Final answer:
The question involves determining the area of a gymnasium as represented by a quadratic expression. References indicate a context of kinematic equations and algebraic solutions. We lack sufficient information to directly answer the student's question, but it involves mathematical representations of area with one dimension potentially being 100 feet.
Step-by-step explanation:
The question appears to involve determining the dimension of an area (represented by a quadratic expression) related to a gymnasium, potentially as part of a physics or algebra problem involving kinematics. The provided references mention rotational kinematics equations and solving for unknown values. However, these references lack enough context to directly answer the student's question about the total area of the new gym. Nonetheless, the expression for the area is quadratic in nature, and the relevant solution method would often involve factoring, completing the square, or using the quadratic formula to solve for potential lengths (w) that would satisfy the area expression.
Given the dimensions of the gymnasium as 140 feet by 100 feet, we can establish that this relates to a rectangular area calculation. Therefore, the total area would be the product of the two lengths, which suggests that the expressions are mathematical representations of that area, likely with the variable w representing one of the dimensions.
In this context, if w were 100 feet, we could calculate the area with one of the provided expressions by substituting 100 for w, although we are not given sufficient information to determine which expression is correct. The references to kinematics and quadratic equations imply that students may need to understand concepts such as acceleration, initial velocity, and angular velocity to approach certain problems.