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
Let the height of the taller pole be H_T = 8 m and its top be T. Let its foot be A. Let the height of the smaller pole be H_R = 7 m and its top be R. Let its foot be C. Point P is on the ground. The smaller pole is between the taller pole and point P. The distance from point P to the smaller pole is PC = 20 m. We can visualize this with a diagram: Let P be the origin (0,0) on the ground. The foot of the smaller pole C is at (20,0). The top of the smaller pole R is at (20,7). Let the horizontal distance between the feet of the poles be AC = x. The foot of the taller pole A is at (20+x, 0). The top of the taller pole T is at (20+x, 8). Now, consider the triangle formed by R, T, and a point E directly below T on the horizontal line passing through R. The coordinates of R are (20,7). The coordinates of T are (20+x, 8). The coordinates of E are (20+x, 7). This forms a right-angled triangle RET with the right angle at E. The horizontal distance RE = (20+x) - 20 = x. The vertical distance ET = 8 - 7 = 1 m. The angle of elevation of T from R is TRE = 30^. a) The distance RT. Step 1: Use the sine function in RET. ( TRE) = (ET)/(RT) (30^) = (1)/(RT) Step 2: Solve for RT. RT = (1)/((30^)) RT = (1)/(0.5) RT = 2 m The distance RT is 2 m. b) The distance of the foot of the taller pole from P (PA). Step 1: Find the horizontal distance x (which is AC) using the tangent function in RET. ( TRE) = (ET)/(RE) (30^) = (1)/(x) Step 2: Solve for x. x = (1)/((30^)) x = (1)/(1/sqrt(3)) x = sqrt(3) m Step 3: Calculate the distance PA. The distance PA = PC + AC = 20 + x. PA = 20 + sqrt(3) m PA ≈ 20 + 1.7320508 PA ≈ 21.7320508 m Step 4: Round to 3 significant figures. PA ≈ 21.7 m The distance PA is 21.7 m. c) The angle of elevation of T from P, correct to one decimal point. Step 1: Consider the right-angled triangle PAT. The height of the taller pole is AT = 8 m. The distance PA = 20 + sqrt(3) m. Step 2: Use the tangent function to find TPA. ( TPA) = (AT)/(PA) ( TPA) = (8)/(20 + sqrt(3)) ( TPA) ≈ (8)/(21.7320508) ( TPA) ≈