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?

Rationale

x² - 24x - 4,081 = 0.

To find the dimensions of the painting, we can set up an equation based on the relationship between the length and width, where the width is represented as (x - 24) and the area is the product of length and width, giving us the equation x(x - 24) = 4,081. Rearranging this leads to the quadratic equation x² - 24x - 4,081 = 0.

A (x² - 24x - 4,081 = 0) CORRECT

This equation accurately represents the relationship between the length (x) and width (x - 24) of the painting. The area is calculated as length times width, which simplifies to x² - 24x - 4,081 = 0, making it the correct choice for finding the dimensions.

B (x² + 24x - 4,081 = 0) YOUR ANSWER

This equation incorrectly adds 24x instead of subtracting it. The width must be shorter than the length, making it necessary to subtract 24 from x. Thus, this formulation does not represent the dimensions correctly.

C (x² + 24x + 4,081 = 0) YOUR ANSWER

This equation incorrectly includes a positive sign for both the 24x and 4,081 terms. Since the area of the painting is a product that results in a negative when rearranging, this formulation does not correspond to the situation described.

D (x² - 24x + 4,081 = 0) YOUR ANSWER

While this equation correctly subtracts 24x, the inclusion of a positive 4,081 is erroneous. The area of the painting must result in a negative constant when rearranging, thus making this equation incorrect.

Conclusion

To determine the dimensions of the painting, the equation x² - 24x - 4,081 = 0 accurately reflects the relationship between length and width based on the area. The other options misrepresent the mathematical relationships necessary to find the correct dimensions, reinforcing the understanding of how area is derived from the length and width of geometric figures.

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?

x² - 24x - 4,081 = 0.

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