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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Here's the solution to question 14: Given: Height of view tower AB = 45 m Distance from tower to house BD = 15 m Angle of depression from the top of the tower (A) to the roof of the house (C) = 60° Let's denote the height of the house as CD. Draw a horizontal line CE from the roof of the house (C) to the tower (AB), such that E is on AB. Then, BCDE forms a rectangle. Therefore, the horizontal distance EC = BD = 15 m. The height of the house CD = BE. a) Find the height of the house (CD). Step 1: Identify the relevant triangle for the angle of depression. The angle of depression from A to C is 60°. This means that if we draw a horizontal line from A, the angle between this line and AC is 60°. Due to alternate interior angles, ACE = 60^. Step 2: Use trigonometry in AEC. In the right-angled triangle AEC: ( ACE) = OppositeAdjacent = (AE)/(EC) We know ACE = 60^ and EC = 15 m. (60^) = (AE)/(15) sqrt(3) = (AE)/(15) AE = 15sqrt(3) m Step 3: Calculate the height of the house (CD). The total height of the tower AB = AE + EB. We know AB = 45 m and AE = 15sqrt(3) m. 45 = 15sqrt(3) + EB EB = 45 - 15sqrt(3) m Since CD = EB, the height of the house is: CD = 45 - 15sqrt(3) m Using sqrt(3) ≈ 1.732: CD ≈ 45 - 15(1.732) CD ≈ 45 - 25.98 CD ≈ 19.02 m The height of the house is 45 - 15sqrt(3) m or approximately 19.02 m. b) Find the angle of elevation from the base of the house (D) to the top of the tower (A). Step 1: Identify the relevant triangle for the angle of elevation. We need to find ADB. Consider the right-angled triangle ABD. Step 2: Use trigonometry in ABD. In ABD: ( ADB) = OppositeAdjacent = (AB)/(BD) We know AB = 45 m and BD = 15 m. ( ADB) = (45)/(15) ( ADB) = 3 ADB = (3) Using a calculator, (3) ≈ 71.565^. The angle of elevation from the base of the house to the top of the tower is (3) or approximately 71.57^.