Problem 65 - Precalculus Mathematics for Calculus

Question

An architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of \(870,\) how many rows must the architect use in his design?

Step-by-Step Solution

Answer

The architect must design 20 rows for the theater.

Step 1: Identify the Pattern

Look at the number of seats in each row to find a pattern. The seats in the rows form an arithmetic sequence: 15, 18, 21, ... where the first term, \(a_1\), is 15 and the common difference, \(d\), is 3.

Step 2: Set Up the Formula

The formula for the sum of the first \(n\) terms of an arithmetic sequence is: \(S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d)\). We know \(S_n = 870\), \(a_1 = 15\), and \(d = 3\). Substitute these values into the formula: \(870 = \frac{n}{2} \cdot (30 + (n-1) \times 3)\).

Step 3: Simplify the Expression

Simplify the expression obtained: \(870 = \frac{n}{2} \cdot (30 + 3n - 3)\). This simplifies further to \(870 = \frac{n}{2} (3n + 27)\). Multiply both sides by 2 to eliminate the fraction: \(1740 = n(3n + 27)\).

Step 4: Solve the Quadratic Equation

Reorganize the equation: \(3n^2 + 27n - 1740 = 0\). Use the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 27\), \(c = -1740\). Calculate the discriminant \(b^2 - 4ac = 27^2 - 4 \times 3 \times (-1740)\).

Step 5: Calculate the Discriminant

Compute the discriminant: \(729 + 20880 = 21609\). Since the discriminant is a perfect square, we find \(\sqrt{21609} = 147\).

Step 6: Apply the Quadratic Formula

Substitute into the quadratic formula: \(n = \frac{-27 \pm 147}{6}\). Calculate the two solutions: \(n = 20\) (as \(n = -29\) is not possible for the number of rows).

Step 7: Confirm the Answer

Verify that using 20 rows gives a total of 870 seats by calculating the sum again using the arithmetic sequence sum formula.

Key Concepts

Sum of Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
To find the sum of an arithmetic series, we use the formula for the sum of the first \(n\) terms:

• \( S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d) \)

In this formula:

• \( S_n \) is the sum of the first \(n\) terms.
• \( n \) is the number of terms.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference.

This formula helps calculate the total sum of all seats in the theater, given as 870. Understanding how to apply the formula can solve many practical problems involving sums of arithmetic sequences.

Quadratic Equation

Quadratic equations are essential in algebra and they come in the form \( ax^2 + bx + c = 0 \). Solving a quadratic equation requires finding the values of \( x \) that satisfy this equation.
In the theater seating problem, the expression simplifies to a quadratic form: \( 3n^2 + 27n - 1740 = 0 \).
To solve this, use the quadratic formula:

• \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here:

• \( a = 3 \),
• \( b = 27 \),
• \( c = -1740 \).

The solution requires calculating the discriminant \( b^2 - 4ac \), which helps determine the nature of the roots. A positive discriminant, as seen here, indicates the presence of real and distinct solutions.

Common Difference

The common difference in an arithmetic sequence is the constant gap between consecutive terms. It dictates how the sequence progresses.
For the theater seating plan:

• The first term \( a_1 \) is 15 (seats in the first row).
• The common difference \( d \) is found as 3, since each subsequent row increases by three seats (18 – 15 = 3).

Understanding the common difference is crucial as it directly impacts the calculation of the number of terms \( n \) needed to reach the seating capacity. It can help solve other problems with sequences by setting the pace of growth or decline in the series.

Arithmetic Series Formula

The arithmetic series formula finds vast applications beyond just calculating seats in a theater. It is vital for adding up any series of numbers that change consistently.This formula:

• \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \)

allows quick summation of the series when you know:

• \( a_1 \) as the starting point,
• \( d \) as the common difference,
• \( n \) the number of terms.

With the arithmetic series formula, real-world problems like determining how many rows of seats fit a set capacity become manageable. This formula provides a structured way to sum a sequence efficiently and is a key tool in many branches of science and engineering.