Final answer:
The area of the picture is found by solving for the side length of the picture from the given area of the frame. The side length is determined to be 19 inches, resulting in an area of 361 square inches for the picture.
Step-by-step explanation:
To find the area of the picture within a frame when given the area of the frame alone, we must first determine the dimensions of the entire framed area (picture plus frame) and then subtract the area of the frame to find the area of the picture itself.
Let's assume the side length of the picture is x inches. Since the frame is 1 inch wide all around the picture, it adds 2 inches to each dimension of the picture (1 inch on each side). Thus, the side length of the framed area is x + 2 inches. The area of the frame is provided as 80 square inches. This area can be found by calculating the area of the larger square (the framed picture) and subtracting the area of the smaller square (the picture itself).
The formula for the total area of the framed picture is (x + 2)^2, and the area of just the picture is x^2. The frame's area is the difference between the two: (x + 2)^2 - x^2 = 80.
To solve for x, we expand the equation: x^2 + 4x + 4 - x^2 = 80.
Cancel out x^2 and we get 4x + 4 = 80. When we solve for x, we subtract 4 from both sides and get 4x = 76, then divide both sides by 4 to get x = 19.
Therefore, the side length of the picture is 19 inches, and the area of the picture is 19 inches × 19 inches = 361 square inches.