A coin sold for $\$ 218$ in 1975 and was sold again in 1989 for $\$ 440$. Assume that the growth in the value $\mathrm{V}$ of the collector's item was exponential.
a) Find the value $k$ of the exponential growth rate. Assume $V_{0}=218$.
\[
\mathrm{k}=.050
\]
(Round to the nearest thousandth.)
b) Find the exponential growth function in terms of $t$, where $t$ is the number of years since 1975 .
\[
V(t)=218 \times e^{0.050 t}
\]
c) Estimate the value of the coin in 2014.
\[
\$ \square
\]
(Round to the nearest dollar.)

Step 1 ：Given the formula for exponential growth: \(V = V0 * e^{kt}\), where \(V\) is the final value, \(V0\) is the initial value, \(k\) is the rate of growth, and \(t\) is the time.

Step 2 ：We can rearrange the formula to solve for \(k\): \(k = \frac{ln(V/V0)}{t}\)

Step 3 ：Substitute the given values into the formula: \(k = \frac{ln(440/218)}{14}\)

Step 4 ：Calculate the value of \(k\) to get \(k = 0.050\) (rounded to the nearest thousandth)

Step 5 ：The exponential growth function in terms of \(t\) is: \(V(t) = V0 * e^{kt}\)

Step 6 ：Substitute the given values into the function: \(V(t) = 218 * e^{0.050t}\)

Step 7 ：To estimate the value of the coin in 2014, substitute \(t = 2014 - 1975 = 39\) years into the function: \(V(39) = 218 * e^{0.050*39}\)

Step 8 ：Calculate the value of \(V(39)\) to get \(V(39) = 218 * e^{1.95}\)

Step 9 ：Further calculate the value of \(V(39)\) to get \(V(39) = 218 * 7.02\)

Step 10 ：Finally, calculate the value of \(V(39)\) to get \(V(39) = 1532.36\)

Step 11 ：Rounding to the nearest dollar, the estimated value of the coin in 2014 is \(\boxed{1532}\)