Question Video: Solving Word Problems Involving Geometric Series Mathematics

The production of an oil well every year is 20% less than the year before. Find the total number of barrels produced in the first eight years given the production in the first year was 28,450. Then, find the maximum production of the well. Give all answers to the nearest unit.

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Video Transcript

The production of an oil well every year is 20 percent less than the year before. Find the total number of barrels produced in the first eight years given the production in the first year was 28,450. Then, find the maximum production of the well. Give all answers to the nearest unit.

We are told in this question that a well produces 28,450 barrels of oil in the first year. In the second year, there is a 20 percent reduction. 20 percent of 28,450 is 5,690. We can subtract this from the number of barrels produced in the first year, giving us 22,760 barrels of oil in the second year. We can perform this calculation in one step by multiplying by 0.8. The amount of oil produced in year two is 80 percent of the amount of oil produced in year one. And since 80 percent is equivalent to 0.8, we have a multiplier of 0.8.

We can repeat this process to show that 18,208 barrels of oil were produced in year three. At this stage, we notice that we have a geometric sequence. This is any sequence that has a common ratio or multiplier between consecutive terms. If we let the first term of the sequence be 𝑎 sub one, the second term 𝑎 sub two, the third term 𝑎 sub three, and so on, then we have a geometric sequence with first term 28,450 and common ratio equal to 0.8.

We are asked to calculate the total number of barrels produced in the first eight years. The sum of the first 𝑛 terms of any geometric sequence is equal to 𝑎 sub one multiplied by one minus 𝑟 to the power of 𝑛 all divided by one minus 𝑟. In this question, we want to find the sum of the first eight terms or 𝑆 sub eight. Substituting in our values of 𝑎 sub one and 𝑟, this is equal to 28,450 multiplied by one minus 0.8 to the eighth power all divided by one minus 0.8. Typing this into our calculator, we get 118,384.41 and so on. As we are asked to give our answer to the nearest unit, this is equal to 118,384 barrels.

The second part of the question asks us to find the maximum production of the well. Since the absolute value of the common ratio is less than one, we can find this by calculating the sum to ∞. If 𝑆 sub 𝑛 tends to a limit as 𝑛 tends to ∞, we can calculate this by dividing 𝑎 sub one by one minus 𝑟. In this question, we need to divide 28,450 by one minus 0.8. This is the same as dividing 28,450 by 0.2, which is 142,250. The maximum production of the well is 142,250 barrels of oil.

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