Question

the equation below shows that the value of a $4,800 certificate of deposit (cd) n years in the future will be $5,000.
$5,000 = 4,800(1 + 0.03)^n$
what is the approximate amount of time that must pass before the cd will have a value of $5,000?

a. 1.4 years

b. 1.8 years

c. 2.6 years

d. 3.7 years

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation \( 5000 = 4800(1 + 0.03)^n \) by 4800:
\( \frac{5000}{4800}=(1.03)^n \)
Simplify \( \frac{5000}{4800} \) to \( \frac{25}{24}\approx1.0417 \), so we have \( 1.0417=(1.03)^n \)

Step2: Take the logarithm of both sides

Take the natural logarithm (ln) of both sides:
\( \ln(1.0417)=\ln((1.03)^n) \)
Using the logarithm power rule \( \ln(a^b)=b\ln(a) \), we get:
\( \ln(1.0417)=n\ln(1.03) \)

Step3: Solve for \( n \)

Divide both sides by \( \ln(1.03) \):
\( n = \frac{\ln(1.0417)}{\ln(1.03)} \)
Calculate the values: \( \ln(1.0417)\approx0.0409 \) and \( \ln(1.03)\approx0.0296 \)
Then \( n=\frac{0.0409}{0.0296}\approx1.4 \)

Answer:

A. 1.4 years