Question

In 2011 a​ country's federal receipts​ (money taken​ in) totaled ​$2.32 trillion. In​ 2013, total federal receipts were ​$2.75 trillion. Assume that the growth of federal​ receipts, F, can be modeled by an exponential function and use 2011 as the base year ​(tequals​0). ​a) Find the growth rate k to six decimal​ places, and write the exponential function​ F(t), for total receipts in trillions of dollars. ​b) Estimate total federal receipts in 2015. ​c) When will total federal receipts be ​$14 ​trillion? ​a) Find the growth rate k.

Answer 1

Solution:

general form of exponential function is y=ae^kt

a) in 2011, t=0

In​ 2011, total federal receipts were ​$2.32 trillion

2.32 =ae⁰

a=2.32

y=2.32e^kt

In​ 2013, total federal receipts were ​$2.75 trillion

in 2013, t=2

2.75=2.32e^k*2

e^k*2=2.75/2.32

2k=ln(2.75/2.32)

k=(1/2)ln(2.75/2.32)

k=0.085016863

f(t)=y=2.32e^0.085016863t

b)

t=2015-2011=4

f(4)=y=2.32e^{0.085016863*4}

f(4)=3.25trillion

c)

14=2.32e^{0.085016863​​​​​​​t}

e^{0.085016863​​​​​​​t}=14/2.32

0.085016863^{​​​​​​​}t=ln(14/2.32)

t =[ln(14/2.32)]/0.085016863​​​​​​​

t =21.1427

t =21

in the year 2011+21=2031 total federal receipts be $14 trillion