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 (tequals0). 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 =ae^{0}

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^{0.085016863t}

e^{0.085016863t}=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

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