6. 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?
Answer: B
The equation that can be used to find x, the width, in feet, of the main room is 2x² - 10x - 300 = 0.
To find the width of the main room, the equation must represent the area of the room accurately. Since the length of the main room is twice the width minus the reserved space, the equation that incorporates these dimensions correctly is 2x² - 10x - 300 = 0.
A) 2x² + 10x - 300 = 0
This equation incorrectly adds a positive term instead of subtracting the reserved length for the kitchen and bathroom. Therefore, it does not accurately reflect the relationship between the length, width, and the area of the main room.
B) 2x² - 10x - 300 = 0
This equation is correct as it represents the area of the main room. By setting the length of the main room as 2x (twice the width) and subtracting the reserved 10 ft for the kitchen and bathroom, it leads to the correct formulation of the area being 300 square feet.
C) 2x² - 20x - 300 = 0
This equation incorrectly uses -20x for the linear term, which does not correspond to the actual dimensions of the room. The length should be represented as 2x - 10, not as -20x, making this equation inaccurate.
D) 2x² + 20x - 300 = 0
This equation is incorrect as it adds a positive linear term, which does not reflect the reserved space for the kitchen and bathroom. The correct relationship should involve a subtraction, not an addition, leading to an inaccurate representation of the area.
Conclusion
The correct equation, 2x² - 10x - 300 = 0, accurately represents the area relationship of the cabin's main room by correctly accounting for the reserved space. All other options fail to reflect the correct dimensions and relationships, leading to incorrect formulations that do not yield the required area of 300 square feet.