Question

For a specific design of framed art, the width of the frame is 10 inches more than the width of the art inside the frame, and the length of the frame is twice as long as the frame's width. If x represents the width of the art inside the frame, what function could be used to model the area of the entire design, frame and art included?

1) f(x) = 2x² + 20x
2) f(x) = 2x² + 40x + 200
3) f(x) = 2x² + 40x + 100
4) f(x) = 2x² + 10x

Answer 1

The area of the entire design, including the art and the frame, is represented by the function f(x) = 2x² + 40x + 200, where x is the width of the art inside the frame.

Step-by-step explanation:

For a specific design of framed art, we need to model the area of the entire design, which includes both the frame and the art inside it. If x represents the width of the art inside the frame, then the width of the frame is x + 10 (since the frame's width is 10 inches more than the width of the art). The length of the frame is given as twice the frame's width, which would be 2(x + 10).

To find the area of the entire design, we multiply the width of the framed art by its length, which is (x + 10)(2(x + 10)). This simplifies to f(x) = 2x² + 40x + 200. Therefore, the correct function to model the area of the entire design is option 2).