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 \(\boxed{1532}\)