Respuesta :
Answer:
Interest compounded quarterly ≈ 8.7 years
Interest compounded continuously ≈ 8.6 years
Step-by-step explanation:
To determine how many years it will take for an investment of $4,000 to reach $7,500, given that the interest rate of 7.3% is compounded quarterly, we can use the compound interest formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]
In this case:
- A = $7,500
- P = $4,000
- r = 7.3% = 0.073
- n = 4 (quarterly)
Substitute the values into the formula:
[tex]7500=4000\left(1+\dfrac{0.073}{4}\right)^{4t}[/tex]
Now, solve for t:
[tex]\begin{aligned}7500&=4000\left(1+0.01825\right)^{4t}}\\\\7500&=4000\left(1.01825\right)^{4t}}\\\\\dfrac{7500}{4000}&=\dfrac{4000\left(1.01825\right)^{4t}}{4000}\\\\1.875&=\left(1.01825\right)^{4t}\\\\\ln(1.875)&=\ln\left(\left(1.01825\right)^{4t}\right)\\\\\ln(1.875)&=4t\ln(1.01825)\\\\\dfrac{\ln(1.875)}{4\ln(1.01825)}&=\dfrac{4t\ln(1.01825)}{4\ln(1.01825)}\\\\t&=\dfrac{\ln(1.875)}{4\ln(1.01825)}\\\\t&=8.6894167625...\\\\t&=8.7\; \sf years\end{aligned}[/tex]
Therefore, it will take approximately 8.7 years to earn enough money if the interest was compounded quarterly.
To determine how many years it would take if the interest was compounded continuously, we can use the continuous compounding interest formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Continuous Compounding Interest Formula}}\\\\A=Pe^{rt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$e$ is Euler's number (constant).}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]
In this case:
- A = $7,500
- P = $4,000
- r = 7.3% = 0.073
Substitute the values into the formula:
[tex]7500=4000\cdot e^{0.073t}[/tex]
Now, solve for t:
[tex]\begin{aligned}\dfrac{7500}{4000}&=\dfrac{4000\cdot e^{0.073t}}{4000}\\\\1.875&=e^{0.073t}\\\\\ln(1.875)&=\ln(e^{0.073t})\\\\\ln(1.875)&=0.073t\ln(e)\\\\\ln(1.875)&=0.073t\\\\t&=\dfrac{\ln(1.875)}{0.073}\\\\t&=8.611077526....\\\\t&=8.6\;\sf years\end{aligned}[/tex]
Therefore, it will take approximately 8.6 years to earn enough money if the interest was compounded continuously.
