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
Step 1: Identify the given lengths. The lengths of the parallel bases are 10 m (top) and 20 m (bottom). The height is h = 4 m. One base angle is = 45^ (between the longer base and one leg). Step 2: Find the difference in base lengths. Difference = 20\,m - 10\,m = 10\,m This difference equals the sum of the horizontal projections of the two legs: p_1 + p_2 = 10 m. Step 3: Calculate the left projection p_1 (for the 45^ angle). Formula: = (h)/(p_1) p_1 = (h)/( 45^) = 4\,m1 = 4\,m Step 4: Calculate the right projection p_2. p_2 = 10\,m - 4\,m = 6\,m Step 5: Calculate the left leg length l_1. Formula: l_1 = sqrt(h^2 + p_1^2) l_1 = sqrt((4\,m))^2 + (4\,m)^2 = sqrt(16\,m)^2 + 16\,m^2 = sqrt(32\,m)^2 = sqrt(16 · 2)\,m = 4sqrt(2)\,m Step 6: Calculate the right leg length l_2. Formula: l_2 = sqrt(h^2 + p_2^2) l_2 = sqrt((4\,m))^2 + (6\,m)^2 = sqrt(16\,m)^2 + 36\,m^2 = sqrt(52\,m)^2 = sqrt(4 · 13)\,m = 2sqrt(13)\,m Step 7: Calculate the perimeter P. Formula: P = 10\,m + 20\,m + l_1 + l_2 P = 10\,m + 20\,m + 4sqrt(2)\,m + 2sqrt(13)\,m = 30 + 4sqrt(2) + 2sqrt(13)\,m **Final answer: 30 + 4sqrt(2) + 2sqrt(13) meters