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You decide to put $2,000 in a savings account to save for a $3,000 downpayment on a new car. If the account has an interest rate of 4% per year and is compounded monthly, how long does it take until you have $3,000 without depositing any additional funds? 121.862 years 12.1862 years 10.155 years 1.0155 years

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sqdancefan

Answer:

10.153 years

Step-by-step explanation:

The future value of such an investment is given by ...

FV = P·(1 +r/12)^(12t)

where P is the principal invested, FV is the future value of it, r is the annual interest rate, and t is the number of years.

Dividing by P and taking the log, we have ...

FV/P = (1 +r/12)^(12t)

log(FV/P) = 12t·log(1 +r/12)

Dividing by the coefficient of t gives ...

t = log(FV/P)/log(1 +r/12)/12 = log(3000/2000)/log(1 +.003333...)/12 ≈ 121.842/12

t ≈ 10.153 . . . years

Ver imagen sqdancefan
abidemiokin

The time it takes until you have $3,000 without depositing any additional funds is 1.015 years. Option D is correct.

The formula for calculating compound interest is expressed as:

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

Given the following parameters

A = $3000

P = $2000

rate = 4% = 0.04

Time t = ?

Time of compounding = 12 (monthly)

Substitute the given parameters into the formula to get "t"

[tex]3000 = 2000(1+\frac{0.04}{12} )^{12t}\\1.5 = (1 + 0.0333)^{12t}\\1.5 = (1.033)^{12t}\\log 1.5 =12tlog1.0333\\0.17609 =12 (0.01422)t\\t = \frac{0.17609}{0.1707} \\t = 1.015 years[/tex]

Hence the time it takes until you have $3,000 without depositing any additional funds is 1.015 years

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