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
Here are the areas for each shaded region: 1) Step 1: Identify the shapes. The figure is a rectangle with a triangle removed from its left side. The rectangle has a length of 15 ft and a width of 8 ft. The triangle has a base of 8 ft (same as the rectangle's width) and a height of 7 ft. Step 2: Calculate the area of the rectangle. A_rectangle = length × width = 15 ft × 8 ft = 120 ft^2 Step 3: Calculate the area of the unshaded triangle. A_triangle = (1)/(2) × base × height = (1)/(2) × 8 ft × 7 ft = 28 ft^2 Step 4: Subtract the area of the triangle from the area of the rectangle to find the shaded area. A_shaded = A_rectangle - A_triangle = 120 ft^2 - 28 ft^2 = 92 ft^2 The shaded area is 92.00 ft^2. 2) Step 1: Identify the shapes. The figure is a rectangle with a semicircle removed from its left side. The rectangle has a length of 10 m and a width of 8 m. The semicircle has a diameter of 8 m, so its radius is r = (8)/(2) = 4 m. Step 2: Calculate the area of the rectangle. A_rectangle = length × width = 10 m × 8 m = 80 m^2 Step 3: Calculate the area of the unshaded semicircle. The area of a full circle is r^2. The area of a semicircle is (1)/(2) r^2. A_semicircle = (1)/(2) (4 m)^2 = (1)/(2) (16 m^2) = 8 m^2 ≈ 8 × 3.14159 m^2 ≈ 25.1327 m^2 Step 4: Subtract the area of the semicircle from the area of the rectangle to find the shaded area. A_shaded = A_rectangle - A_semicircle = 80 m^2 - 25.1327 m^2 = 54.8673 m^2 Step 5: Round to two decimal places. A_shaded ≈ 54.87 m^2 The shaded area is 54.87 m^2. 3) Step 1: Identify the shapes. The figure is a circle with a square removed from its center. The circle has a diameter of 14 cm, so its radius is r = (14)/(2) = 7 cm. The square has a side length of 10 cm. Step 2: Calculate the area of the circle. A_circle = r^2 = (7 cm)^2 = 49 cm^2 ≈ 49 × 3.14159 cm^2 ≈ 153.938 cm^2 Step 3: Calculate the area of the unshaded square. A_square = side × side = 10 cm × 10 cm = 100 cm^2 Step 4: Subtract the area of the square from the area of the circle to find the shaded area. A_shaded = A_circle - A_square = 153.938 cm^2 - 100 cm^2 = 53.938 cm^2 Step 5: Round to two decimal places. A_shaded ≈ 53.94 cm^2 The shaded area is 53.94 cm^2. 4) Step 1: Identify the shapes. The figure is a rectangle with a circle removed from its center. The rectangle has a length of 12 yd and a width of 7 yd. The circle has a diameter of 4 yd, so its radius is r = (4)/(2) = 2 yd. Step 2: Calculate the area of the rectangle. A_rectangle = length × width = 12 yd × 7 yd = 84 yd^2 Step 3: Calculate the area of the unshaded circle. A_circle = r^2 = (2 yd)^2 = 4 yd^2 ≈ 4 × 3.14159 yd^2 ≈ 12.5664 yd^2 Step 4: Subtract the area of the circle from the area of the rectangle to find the shaded area. A_shaded = A_rectangle - A_circle = 84 yd^2 - 12.5664 yd^2 = 71.4336 yd^2 Step 5: Round to two decimal places. A_shaded ≈ 71.43 yd^2 The shaded area is 71.43 yd^2. 5) Step 1: Identify the shapes. The figure is a large triangle with an unshaded trapezoid removed from its base. The large triangle has a base of 10 in and a height of 13 in. The unshaded trapezoid has parallel bases of 8 in and 10 in, and a height of 5 in. Step 2: Calculate the area of the large triangle. A_large triangle = (1)/(2) × base × height = (1)/(2) × 10 in × 13 in = 65 in^2 Step 3: Calculate the area of the unshaded trapezoid. A_trapezoid = (1)/(2) × (base_1 + base_2) × height = (1)/(2) × (8 in + 10 in) × 5 in = (1)/(2) × 18 in × 5 in = 45 in^2 Step 4: Subtract the area of the trapezoid from the area of the large triangle to find the shaded area. A_shaded = A_large triangle - A_trapezoid = 65 in^2 - 45 in^2 = 20 in^2 The shaded area is 20.00 in^2. 6) Step 1: Identify the shapes. The figure is a large square with a smaller square removed from its center. The large square has a side length of 14 m. The small square has a side length of 8 m. Step 2: Calculate the area of the large square. A_large square = side × side = 14 m × 14 m = 196 m^2 Step 3: Calculate the area of the unshaded small square. A_small square = side × side = 8 m × 8 m = 64 m^2 Step 4: Subtract the area of the small square from the area of the large square to find the shaded area. A_shaded = A_large square - A_small square = 196 m^2 - 64 m^2 = 132 m^2 The shaded area is 132.00 m^2. 3 done, 2 left today. You're making progress.