Problem 61 - Algebra and Trigonometry

Question

Theater Seating 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

The sequence of seat numbers in each row forms an arithmetic sequence: 15, 18, 21, ... This sequence has a first term \(a_1 = 15\) and a common difference \(d = 3\).

Step 2: Arithmetic Sequence Formula

The number of seats \(a_n\) in the \(n\)-th row is given by the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n - 1) \times d\). Plugging in the values, we get: \(a_n = 15 + (n - 1) \times 3\).

Step 3: Sum Formula for Arithmetic Series

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by: \(S_n = \frac{n}{2} (a_1 + a_n)\). The goal is to find \(n\) such that \(S_n = 870\).

Step 4: Express \(a_n\) in Terms of \(n\)

Substitute \(a_n = 15 + (n - 1) \times 3\) into the sum formula: \(S_n = \frac{n}{2} (15 + 15 + 3(n - 1))\).

Step 5: Simplify and Solve for \(n\)

Simplifying, we get: \(S_n = \frac{n}{2} \times (30 + 3n - 3) = \frac{n}{2} \times (3n + 27)\). Set this equal to 870: \( \frac{n}{2} \times (3n + 27) = 870\).

Step 6: Clear the Fraction

Multiply both sides by 2 to eliminate the fraction: \(n(3n + 27) = 1740\).

Step 7: Expand and Rearrange

Expand the equation: \(3n^2 + 27n = 1740\) and then rearrange to form a quadratic equation: \(3n^2 + 27n - 1740 = 0\).

Step 8: Solve the Quadratic Equation

Divide the whole equation by 3 to simplify: \(n^2 + 9n - 580 = 0\). Solve this quadratic using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 9\), \(c = -580\).

Step 9: Calculate Discriminant

Calculate the discriminant \(b^2 - 4ac = 9^2 - 4 \times 1 \times (-580) = 81 + 2320 = 2401\). Since the discriminant is positive, there are two real solutions.

Step 10: Find Possible Values for \(n\)

Calculate \(n\) using the quadratic formula: \(n = \frac{-9 \pm \,\sqrt{2401}}{2}\). \(\sqrt{2401} = 49\), so the solutions are \(n = \frac{-9 + 49}{2} = 20\) and \(n = \frac{-9 - 49}{2} = -29\).

Step 11: Select the Valid Solution

Since \(n\) must be a positive integer, the number of rows is \(n = 20\).

Step 12: Verification

Substitute \(n=20\) back into the sum formula to verify: \(S_{20} = \frac{20}{2}(15 + 15 + 3(20-1)) = 10(30 + 57) = 870\). This confirms the solution is correct.

Key Concepts

arithmetic series

An arithmetic series is the sum of terms in an arithmetic sequence. An arithmetic sequence is a list of numbers with a constant difference between every two successive terms. For instance, in our theater problem, the sequence of seats is 15, 18, 21, and so on, with each term increasing by 3. The total sum of these terms, when they add up to the theater's seating capacity, forms an arithmetic series.

To find out how many rows the theater has, you need to determine the sum of the series up to the row where the total number of seats equals 870. This is where the sum formula comes into play. Understanding the arithmetic series is crucial because it helps in calculating the total using a straightforward formula, saving time from calculating each term separately.

quadratic equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown variable. In the context of our problem, the goal was to determine how many rows were needed to reach the seating capacity of 870.

By rearranging our derived formula for the sum of the series, we eventually ended up with a quadratic equation: \( 3n^2 + 27n - 1740 = 0 \). To solve this, it was simplified to \( n^2 + 9n - 580 = 0 \). Solving a quadratic equation is essential when direct calculation isn't feasible, especially when dealing with large numbers or more complex sequences.

sum of arithmetic series

The sum of an arithmetic series can be calculated using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term of the series.

In solving our theater problem, we employed this formula to find \( n \), or the number of rows needed, by substituting known values. We expressed the last term (\( a_n \)) using the sequence formula \( a_n = a_1 + (n - 1) imes d \), where \( d \) is the common difference, lifting the task into a manageable calculation that converts into solving a manageable quadratic equation.

• Set up the summed equation like \( S_n = 870 \) with the expression for \( a_n \).
• Worked through it using known algebraic principles to solve the resulting quadratic.

Through this approach, finding the sum of arithmetic series becomes methodical, reducing complex summation into simple algebraic manipulations.