To model the area of the entire design, including both the framed art and the frame, we need to express the dimensions of the frame in terms of \( x \), the width of the art inside the frame.
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Width of the Frame:
- Since the width of the frame is 10 inches more than the width of the art inside the frame, we can express the width of the frame as: \[ \text{Width of the frame} = x + 10 \]
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Length of the Frame:
- The length of the frame is given as twice the width of the frame. Therefore, the length of the frame can be calculated as: \[ \text{Length of the frame} = 2 \cdot (x + 10) = 2x + 20 \]
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Dimensions of the Entire Design:
- The dimensions of the entire design include both the art and the frame around it.
- The total width of the framed artwork is the width of the art plus the frame on either side (the frame adds 10 inches on each side): \[ \text{Total Width} = x + 10 + 10 = x + 20 \]
- The total length of the framed artwork is the length of the art plus the frame on the top and bottom (the frame adds the double length of the frame): \[ \text{Total Length} = (2x) + (2 \cdot 10) = 2x + 20 \]
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Area of the Entire Design:
- The area \( A \) of the entire design (art and frame) can therefore be expressed as: \[ A = \text{Total Width} \times \text{Total Length} = (x + 20)(2x + 20) \]
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Expanded Area Function:
- We can expand this expression: \[ A = (x + 20)(2x + 20) = x(2x + 20) + 20(2x + 20) \] \[ = 2x^2 + 20x + 40x + 400 \] \[ = 2x^2 + 60x + 400 \]
Thus, the function that models the area of the entire design, frame and art included, is: \[ A(x) = 2x^2 + 60x + 400 \]