Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?

Rationale

2x² - 10x - 300 = 0

To find the width \( x \) of the main room, we can set up the equation for the area based on the given dimensions. The length of the main room is \( 2x - 10 \) (since 10 ft is reserved for the kitchen and bathroom), and the area is given as 300 square feet, leading to the equation \( 2x(x - 10) = 300 \), which simplifies to \( 2x² - 10x - 300 = 0 \).

A (2x² + 10x - 300 = 0) YOUR ANSWER

This equation incorrectly adds \( 10x \) instead of subtracting it, which does not represent the relationship between the width and the reserved area for the kitchen and bathroom. The correct formulation involves a subtraction, not addition, of the reserved space.

B (2x² - 10x - 300 = 0) CORRECT

This is the correct equation, derived from the area of the main room being \( 2x(x - 10) = 300 \). It accurately reflects the relationship between the dimensions of the main room, accounting for the length being twice the width and the reserved space.

C (2x² - 20x - 300 = 0) YOUR ANSWER

This equation incorrectly uses \( 20x \) instead of \( 10x \). The length of the main room should be \( 2x - 10 \), not \( 2x - 20 \), which misrepresents the problem's conditions regarding the reserved area.

D (2x² + 20x - 300 = 0) YOUR ANSWER

This choice incorrectly adds \( 20x \), which misrepresents the dimensions of the room. The addition of \( 20x \) does not align with the given parameters, as the length is supposed to be \( 2x - 10 \).

Conclusion

To solve for the width \( x \) of Tina's main room, the equation \( 2x² - 10x - 300 = 0 \) accurately represents the geometric relationship based on the problem's constraints. The other options fail to adhere to the conditions specified, either through incorrect arithmetic or misinterpretation of the dimensions, underscoring the importance of precise formulation in geometric problems.

2x² - 10x - 300 = 0

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