Solving Real-World Math Problems: Balcony Seating Arrangements

Mathematics is not just a theoretical discipline; it's a powerful tool for solving real-world problems. This article will tackle a practical problem involving the arrangement of seats in a theater's balcony, utilizing algebraic equations to find the solution.

Problem Statement

The balcony of a theater has a total of 240 seats. The number of seats in each row is 14 more than the number of rows. Our task is to determine the number of rows in the balcony.

Mathematical Formulation

To solve this problem, we can use algebraic equations. Let's denote the number of rows as R.

Given:

Total number of seats 240 Number of seats in each row R 14

The total number of seats can be expressed as:

R(R 14) 240

Solving the Equation

Let's simplify and solve the equation step-by-step:

Start with the equation: R(R 14) 240. Expanding the equation, we get: R2 14R - 240 0. This is a quadratic equation in the form of a2 bR c 0, where a 1, b 14, and c -240. To solve the quadratic equation, we use the quadratic formula: R [-b ± sqrt(b2 - 4ac)] / (2a).

Applying the Quadratic Formula

Substitute the values of a, b, and c into the quadratic formula:

R [-14 ± sqrt(196 - 4(1)(-240))] / (2 * 1)

Calculate the discriminant:

sqrt(196 960) sqrt(1156) 34

Substitute back into the formula:

R [-14 ± 34] / 2

Thus, we have two possible values for R:

R (-14 34) / 2 20 / 2 10 R (-14 - 34) / 2 -48 / 2 -24

Interpreting the Results

Since the number of rows cannot be negative, we discard R -24. Therefore, the number of rows is 10.

Now, let's calculate the number of seats in each row:

Seats per row 10 14 24

Verification

To verify, we can multiply the number of rows by the number of seats per row:

10 rows * 24 seats/row 240 seats

Conclusion

The number of rows in the balcony is 10, and each row contains 24 seats. This solution confirms that the problem is correctly solved using algebraic equations.

We hope this detailed explanation has helped you understand how to solve similar real-world problems. If you have any questions or need further assistance, feel free to contact us.