Three rectangles share a common width. The first rectangle has area 25, the third has area 32. The combined lengths of the first and second is 8, and the second and third is 9. Find the width.

Step 1: Define variables for the dimensions of the rectangles. Let w be the common width of all three rectangles. Let l_1 be the length of the first (leftmost) rectangle. Let l_2 be the length of the second (shaded) rectangle. Let l_3 be the length of the third (rightmost) rectangle. Step 2: Formulate equations based on the given information. The area of the first rectangle is 25 cm^2: l_1 × w = 25 (1) The area of the third rectangle is 32 cm^2: l_3 × w = 32 (2) The combined length of the first and second rectangles is 8 cm: l_1 + l_2 = 8 (3) The combined length of the second and third rectangles is 9 cm: l_2 + l_3 = 9 (4) Step 3: Express l_1 and l_3 in terms of w using equations (1) and (2). From (1): l_1 = (25)/(w) From (2): l_3 = (32)/(w) Step 4: Substitute these expressions into equations (3) and (4). Substitute l_1 into (3): (25)/(w) + l_2 = 8 (5) Substitute l_3 into (4): l_2 + (32)/(w) = 9 (6) Step 5: Solve the system of equations (5) and (6) for l_2 and w. From (5), express l_2: l_2 = 8 - (25)/(w) Substitute this expression for l_2 into (6): (8 - (25)/(w)) + (32)/(w) = 9 8 + (32 - 25)/(w) = 9 8 + (7)/(w) = 9 Subtract 8 from both sides: (7)/(w) = 9 - 8 (7)/(w) = 1 Multiply by w: 7 = w So, the common width is w = 7 cm. Step 6: Calculate the length of the shaded rectangle, l_2. Substitute w = 7 into the expression for l_2 from Step 5: l_2 = 8 - (25)/(7) l_2 = (8 × 7 - 25)/(7) l_2 = (56 - 25)/(7) l_2 = (31)/(7) cm Step 7: Calculate the area of the shaded region. The area of the shaded region is l_2 × w: Area = (31)/(7) cm × 7 cm Area = 31 cm^2 The area of the shaded region is 31 cm^2.