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.
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Let's knock this out. Problem 3: To calculate the inner surface area of the dust pan that was painted, we need to find the area of the two triangular faces, the bottom rectangular face, the vertical rectangular face, and the slanted rectangular face. The top opening is not painted. Step 1: Calculate the area of the two triangular faces. The base of the triangle is 4 cm and the height is 3 cm. A_triangles = 2 × (1)/(2) × base × height = 2 × (1)/(2) × 4 cm × 3 cm = 12 cm^2 Step 2: Calculate the area of the bottom rectangular face. The dimensions are 4 cm by 7 cm. A_bottom = length × width = 4 cm × 7 cm = 28 cm^2 Step 3: Calculate the area of the vertical rectangular face. The dimensions are 3 cm by 7 cm. A_vertical = height × length = 3 cm × 7 cm = 21 cm^2 Step 4: Calculate the area of the slanted rectangular face. The dimensions are 5 cm (hypotenuse of the triangle) by 7 cm. A_slanted = hypotenuse × length = 5 cm × 7 cm = 35 cm^2 Step 5: Calculate the total inner surface area. A_total = A_triangles + A_bottom + A_vertical + A_slanted A_total = 12 cm^2 + 28 cm^2 + 21 cm^2 + 35 cm^2 = 96 cm^2 The surface area of the dust pan that was painted is 96 cm^2. Problem 4: To calculate the external surface area of the wedge, we need to find the area of all its faces: two triangular faces, the bottom rectangular face, the vertical rectangular face, and the slanted rectangular face. Step 1: Calculate the area of the two triangular faces. The base of the triangle is 12 cm and the height is 5 cm. A_triangles = 2 × (1)/(2) × base × height = 2 × (1)/(2) × 12 cm × 5 cm = 60 cm^2 Step 2: Calculate the area of the bottom rectangular face. The dimensions are 12 cm by 20 cm. A_bottom = length × width = 12 cm × 20 cm = 240 cm^2 Step 3: Calculate the area of the vertical rectangular face. The dimensions are 5 cm by 20 cm. A_vertical = height × length = 5 cm × 20 cm = 100 cm^2 Step 4: Calculate the area of the slanted rectangular face. The dimensions are 13 cm (hypotenuse of the triangle) by 20 cm. A_slanted = hypotenuse × length = 13 cm × 20 cm = 260 cm^2 Step 5: Calculate the total external surface area. A_total = A_triangles + A_bottom + A_vertical + A_slanted A_total = 60 cm^2 + 240 cm^2 + 100 cm^2 + 260 cm^2 = 660 cm^2 The external surface area of the wedge is 660 cm^2. Problem 5: To calculate the surface area of the canvas used for the tent, we need to find the area of the two triangular faces and the two slanted rectangular faces. The base of the tent (on the ground) is not made of canvas. Step 1: Determine the height of the triangular face. The base of the triangle is 1.8 m, so half the base is 0.9 m. The slanting edge is 1.5 m. Using the Pythagorean theorem: height^2 + (0.9 m)^2 = (1.5 m)^2 height^2 + 0.81 m^2 = 2.25 m^2 height^2 = 2.25 m^2 - 0.81 m^2 = 1.44 m^2 height = sqrt(1.44 m)^2 = 1.2 m Step 2: Calculate the area of the two triangular faces. A_triangles = 2 × (1)/(2) × base × height = 2 × (1)/(2) × 1.8 m × 1.2 m = 2.16 m^2 Step 3: Calculate the area of the two slanted rectangular faces. The dimensions of each slanted face are the slanting edge (1.5 m) by the length of the tent (3 m). A_slanted faces = 2 × (slanting edge × length) A_slanted faces = 2 × (1.5 m × 3 m) = 2 × 4.5 m^2 = 9 m^2 Step 4: Calculate the total surface area of the canvas used. A_total canvas = A_triangles + A_slanted faces A_total canvas = 2.16 m^2 + 9 m^2 = 11.16 m^2 The surface area of the canvas used is 11.16 m^2. Send me the next one 📸