Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Manufacturing The manufacturer of a new food processor plans to produce and sell 8000 units per year. Each year, \(10 \%\) of all units sold become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. (a) Write a formula in series notation for the number of units that will be operative after \(n\) years. (b) Find the numbers of units that will be operative after 10 years, 15 years, and 20 years. (c) If this trend continues indefinitely, will the number of units that will be operative be finite? If so, how many? If not, explain your reasoning.

Step-by-Step Solution

Answer

a) The formula in series notation for the number of units that will be operative after \(n\) years is given by \(S_n = 8000*(1 - 0.9^n) / (1 - 0.9)\). \n b) After substituting \(n = 10, 15, 20\) into the formula, we can calculate the number of operational units remaining. \n c) The total number of units that will be operative in the long run can be found by applying the formula for an infinite geometric series, \(S_{\infty} = 8000/(1-0.9)\).

Step 1: Understand and Formulate the Geometric Series

Given the manufacturer produces and sells 8000 units per year and \(10%\) breakdown each year, each successive term is given by multiplying the previous term by \(0.9\) (or \(90%\) of the previous term). So, after the first year, there are \(8000\) units; after the second year, there are \([8000 + 0.9*(8000)]\) units and so on. This is a geometric series with first term \(a = 8000\), and common ratio \(r = 0.9\). So, the formula in series notation for the number of units that will be operative after \(n\) years is \(S_n = a + ar + ar^2 + ... + ar^{(n-1)}\) for each \(n \geq 1\).

Step 2: Calculate the number of operative units after specified years

We can use the formula for the sum of a geometric series to find the number of units that will be operative after 10, 15 and 20 years. The formula for the sum \(S_N\) of the first \(N\) terms of a geometric series where \(a\) is the first term, \(r\) is the common ratio is:\[S_N = a * \frac{1 – r^N}{1 - r} \]Substitute \(N = 10, 15, 20\), \(a = 8000\), and \(r = 0.9\) into the formula and calculate to find the number of operational units remaining.

Step 3: Determine if the number of operative units is finite

If the trend continues indefinitely (i.e., \(n\) approaches infinity), we use the formula for the sum of an infinite geometric series to determine if the number of units that will be operative is finite. The formula for an infinite geometric series where \(|r| < 1\) is:\[S_{\infty} = \frac{a}{1- r} \]Substitute \(a = 8000\) and \(r = 0.9\) into the formula to find the total number of units that will be operative in the long run.

Key Concepts

Sum of Geometric Series

Geometric series play a crucial role in various mathematical and real-world applications, like modeling the scenario of a manufacturer's food processor unit longevity. A geometric series is the sum of the terms of a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For a geometric series with first term 'a' and common ratio 'r', the sum of the first 'n' terms is determined using the formula:

• \[ S_n = a + ar + ar^2 + ... + ar^{(n-1)} \]
• \[ S_n = a \times \frac{1 - r^n}{1 - r} \] for \( r eq 1 \)

This provides us a straightforward way to calculate the accumulation of all terms up to a certain point, which can be invaluable in projections and planning, such as determining how many food processors from the manufacturer will remain operational over the years.

In our example, the first term 'a' is the initial number of units sold (8000) and the common ratio 'r' is 0.9, indicating that there is a 10% reduction each year in operative units. Using the sum formula, we can predict the operative units for any given year 'n'.

Geometric Sequence

Understanding the components of a geometric sequence is essential for interpreting patterns and predicting future events in sequences that exhibit exponential growth or decay. A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, represented by 'r'.

The general form of a geometric sequence can be written as:

• \[ a, ar, ar^2, ar^3, ..., ar^{(n-1)} \]

where 'a' is the first term and 'n' is the number of terms.

In our exercise's context, every subsequent year features a multiplication of the remaining operative units by 0.9, effectively creating a geometric sequence with a common degradation factor. Recognizing this allows us to apply formulas from geometric series to accurately quantify future outcomes, like the number of food processors in use over a period.

Infinite Geometric Series

Sometimes it's necessary to consider a geometric series that goes on forever which is particularly relevant when dealing with continuous processes or phenomena that do not have a clear end. An infinite geometric series is the sum of an infinite geometric sequence with terms that approach zero as they continue indefinitely. The sum can be finite, but only if the absolute value of the common ratio is less than 1 (\(|r| < 1\)). The sum is given by the formula:

• \[ S_{\text{\infty}} = \frac{a}{1 - r} \] where \( |r| < 1 \)

This formula is derived from the finite geometric series sum formula as 'n' approaches infinity, causing the \( r^n \) term to vanish due to the diminishing size of 'r'.

In the long-term projection for the manufacturer's food processors, we apply this concept to assess whether a finite number of units will remain in operation indefinitely. Since the common ratio is 0.9, a value less than 1, we can expect a finite sum, which in this case represents the total number of units that would remain operational indefinitely.