Question

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?

Answer 1

a) k = 0.063578

F(t) = F₀ e⁰•⁰⁶³⁵⁷⁸ᵗ

b) $3.04 trillion

c) 25.6 years

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