Answer: To find the exponential growth of the coin's value over the given period, we can use the formula for exponential growth:
V(t) = V0 × e^(kt)
where V(t) is the value of the coin at time t, V0 is the initial value of the coin, e is the mathematical constant e (approximately equal to 2.71828), k is the growth rate, and t is the time period.
We can use the information given to find the value of k. Let's use 1977 as the initial time period and 1990 as the final time period. The value of the coin in 1977 is V0 = $279, and the value of the coin in 1990 is V(13) = $434 (since 1990 - 1977 = 13 years).
Substituting these values into the formula, we get:
$434 = $279 × e^(k × 13)
Dividing both sides by $279, we get:
e^(k × 13) = $434 / $279
Taking the natural logarithm of both sides, we get:
k × 13 = ln($434 / $279)
Simplifying, we get:
k = ln($434 / $279) / 13
Using a calculator, we find that k is approximately 0.0556.
Now that we have found the growth rate, we can use the formula to find the value of the coin at any time t. For example, to find the value of the coin in 1985 (8 years after 1977), we can use the formula:
V(8) = $279 × e^(0.0556 × 8)
Using a calculator, we find that V(8) is approximately $425.46.
Therefore, we can conclude that the value of the coin grew exponentially at a rate of approximately 5.56% per year, and was worth about $425.46 in 1985.