This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) Step 1: Identify the radius of the quarter-circle. From the diagram, the radius of the quarter-circle is equal to the width of the rectangle, which is 7 \, cm. So, r = 7 \, cm. Step 2: Use the formula for the area of a quarter-circle. The area of a full circle is A = r^2. For a quarter-circle, the area is (1)/(4) r^2. Given = (22)/(7). Step 3: Substitute the values and calculate the area. Area of quarter-circle = (1)/(4) × (22)/(7) × (7 \, cm)^2 Area of quarter-circle = (1)/(4) × (22)/(7) × 49 \, cm^2 Area of quarter-circle = (1)/(4) × 22 × 7 \, cm^2 Area of quarter-circle = (154)/(4) \, cm^2 Area of quarter-circle = 38.5 \, cm^2 The area of the quarter-circle is indeed 38.5 cm^2. b) Step 1: Calculate the area of the rectangle. The length of the rectangle is 12 \, cm and the width is 7 \, cm. Area of rectangle = length × width Area of rectangle = 12 \, cm × 7 \, cm Area of rectangle = 84 \, cm^2 Step 2: Calculate the area of the shaded part. The shaded part is the area of the rectangle minus the area of the quarter-circle. Area of shaded part = Area of rectangle - Area of quarter-circle Area of shaded part = 84 \, cm^2 - 38.5 \, cm^2 Area of shaded part = 45.5 \, cm^2 The area of the shaded part is 45.5 cm^2. 3 done, 2 left today. You're making progress.