Homework 8 sol PDF

Preview

Summary

Homework assignment 8...

Description

MATH 1108 Homework

Han-Bom Moon

Homework 8 Section 3.2. • Write your answer neatly. • You may discuss with your classmates. But do not copy directly. • You have to explain how you deduced your answer. Explain your notations, and show computational steps.

1. In 1997 article in The New York Times discussed how long it would take for Bill Gates, the world’s second richest person at the time (behind the Sultan of Brunei), to become the world’s first trillionaire. He was born in 1955, and on 1997, he was worth $42 billion. (In this problem, the interest is compounded annually.) (a) Assume that Bill Gates’ fortune grows at an annual rate of 58%, the historical growth rate through 1997 of Microsoft stock, which made up most of his wealth in 1997. Find the age at which he becomes a trillionaire. We will use billion dollars as the unit. P = 42,

A = 1000,

m = 1,

i = r = 0.58

A = P (1 + i)n ⇒ 1000 = 42(1 + 0.58)n = 42(1.58)n 1000 1000 ⇒ (1.58)n = ⇒ ln(1.58)n = ln 42 42 1000 ⇒ 0.457425n ≈ 3.170086 ⇒ n ln 1.58 = ln 42 3.170086 ≈ 6.930285 ⇒n≈ 0.457425 It takes 7 years to be a trillionaire. At 2004, he is 49. (b) What rate of growth would be necessary for Bill Gates to become a trillionaire by the time he is eligible for Social Security on 2022, after he has turned 66? n = 2022 − 1997 = 25,

P = 42,

A = 1000,

m = 1,

A = P (1 + i)n ⇒ 1000 = 42(1 + i)25 r 1000 25 1000 25 ≈ 1.135194 ⇒ 1+i = ⇒ (1 + i) = 42 42 ⇒ r = i ≈ 1.135194 − 1 = 0.135194 The growth rate would be 13.52%. 1

i=r

MATH 1108 Homework

Han-Bom Moon

(c) Forbes magazine’s listings of billionaires for 2006 and 2010 have given Bill Gates’ worth as roughly $50.0 billion and $53.0 billion, respectively. What was the rate of growth of his wealth between 2006 and 2010? n = 2010 − 2006 = 4,

P = 50,

A = 53,

= m = 1,

A = P (1 + i)n ⇒ 53 = 50(1 + i)4 r 53 4 53 4 ⇒ 1+i = ≈ 1.014674 ⇒ (1 + i) = 50 50 r = i ≈ 1.014674 − 1 = 0.014674 The growth rate is 1.47%.

2

i=r...