Respuesta :
Answer:
After 7 years and 5 months.
Step-by-step explanation:
Let x represent number of years.
We have been given that a certain company recently sold five-year $1000 bonds with an annual yield of 9.75%.
We can see that the value of bond is increasing exponentially, so we will use exponential growth formula to solve our given problem.
[tex]y=a\cdot (1+r)^x[/tex], where,
y = Final value,
a = Initial value,
r = Rate in decimal form,
x = Time
[tex]9.75\%=\frac{9.75}{100}=0.0975[/tex]
Substituting given values:
[tex]y=1000\cdot (1+0.0975)^x[/tex]
[tex]y=1000\cdot (1.0975)^x[/tex]
Since we need the selling price to be twice the original price, so we will substitute [tex]y=2000[/tex] in above equation as:
[tex]2000=1000\cdot (1.0975)^x[/tex]
[tex]\frac{2000}{1000}=\frac{1000\cdot (1.0975)^x}{1000}[/tex]
[tex]2=1.0975^x[/tex]
Switch sides:
[tex]1.0975^x=2[/tex]
Take natural log of both sides:
[tex]\text{ln}(1.0975^x)=\text{ln}(2)[/tex]
Applying rule [tex]\text{ln}(a^b)=b\cdot \text{ln}(a)[/tex]:
[tex]x\cdot \text{ln}(1.0975)=\text{ln}(2)[/tex]
[tex]\frac{x\cdot \text{ln}(1.0975)}{\text{ln}(1.0975)}=\frac{\text{ln}(2)}{\text{ln}(1.0975)}[/tex]
[tex]x=\frac{0.6931471805599453}{0.0930348659671894}[/tex]
[tex]x=7.45040231265[/tex]
[tex]x\approx 7.4504[/tex]
Since x represents time in years, so we need to convert decimal part into months by multiplying .4504 by 12 as 1 year equals 12 months.
7 years and 12*0.4504023 months = 7 years 5.4 months = 7 years 5 months
Therefore, after 7 years and 5 months the company could sold the bonds for twice their original price.
