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morpeh [17]
3 years ago
9

If you invest $100,000 in an account earning 8% interest compounded annually, how long will it take until the account holds $300

,000?
(Round to the nearest 0.1 of a year.) 



( answergoes here ) ______________________




Compound interest formulas:  

A=P(1+rn)ntA=P(1+r/n)^nt and A=Pe^rt

​
Mathematics
1 answer:
Monica [59]3 years ago
7 0

We have been given that you invest $100,000 in an account earning 8% interest compounded annually. We are asked to find the time it will take the amount to reach $300,000.

We will use compound interest formula to solve our given problem.

A=P(1+\frac{r}{n})^{nt}, where,

A = Final amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form,

n = Number of times interest is compounded per year,

t = Time in years.  

8\%=\frac{8}{100}=0.08

300,000=100,000(1+\frac{0.08}{1})^{1\cdot t}

300,000=100,000(1.08)^{t}

\frac{300,000}{100,000}=\frac{100,000(1.08)^{t}}{100,000}

3=(1.08)^{t}

(1.08)^{t}=3

Let us take natural log on both sides of equation.

\text{ln}((1.08)^{t})=\text{ln}(3)

Using natural log property \text{ln}(a^b)=b\cdot \text{ln}(a), we will get:

t\cdot \text{ln}(1.08)=\text{ln}(3)

\frac{t\cdot \text{ln}(1.08)}{\text{ln}(1.08)}=\frac{\text{ln}(3)}{\text{ln}(1.08)}

t=\frac{1.0986122886681097}{0.0769610411361283}

t=14.274914586

Upon rounding to nearest tenth of year, we will get:

t\approx 14.3

Therefore, it will take approximately 14.3 years until the account holds $300,000.

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