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
To find the surface area of the frustum, we need to calculate the area of its top base, bottom base, and the four trapezoidal lateral faces. Step 1: Determine the side length of the top square base. The frustum is formed from similar pyramids. Let s_1 be the side length of the top base, and s_2 be the side length of the bottom base. From the image, the slant height of the removed small pyramid is L_1 = 41 mm. The slant height of the frustum's lateral face is L_2 = 41 mm. The total slant height of the original large pyramid is L_total = L_1 + L_2 = 41 mm + 41 mm = 82 mm. The side length of the bottom base is s_2 = 36 mm. Due to similarity, the ratio of corresponding lengths is equal: (s_1)/(s_2) = (L_1)/(L_total) (s_1)/(36 mm) = 41 mm82 mm (s_1)/(36 mm) = (1)/(2) s_1 = (1)/(2) × 36 mm = 18 mm Step 2: Calculate the areas of the top and bottom square bases. Area of top base (A_top) = s_1^2 A_top = (18 mm)^2 = 324 mm^2 Area of bottom base (A_bottom) = s_2^2 A_bottom = (36 mm)^2 = 1296 mm^2 Step 3: Calculate the area of one trapezoidal lateral face. Each lateral face is a trapezoid with parallel sides s_1 = 18 mm and s_2 = 36 mm, and height (slant height of the frustum) L_2 = 41 mm. The area of one trapezoid (A_trap) is given by the formula (1)/(2)(a+b)h. A_trap = (1)/(2)(s_1 + s_2)L_2 A_trap = (1)/(2)(18 mm + 36 mm)(41 mm) A_trap = (1)/(2)(54 mm)(41 mm) A_trap = 27 mm × 41 mm = 1107 mm^2 Step 4: Calculate the total area of the four trapezoidal lateral faces. Since there are four identical lateral faces: A_lateral = 4 × A_trap A_lateral = 4 × 1107 mm^2 = 4428 mm^2 Step 5: Calculate the total surface area of the frustum. The total surface area (A_total) is the sum of the areas of the top base, bottom base, and the four lateral faces. A_total = A_top + A_bottom + A_lateral A_total = 324 mm^2 + 1296 mm^2 + 4428 mm^2 A_total = 1620 mm^2 + 4428 mm^2 A_total = 6048 mm^2 The surface area of the frustum is 6048 mm^2. Send me the next one 📸