Question

WILL GIVE BRAINLIEST Chang deposited $5000 into an account with a 4.8% annual interest rate, compounded monthly. Assuming that no withdrawals are made, how long will it take for the investment to grow to $7850 ?
Do not round any intermediate computations, and round your answer to the nearest hundredth.

Answer 1

Answer:

9.42 years (= 113 months)

Step-by-step explanation:

Use the compound rate interest formula:

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

where:

• A = amount
• P = principal
• r = interest rate (in decimal format)
• n = number of times interest is compounded per unit t
• t = time

Given:

• A = $7850
• P = $5000
• r = 4.8% = 0.048
• n = 12
• t = years

$\implies 7850=5000(1+\frac{0.048}{12})^{12t}$

$\implies 7850=5000(1.004)^{12t}$

$\implies \dfrac{7850}{5000}=(1.004)^{12t}$

$\implies 1.57=(1.004)^{12t}$

Take natural logs:

$\implies \ln1.57=\ln(1.004)^{12t}$

$\implies \ln1.57=12t\ln(1.004)$

$\implies t=\dfrac{\ln 1.57}{12 \ln1.004}$

$\implies t=9.42\textsf{ years (nearest hundredth)}$

$\implies t=113 \textsf{ months}$