Here is the step-by-step solution to the problem.
Step 1: Identify the Type of Problem
The problem states that the production decreases by a constant percentage (20%) every year. This means the production amounts form a
Geometric Sequence.
*
First term ($a_1$): The production in Year 1 is
28,450 barrels.
*
Common Ratio ($r$): Since production drops by 20%, the well produces 80% of what it did the previous year.
* $r = 1 - 0.20 = \mathbf{0.8}$
Step 2: Find Total Production in the First Eight Years
To find the total sum of the first 8 years ($S_8$), we use the geometric series sum formula:
$$S_n = \frac{a_1(1 - r^n)}{1 - r}$$
Substitute the values for $n = 8$:
$$S_8 = \frac{28,450(1 - 0.8^8)}{1 - 0.8}$$
Calculation:
1. Calculate $0.8^8$:
$$0.8^8 \approx 0.167772$$
2. Subtract from 1:
$$1 - 0.167772 = 0.832228$$
3. Multiply by the first term ($28,450$):
$$28,450 \times 0.832228 \approx 23,676.89$$
4. Divide by $(1 - 0.8)$, which is $0.2$:
$$\frac{23,676.89}{0.2} = 118,384.45...$$
Rounding to the nearest unit, the total is
118,384 barrels.
Step 3: Find the Maximum Production of the Well
The "maximum production" refers to the total amount the well will produce over its entire lifetime if it continues producing forever (until the amount becomes negligible). This is the sum to infinity ($S_\infty$).
Formula for sum to infinity (valid because $|r| < 1$):
$$S_\infty = \frac{a_1}{1 - r}$$
Calculation:
$$S_\infty = \frac{28,450}{1 - 0.8}$$
$$S_\infty = \frac{28,450}{0.2}$$
$$S_\infty = 142,250$$
The maximum production is exactly
142,250 barrels.
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Final Answer:
Total production in the first eight years:
118,384 barrels
Maximum production of the well:
142,250 barrels
Parent Tip: Review the logic above to help your child master the concept of arithmetic and geometric sequences word problems worksheet.
