27. The width of a painting is 24 centimeters shorter than its length, x. The area of the painting is 4,081 square centimeters. Which equation could be used to find the dimensions of the painting?
Answer: A
x² - 24x - 4,081 = 0
To find the dimensions of the painting, we can set up the equation based on the relationship between the length and width. The width is 24 centimeters shorter than the length, which leads to the equation x² - 24x - 4,081 = 0 when we express the area as length times width.
A) x² - 24x - 4,081 = 0
This equation correctly represents the problem. Here, x is the length of the painting, and the width is x - 24. The area is calculated as length times width: x(x - 24) = 4,081, which simplifies to x² - 24x - 4,081 = 0.
B) x² + 24x - 4,081 = 0
This equation incorrectly adds 24x instead of subtracting it. It suggests that the width is longer than the length, which contradicts the information provided in the question about the width being shorter.
C) x² + 24x + 4,081 = 0
This equation contains both an incorrect sign for the linear term and a positive constant term. The positive terms would not align with the area calculation of a rectangle, where the width is less than the length.
D) x² - 24x + 4,081 = 0
This equation incorrectly includes a positive constant term. The area of the painting should result in a negative constant when set to zero, as per the derived equation from the area formula.
Conclusion
The equation x² - 24x - 4,081 = 0 accurately represents the relationship between the length and width of the painting, confirming it as the correct choice. All other options fail due to incorrect signs or terms that do not reflect the stated dimensions and area of the painting.