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

Answer 1

The correct function to model the area of the entire framed art design is f(x) = 2x² + 40x + 100, with the frame's width being x + 10 inches and its length being twice the frame's width.

Step-by-step explanation:

The question asks for a function to model the area of an entire framed art design, where the width of the frame is 10 inches more than the width of the art (x), and the length of the frame is twice as long as the frame's width. To determine the area of the entire design, we need to calculate the dimensions of the frame and then use these to find the area.

First, we calculate the width of the frame as x + 10 inches. Since the length of the frame is twice the width, it would be 2(x + 10). The area of the frame, including the art, can be calculated by multiplying the length by the width, which provides the function: A(x) = (x + 10) × 2(x + 10). Simplifying this expression, we get:

A(x) = 2x2 + 20x + 100, therefore, the correct function that models the area of the entire design is option 2: f(x) = 2x2 + 40x + 100.