Question

13. Problem 5.13 (Time for a Lump Sum to Double) How long will it take $300 to double if it earns the following rates? Compounding occurs once a year. Round your answers to two decimal places. a. 4%. year(s) b. 11%. year(s) c. 20%. year(s) d. 100%. year(s)

Answer 1

Hi,

Explanation:

`Year 0: u_0=a\\rate= t\%\\\\u_1=a*(1+(t)/(100) )\\u_2=a*(1+(t)/(100) )^2\\u_3=a*(1+(t)/(100) )^3\\\\...\\u_n=a*(1+(t)/(100) )^n=2a\\\\(1+(t)/(100) )^n=2\\\\Let's\ take\ the\ logarithm (ln)\\\\ln( (1+(t)/(100) )^n) =ln(2)\\\\n=\frac{ln(2)}{ *ln(1+\frac{t} {100} ) }\\\\a)4\%:\\n=(ln(2))/(ln(1.04))=17,67...\approx{18}\\\\b)11 \%:\\n=(ln(2))/(ln(1.11))=6,64...\approx{7}\\\\`

`c)20 \%:\\n=(ln(2))/(ln(1.20))=3,80...\approx{4}\\\\d)100 \%:\\n=(ln(2))/(ln(2))=1\\`