Respuesta :
Answer:
4.14 years.
Step-by-step explanation:
We are asked to find time it will take for an investment of 5000 dollars to grow to 6800 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
We will use compound interest formula to solve our given problem.
[tex]A=P(1+\frac{r}{n})^{nt}[/tex], where,
A = Final amount,
P = Principal amount,
r = Annual interest rate in decimal form,
n = Number of times interest is compounded per year,
t = Time in years
[tex]7.5\%=\frac{7.5}{100}=0.075[/tex]
Upon substituting our given values in above formula, we will get:
[tex]\$6800=\$5000(1+\frac{0.075}{4})^{4*t}[/tex]
[tex]\frac{\$6800}{\$5000}=\frac{\$5000(1+\frac{0.075}{4})^{4t}}{\$5000}[/tex]
[tex]1.36=(1+0.01875)^{4t}[/tex]
[tex]1.36=(1.01875)^{4t}[/tex]
Take natural log of both sides:
[tex]\text{ln}(1.36)=\text{ln}((1.01875)^{4t})[/tex]
[tex]\text{ln}(1.36)=4t\cdot \text{ln}(1.01875)[/tex]
[tex]0.3074846997479606=4t\cdot 0.0185763855729354[/tex]
[tex]0.3074846997479606=t\cdot 0.0743055422917416[/tex]
[tex]t=\frac{0.3074846997479606}{0.0743055422917416}[/tex]
[tex]t=4.13811258574[/tex]
[tex]t\approx 4.14[/tex]
Therefore, it will take approximately 4.14 years for an investment of 5000 dollars to grow to 6800 dollars.
Answer:
4.14 years required for an investment of 5000 dollars to grow to 6800 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Step-by-step explanation:
We are asked to find time it will take for an investment of 5000 dollars to grow to 6800 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
We will use compound interest formula to solve our given problem.
[tex]A=P(1+\frac{r}{n}) ^{nt}[/tex]
A = Final amount,
P = Principal amount,
r = Annual interest rate in decimal form,
n = Number of times interest is compounded per year,
t = Time in years
Upon substituting our given values in above formula, we will get:
[tex]$6800=5000((1+\frac{0.075}{4} )^{4t}\\[/tex]
[tex]\frac{6800}{5000} =\frac{5000}{5000} ((1+\frac{0.075}{4}) ^{4t} )[/tex]
[tex]1.36=(1+0.01865)^{4t}[/tex]
[tex]1.36=(1.01875)^{4t}[/tex]
Take natural log of both sides:
[tex]ln(1.36)=ln (1.01875)^{4t}[/tex]
[tex]ln(1.36)=4t .(1.01875)[/tex]
[tex]0.3074846997479606=4t. 0.01857638557293\\t=\frac{0.3074846997479606}{4. 0.01857638557293}[/tex]
[tex]t=4.13811258574[/tex]
t ≈ 4.14 (up to two decimal places)
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