Problem 98 - Precalculus with Limits

Question

A city bowling league is holding a tournament in which the top \( 12 \) bowlers with the highest three-game totals are awarded cash prizes. First place will win \( \$1200 \), second place \( \$1100 \), third place \( \$1000 \), and so on. (a) Write a sequence that represents the cash prize awarded in terms of the place in which the bowler finishes. (b) Find the total amount of prize money awarded at the tournament.

Step-by-Step Solution

Answer

The sequence representing the cash prize in terms of the place the bowler finishes can be expressed in the form of \(a_n = 1200 + (n-1)(-100)\), where \(n\) represents the place in which the player finishes. The total prize money awarded at the tournament is 9000 dollars.

Step 1: Formulate the sequence

The prize money forms an arithmetic sequence where the first term \(a_1\) is $1200 (the prize for the first player) and the common difference \(d\) is -$100 (because the prize is decreasing by $100 for each position after the first one). This sequence is represented by \(a_n = a_1 + (n - 1) * d\) where \(n\) is the position of the player.

Step 2: Find out the last term

The last term of the sequence is the prize for the twelveth player. Use the arithmetic sequence formula \(a_n = a_1 + (n - 1) * d\) to calculate. Substitute \(a_1 = 1200\), \(d = -100\), and \(n = 12\) into the formula, resulting in \(a_{12} = 1200 + (12 - 1) * -100\). Calculate to find \(a_{12} = $300\).

Step 3: Calculate the total prize

To find the total prize, we should calculate the sum of the arithmetic sequence. The sum \(S\) of an arithmetic sequence with \(n\) terms can be given by the formula \(S_n = n/2 * (a_1 + a_n)\). Substitute \(n = 12\), \(a_1 = 1200\), and \(a_n = 300\) into the formula, gives \(S_{12} = 12/2 * (1200 + 300)\). Calculate to find \(S_{12} = 9000\) dollars, which is the total prize for the tournament.

Key Concepts

Prize Distribution

In many competitions and tournaments, the allocation of rewards or prizes often follows a structured sequence. For our bowling league tournament, the prize distribution follows an arithmetic progression. The purpose of using such a sequence is to create a fair and predictable method of rewarding participants based on their performance.

• The first-place winner garners the highest prize, which in this case is \( \\(1200 \).
• Second place receives \( \\)1100 \), and each subsequent position receives \( \\(100 \) less until the twelfth place.
• The sequence of cash prizes is \( 1200, 1100, 1000, \ldots, 300 \).

The consistency in prize decrement ensures that the prizes reduce evenly for every additional position downwards. This arithmetic sequence is defined by its first term, common difference, and the number of terms. The first term \(a_1\) is the prize for the first-place finisher, and each subsequent prize is found by deducting \( \\)100 \), which is the common difference \(d\). The sequence can be expressed as: \[ a_n = a_1 + (n - 1) \times d \]where \(a_n\) represents the prize for the \(n\)-th position.

Tournament Scoring

In any competitive setting, particularly in tournaments like bowling, scoring systems are fundamental in determining outcomes. Scores are pivotal in ranking players and ultimately, in prize distribution based on those rankings.

• The bowling tournament mentioned uses three-game totals to rank the bowlers.
• The rank determines the position in which a player finishes, impacting the prize they receive.
• Understanding the score sequences allows players to gauge their standings effectively.

A precise score calculation ensures fair positioning and transparency, two important pillars in sports and competitions. In our scenario, the prize money awarded is intrinsically tied to the scores. Hence, this method of using scores not only provides a basis for deriving the player rankings but is also crucial in calculating how prize distribution is handled.

Mathematical Modeling

Mathematical modeling plays a pivotal role in simplifying complex ideas into understandable formats through the use of mathematical concepts. In the context of our bowling tournament, modeling the prize decreases as a simple arithmetic sequence allows us to represent the problem smoothly and solve it using formulas.

• This modeling simplifies the distribution of prizes by assuming a constant decrement.
• Equations like \(a_n = a_1 + (n - 1) \times d\) allow calculation of any specific prize.
• For overall prize calculation, utilizing the sum of the arithmetic sequence formula \(S_n = \frac{n}{2} (a_1 + a_n)\) provides efficient summation.

This approach not only streamlines the prize distribution process but also aids in checking the correctness of outcomes. Those familiar with using mathematical models in real-life scenarios understand that they bring clarity and precision, which is evidenced by the effective calculation of the tournament's total prize distribution.