

(a) The value of G(0) is 57. It means the GDF of country in 2000 is 57 billions.

(b) The value of G'(3) is 13.86. It means the marginal GDP in 2003 is 13.86 billion.

(c) The difference be in GDP going from 2001 to 2003 is 18.68 billion.

(d) In 2008 the GDP was increasing.

(e) The marginal GDP in 2005 is 22.5 billion. It means in 2005 the GDP was increases by 22.5 billion.

Explanation:

The given function is

`G(t)=-0.02t^3+2.4t^2+57` .... (1)

where, G(t) is in billions of dollars and t is the number of years past 2000.

(a)

Substitute t=0 in the given function.

`G(0)=-0.02(0)^3+2.4(0)^2+57=57`

The value of G(0) is 57. It means the GDF of country in 2000 is 57 billions.

(b)

Differentiate function (1) with respect to t.

`G'(t)=-0.02(3t^2)+2.4(2t)+(0)`

`G'(t)=-0.06t^2+4.8t` .... (1)

Substitute t=3 in the above function.

`G'(3)=-0.06(3)^2+4.8(3)=13.86`

The value of G'(3) is 13.86. It means the marginal GDP in 2003 is 13.86 billion.

(c)

Substitute t=1 in the given function, to find the GDP in 2001.

`G(1)=-0.02(1)^3+2.4(1)^2+57=59.38`

Substitute t=3 in the given function, to find the GDP in 2003.

`G(3)=-0.02(3)^3+2.4(3)^2+57=78.06`

The difference be in GDP going from 2001 to 2003 is

`G(3)-G(1)=78.06-59.38=18.68`

Therefore the difference be in GDP going from 2001 to 2003 is 18.68 billion.

(d)

Substitute t=8 in function (1).

`G'(8)=-0.06(8)^2+4.8(8)=34.56`

Marginal GDP in 2008 is 34.56 billion. It means in 2008 the GDP was increasing.

(e)

Substitute t=5 in function (1) to find the marginal GDP in 2005.

`G'(5)=-0.06(5)^2+4.8(5)=22.5`

The marginal GDP in 2005 is 22.5 billion. It means in 2005 the GDP was increases by 22.5 billion.