Question 1186907: The world production of gold from 1970 to 1990 can be modelled by G = 5.2t 2 - 76t + 1492, where G is the
number of tonnes of gold and t is the number of years since 1970, t = 1 for 1971 and so on.
a. What was the most amount of gold mined in one year?
b. How much gold was mined in 1978?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **a. Finding the Maximum Gold Production**
The equation G = 5.2t² - 76t + 1492 is a quadratic equation that represents a parabola. Since the coefficient of the t² term is positive (5.2), the parabola opens upwards, meaning it has a minimum value, not a maximum. This indicates the model isn't entirely accurate for representing the *maximum* amount of gold mined. However, we can still find the vertex of the parabola, which represents the year with the *least* amount of gold mined according to this model.
The x-coordinate (in our case, t) of the vertex of a parabola given by ax² + bx + c is -b/2a. In our equation:
* a = 5.2
* b = -76
So, t = -(-76) / (2 * 5.2) = 76 / 10.4 ≈ 7.31
Since 't' represents years since 1970, and we need a whole number, we can check the gold production for t = 7 and t = 8:
* For t = 7: G = 5.2(7)² - 76(7) + 1492 = 1214.8 tonnes
* For t = 8: G = 5.2(8)² - 76(8) + 1492 = 1216.8 tonnes
According to the model, the most gold mined in a single year was approximately **1216.8 tonnes** in 1978 (t=8).
**b. Gold Mined in 1978**
We already calculated this in the previous step. Since 1978 is 8 years after 1970, we use t = 8:
G = 5.2(8)² - 76(8) + 1492 = 1216.8 tonnes
Therefore, according to the model, approximately **1216.8 tonnes** of gold were mined in 1978.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
The world production of gold from 1970 to 1990 can be modelled by G = 5.2t 2 - 76t + 1492,
where G is the number of tonnes of gold and t is the number of years since 1970,
t = 1 for 1971 and so on.
a. What was the most amount of gold mined in one year?
b. How much gold was mined in 1978?
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The problem's formulation is not perfect for a Math problem.
I would edit it in this way (my editing is underlined and crossed)
The world annual production of gold from 1970 to 1990 can be modelled by G = 5.2t 2 - 76t + 1492,
where G is the number of tonnes of gold produced per year and t is the number of years since 1970,
t = 1 for 1971 and so on.
a. What was the  greatest amount of gold mined in one year?
b. How much gold was mined in 1978?
In the post by @CPhill, in the part (a), his reading of the problem is INCORRECT; his interpretation of the problem
is incorrect; his treating of the problem is incorrect; his solution is incorrect and his answer is incorrect.
@CPhill focused on the minimal amount of the gold mined in one year, while the problem in part (a) asks
about the maximum amount.
For it, the amount at the ends of the time interval should be considered.
If you do it, you will find that the greatest amount of the mined gold was achieved in the year 1990 (t = 20),
and this greatest amount was
G(20) = 5.2*20^2 - 76*20 + 1492 = 2052 tons.
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