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 are the solutions to the geometry problems. Question 3.1 3.1.1 Calculate the length of DC. Step 1: Identify the relevant right-angled triangle. In BCD, we have a right angle at C. We are given BC = 12 m and CBD = 44.3^. We need to find the length of DC. Step 2: Use the tangent function. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. ( CBD) = (DC)/(BC) Step 3: Substitute the known values and solve for DC. (44.3^) = (DC)/(12) DC = 12 × (44.3^) DC ≈ 12 × 0.97570 DC ≈ 11.7084 m The length of DC is 11.71 m. 3.1.2 Calculate through how many degrees the spotlight must swing in order to illuminate point A (i.e. calculate ). Step 1: Find the length of AC. Since D is the midpoint of AC, AC = 2 × DC. AC = 2 × 11.7084 m = 23.4168 m Step 2: Find the angle CBA in ABC. In ABC, we have a right angle at C. We know BC = 12 m and AC = 23.4168 m. ( CBA) = (AC)/(BC) ( CBA) = (23.4168)/(12) ( CBA) ≈ 1.9514 CBA = (1.9514) ≈ 62.87^ Step 3: Calculate . The angle is the difference between CBA and CBD. = CBA - CBD = 62.87^ - 44.3^ = 18.57^ The spotlight must swing through 18.57^. Question 3.2 Assume that ABCD is a rectangle in a horizontal plane and EB is a vertical line segment. The angle of elevation from D to E is EDA = 22^. Calculate the length of BE. Step 1: Identify the relevant right-angled triangle and find AE. In right-angled EAD (right-angled at A), we are given DE = 800 cm and EDA = 22^. We can find AE using the sine function: ( EDA) = (AE)/(DE) (22^) = (AE)/(800) AE = 800 × (22^) AE ≈ 800 × 0.374606 AE ≈ 299.68 cm Step 2: Calculate BE. Since ABCD is a rectangle, AB = DC = 155 cm. The length BE is the sum of AB and AE. BE = AB + AE BE = 155 cm + 299.68 cm BE = 454.68 cm The length of BE is 454.68 cm. Determine the distance of BC. Step 1: Find the length of AD. In right-angled EAD, we can find AD using the cosine function: ( EDA) = (AD)/(DE) (22^) = (AD)/(800) AD = 800 × (22^) AD ≈ 800 × 0.927184 AD ≈ 741.75 cm Step 2: Determine BC. Since ABCD is a rectangle, BC = AD. BC = 741.75 cm The distance of BC is 741.75 cm. What's next?