- The area of a rectangle is \(A = l \times w\), so the width \(w\) is obtained by dividing the area \(A\) by the length \(l\), i.e., \(w = \frac{120x^2 + 78x - 90}{12x + 15}\).
- Testing the given options, we multiply \((12x + 15)(10x - 6)\) and verify it equals the area polynomial.
- Expanding the product \((12x + 15)(10x - 6)\) yields \(120x^2 - 72x + 150x - 90 = 120x^2 + 78x - 90\), which matches the original area.
- Thus, the width of the rectangle is \(\boxed{10x - 6}\).
Explanation
- Analyze the Problem
The problem asks for the width \(w\) of a rectangle given its area \(A = 120x^2 + 78x - 90\) and its length \(l = 12x + 15\). We know that the area of a rectangle is the product of its length and width, represented by the formula \(A = l \times w\). To find the width, we must solve for \(w\), which gives \(w = \frac{A}{l}\). Thus, we need to divide the polynomial \(120x^2 + 78x - 90\) by \(12x + 15\).
- Calculate and Verify the Width
To find the width \(w\), we divide the polynomial \(120x^2 + 78x - 90\) by \(12x + 15\). We can test the given options by multiplying each option by the length \(l = 12x + 15\) to see which one results in the given area \(A\).
Let's test option (b): \(w = 10x - 6\).
Calculating the product:
\[(12x + 15)(10x - 6) = 12x(10x) + 12x(-6) + 15(10x) + 15(-6)\]
\[= 120x^2 - 72x + 150x - 90\]
\[= 120x^2 + 78x - 90\]
This matches the given area \(A\). Thus, the width \(w\) must be \(10x - 6\).
- Conclusion
By dividing the area polynomial \(A = 120x^2 + 78x - 90\) by the length polynomial \(l = 12x + 15\), we found the width \(w\) to be \(10x - 6\). Comparing this result with the provided options, it corresponds to choice (b). Therefore, the correct expression for the width is \(\boxed{10x - 6}\).
Examples
In real-world applications, such as construction or interior design, one often needs to determine dimensions based on area requirements. If a room has a fixed length defined by a variable expression (perhaps due to the placement of doors or partitions) and a required total area, this method of dividing polynomials allows for the calculation of the necessary width to ensure the area constraint is satisfied. Understanding how to divide these expressions helps in optimizing layouts and space allocation efficiently.