Question

A mine has a reserve of 150 million tonne (Mt) and a maximum production capacity of 5 Mt/year. First-year production is 2 Mt and increases by 20% each year. Find the reserve remaining at the end of 15 years (rounded off to two decimal places).

Hint:

Increase production geometrically until it hits the capacity limit, then add capped production for remaining years.

Answer 1

Correct Answer: 84.8

First year production:
\[ P_1 = 2. \]
Production increases by 20% per year until it reaches the maximum allowed capacity of 5 Mt/year. Compute yearly production (capped at 5): \[ P_1 = 2.000 \] \[ P_2 = 2(1.2) = 2.400 \] \[ P_3 = 2.88 \] \[ P_4 = 3.456 \] \[ P_5 = 4.1472 \] \[ P_6 = 4.97664 \]
Next year: \[ P_7 = 4.97664 \times 1.2 = 5.971968>5, \] so production is capped at: \[ P_7 = P_8 = \cdots = P_{15} = 5. \]
Total production for first 6 years:
\[ S_6 = 2 + 2.4 + 2.88 + 3.456 + 4.1472 + 4.97664 = 19.85984\ \text{Mt}. \]
Remaining 9 years at full capacity:
\[ S_9 = 9 \times 5 = 45\ \text{Mt}. \]
Total production in 15 years:
\[ S_{15} = 19.85984 + 45 = 64.85984\ \text{Mt}. \]
Remaining reserve:
\[ R = 150 - 64.85984 = 85.14016\ \text{Mt}. \]
Rounded:
\[ \boxed{85.14\ \text{Mt}} \quad (\text{acceptable range: } 84.80\text{–}85.50) \]