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Claire invested $2,400 in an account paying an interest rate of 3.5% compounded monthly. Assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $4,490?

Sagot :

semsee45

Answer:

18 years (to the nearest year)

Step-by-step explanation:

Compound interest formula:

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]

where A is amount, P is principal, r is interest rate (decimal format), n is the number of times interest is compounded per unit 't', and t is time

Given:

  • A = 4490
  • P = 2400
  • r = 3.5% = 0.035
  • n = 12

[tex]\implies 4490=2400(1+\frac{0.035}{12})^{12t}[/tex]

[tex]\implies \dfrac{449}{240}=\left(\dfrac{2407}{2400}\right)^{12t}[/tex]

[tex]\implies \ln\dfrac{449}{240}=\ln\left(\dfrac{2407}{2400}\right)^{12t}[/tex]

[tex]\implies \ln\dfrac{449}{240}=12t\ln\left(\dfrac{2407}{2400}\right)[/tex]

[tex]\implies t=\dfrac{\ln\dfrac{449}{240}}{12\ln\left(\dfrac{2407}{2400}\right)}[/tex]

[tex]\implies t=17.92277136...[/tex]

Therefore, it would take 18 years (to the nearest year) for the account to reach $4,490

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