Final answer:
The minimum amount of gold mined was in 1977, approximately 1214.8 tonnes. This was determined by finding the vertex of the parabola representing the gold production model. In 1985, approximately 1522 tonnes of gold were mined.
Step-by-step explanation:
To find the minimum amount of gold mined in the given period, we need to find the vertex of the parabola represented by the function G = 1492 - 76t + 5.2t^2, where G is the amount of gold in tonnes and t is the number of years since 1970.
The vertex form of a parabola is G(t) = a(t-h)^2 + k, where the vertex is (h, k) and a is the coefficient of the quadratic term. Since the coefficient of the quadratic term is positive (5.2), we know the parabola opens upwards, and the vertex will give us the minimum value. The t-coordinate of the vertex (time at which the minimum gold was mined) can be found using the formula t = -b/(2a), with a being the coefficient of t^2 and b being the coefficient of t:
t = -(-76)/(2*5.2) = 76/10.4 = 7.3077
Since we cannot have a fraction of a year, we round it to 7 years after 1970, which is the year 1977. Substituting t back into our original equation gives us:
G = 1492 - 76(7) + 5.2(7)^2 = 1492 - 532 + 5.2(49) = 1492 - 532 + 254.8 = 1214.8
Therefore, the minimum amount of gold mined was in 1977, and it was approximately 1214.8 tonnes.
To find how much gold was mined in 1985, we replace t with 15 (since 1985 is 15 years after 1970):
G(15) = 1492 - 76(15) + 5.2(15)^2 = 1492 - 1140 + 5.2(225) = 1492 - 1140 + 1170 = 1522 tonnes
Approximately 1522 tonnes of gold were mined in 1985.