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: Define variables and set up trigonometric equations. Let H be the total height of the pylon. Let h be the height of the pylon above the theodolite's eye level. Let x be the initial horizontal distance from the theodolite to the foot of the pylon. The height of the theodolite is 1.5m. Therefore, H = h + 1.5. From the first position, the angle of elevation is 30^: (30^) = (h)/(x) h = x (30^) (1) The surveyor moves 60m nearer, so the new distance is x - 60. The new angle of elevation is 42^: (42^) = (h)/(x - 60) h = (x - 60) (42^) (2) Step 2: Solve the system of equations for x. Equate the expressions for h from equations (1) and (2): x (30^) = (x - 60) (42^) x (30^) = x (42^) - 60 (42^) Rearrange the terms to solve for x: 60 (42^) = x (42^) - x (30^) 60 (42^) = x ((42^) - (30^)) x = (60 (42^))/((42^) - (30^)) Step 3: Calculate the numerical value of x. Using a calculator: (30^) ≈ 0.57735 (42^) ≈ 0.90040 x = (60 × 0.90040)/(0.90040 - 0.57735) x = (54.024)/(0.32305) x ≈ 167.23m Step 4: Calculate the height h. Substitute the value of x back into equation (1): h = x (30^) h = 167.23 × 0.57735 h ≈ 96.52m Step 5: Calculate the total height of the pylon. The total height of the pylon H is the height h plus the height of the theodolite (1.5m): H = h + 1.5m H = 96.52m + 1.5m H = 98.02m The height of the pylon is 98.02 m. Send me the next one 📸