A company's retirement fund has $680.5 million. The fund manager writes the function

Question

F (t) = 680.5e0.0549t to predict its future value, in millions of dollars, after t years. Based on the
given function, after approximately how many years will the value of the retirement fund be
$875 million?
A) t = 1.2 years
B) t = 2.0 years
C) t = 3.2 years
D) t = 4.6 years

Answer 1

To determine how many years it will take for the retirement fund to grow to $875 million using the function \( F(t) = 680.5e^{0.0549t} \), we need to set the function equal to 875 and solve for \( t \):

\[ 680.5 e^{0.0549t} = 875 \]

First, we divide both sides by 680.5:

\[ e^{0.0549t} = \frac{875}{680.5} \]

Calculating the right side:

\[ \frac{875}{680.5} \approx 1.2872 \]

Now we take the natural logarithm of both sides:

\[ \ln(e^{0.0549t}) = \ln(1.2872) \]

Using the property of logarithms, we simplify the left side:

\[ 0.0549t = \ln(1.2872) \]

Calculating \( \ln(1.2872) \):

\[ \ln(1.2872) \approx 0.251 \]

Now, we can solve for \( t \):

\[ t = \frac{\ln(1.2872)}{0.0549} \approx \frac{0.251}{0.0549} \approx 4.57 \]

Thus, the approximate value of \( t \) is about 4.57 years.

Looking at the provided options:

A) t = 1.2 years
B) t = 2.0 years
C) t = 3.2 years
D) t = 4.6 years

The best answer is:

D) t = 4.6 years.