In 2011 a country's federal receipts (money taken in) totaled $2.36trillion. In 2013, total federal receipts were

$2.68 trillion. Assume that the growth of federal receipts, F, can be modeled by an exponential function and use 2011 as the base year (t= 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 $12 trillion?

a) k = 0.063578

F (t) = F₀ e⁰•⁰⁶³⁵⁷⁸ᵗ

b) $3.04 trillion

c) 25.6 years

Step-by-step explanation:

The growth of federal receipts can be modelled as an exponential function F (t) using the value in 2011 as the base year

F (t) = F₀ eᵏᵗ

a) t = 2 (2 years from 2011)

F (t) = F (2) = $2.68 trillion

F₀ = $2.36 trillion

k = ?

2.68 = 2.36 e²ᵏ

e²ᵏ = (2.68/2.36) = 1.1356

2k = In (1.1356) = 0.127155

k = 0.063578

F (t) = F₀ e⁰•⁰⁶³⁵⁷⁸ᵗ

b) F (t) = F₀ e⁰•⁰⁶³⁵⁷⁸ᵗ

In 2015, t = 4,

kt = 0.063478 * 4 = 0.2543

F₀ = $2.36 trillion

F (4) = 2.36 e⁰•²⁵⁴³ = 2.36 * 1.290 = $3.04 trillion

c) At what time will F (t) = $12 trillion

F (t) = F₀ e⁰•⁰⁶³⁵⁷⁸ᵗ

12 = 2.36 e⁰•⁰⁶³⁵⁷⁸ᵗ

e⁰•⁰⁶³⁵⁷⁸ᵗ = (12/2.36) = 5.0847

0.063578 t = In (5.0847) = 1.626

t = 25.6 years