Let's analyze the given problem step-by-step to determine what the constant term represents.
We are given that the total area of the gym and the weight room is expressed by the equation:
[tex]\[ 4x^2 + 480 \, \text{ft}^2 \][/tex]
Here, [tex]\(4x^2\)[/tex] and 480 are terms in the expression. Our goal is to understand what the constant term 480 represents.
1. Identify the constant term:
- The constant term in the given expression is 480. This is a fixed number that does not vary with [tex]\(x\)[/tex].
2. Interpret the meaning of the constant term:
- Since the overall expression [tex]\(4x^2 + 480 \, \text{ft}^2\)[/tex] represents the total area of both the gym and the weight room, each component of the expression must relate to areas.
3. Analyze the choices given:
- A. The gym has an area of 480 ft².
- This choice implies that the fixed area of 480 ft² is associated with the gym. However, we cannot directly infer that from the expression considering [tex]\(4x^2\)[/tex] also contributes to the total area.
- B. The new weight room can hold 480 people.
- This choice describes capacity in terms of people, not area. Since the expression specifically deals with area, this can't be the correct interpretation.
- C. The new weight room has an area of [tex]\(4x^2\)[/tex] ft².
- This choice is describing the variable term [tex]\(4x^2\)[/tex] rather than the constant term. We need to focus on what the constant term, specifically the 480, represents.
- D. The new weight room has an area of 480 ft².
- This implies that the constant term 480 describes the area of the new weight room. Given that 480 is a static part of the total area expression [tex]\(4x^2 + 480\)[/tex], it makes sense to pair this component with the fixed space of the new weight room.
By careful logical analysis, the correct interpretation is:
D. The new weight room has an area of 480 ft².
