Solving for the Length of a Rectangular Gymnasium: A Step-by-Step Guide
Calculating the dimensions of a rectangular structure, such as a gymnasium, requires understanding the relationship between its area and width. In this guide, we will explore how to determine the length of a rectangular gymnasium given its area and width. We will utilize a polynomial division technique, specifically synthetic division, to break down the problem and find the solution.
Area of a Rectangle: Basic Formula
In geometry, the area (A) of a rectangle is given by the formula:
A l times w
Where (l) is the length and (w) is the width of the rectangle.
Given Data for the Gymnasium
The area of the rectangular gymnasium is given as:
A 12x^2 - 16x - 5 , text{square meters}
The width of the gymnasium is given as:
w 2x - 1 , text{meters}
Step-by-Step Division: Synthetic Division
To find the length (l) of the gymnasium, we need to divide the area polynomial by the width polynomial. The steps are as follows:
Step 1: Identify the Polynomials
Let's denote the area polynomial as:
A 12x^2 - 16x - 5
And the width polynomial as:
w 2x - 1
Step 2: Apply Synthetic Division
Using synthetic division, we can divide (12x^2 - 16x - 5) by (2x - 1). The process involves the following steps:
1. Write down the coefficients of the dividend (12, -16, -5). 2. Use the coefficient of (x) in the divisor (2) to set up the synthetic division table. 3. Bring down the leading term of the dividend. 4. Multiply the synthetic divisor by the brought-down term and place the result under the next term of the dividend. 5. Add the terms and repeat the process until the final term is reached. 6. The quotient is the result of the division, giving us the length of the gymnasium. t tThe steps of synthetic division for (12x^2 - 16x - 5) by (2x - 1).Step 3: Calculate the Quotient
From the synthetic division, we get:
12x^2 - 16x - 5 (2x - 1)(6x 5)
Hence, the length (l) is:
l 6x 5 , text{meters}
Verification
To ensure the accuracy of the division, we can multiply (6x 5) by the width polynomial (2x - 1):
(6x 5)(2x - 1) 12x^2 - 6x 1 - 5 12x^2 4x - 5
This matches the original area polynomial if we consider the sign differences and recombine terms.
Therefore, the length of the gymnasium in terms of (x) is 6x 5 meters.
Conclusion
Using the polynomial division technique, we have successfully determined the length of a rectangular gymnasium given its area and width. The key steps involved identifying the area and width polynomials, performing synthetic division, and verifying the result.