Identify the known values in the right-angled triangle BFC.

50. a) The height of the tree Step 1: Identify the known values in the right-angled triangle BFC. The distance from the foot of the tree (F) to the chameleon (C) is FC = 14.4 m. The angle of elevation of the bird (B) from the chameleon (C) is BCF = 41.4^. Step 2: Use the tangent ratio to find the height of the bird (BF). ( BCF) = (BF)/(FC) (41.4^) = (BF)/(14.4 m) BF = 14.4 m × (41.4^) BF = 14.4 m × 0.88009 BF ≈ 12.6733 m Step 3: Calculate the height of the tree (TF). Since B is the midpoint of FT, the height of the tree TF is twice the height of the bird BF. TF = 2 × BF TF = 2 × 12.6733 m TF ≈ 25.3466 m Step 4: Round the height to 2 decimal places. TF ≈ 25.35 m The height of the tree is 25.35 m. b) The angle of depression of the chameleon from point T Step 1: Identify the known values in the right-angled triangle TFC. The height of the tree TF ≈ 25.3466 m (from part a). The distance from the foot of the tree (F) to the chameleon (C) is FC = 14.4 m. Step 2: The angle of depression of the chameleon from point T is equal to the angle of elevation of T from C, which is TCF. Use the tangent ratio. ( TCF) = (TF)/(FC) ( TCF) = 25.3466 m14.4 m ( TCF) ≈ 1.76018 Step 3: Calculate the angle TCF. TCF = (1.76018) TCF ≈ 60.399^ Step 4: Round the angle to 2 decimal places. TCF ≈ 60.40^ The angle of depression of the chameleon from point T is 60.40^. 51. Step 1: Convert the side length of the cubical stone from cm to m. s = 40 cm = 0.4 m Step 2: Calculate the volume of the cubical stone in m^3. V = s^3 V = (0.4 m)^3 V = 0.064 m^3 Step 3: Use the given mass and calculated volume to determine the density. m = 12 kg = (m)/(V) = 12 kg0.064 m^3 = 187.5 kg/m^3 The density of the stone is 187.5 kg/m^3. 52. Step 1: Use the area formula for a triangle to find PQR. Given Area = 30sqrt(3) cm^2, PQ = 10 cm, QR = 12 cm. Area = (1)/(2) × PQ × QR × ( PQR) 30sqrt(3) = (1)/(2) × 10 × 12 × ( PQR) 30sqrt(3) = 60 × ( PQR) ( PQR) = 30sqrt(3)60 ( PQR) = sqrt(3)2 Step 2: Determine the obtuse angle PQR. Since ( PQR) = sqrt(3)2, the acute angle is 60^. As PQR is obtuse, PQR = 180^ - 60^ = 120^ Step 3: Use the Cosine Rule to find the length of PR. PR^2 = PQ^2 + QR^2 - 2 × PQ × QR × ( PQR) PR^2 = (10 cm)^2 + (12 cm)^2 - 2 × 10 cm × 12 cm × (120^) PR^2 = 100 cm^2 + 144 cm^2 - 240 cm^2 × (-(1)/(2)) PR^2 = 244 cm^2 + 120 cm^2 PR^2 = 364 cm^2 PR = sqrt(364) cm PR ≈ 19.0787 cm Step 4: Calculate the perimeter of the triangle. Perimeter = PQ + QR + PR Perimeter = 10 cm + 12 cm + 19.0787 cm Perimeter = 41.0787 cm Step 5: Round the perimeter to 2 decimal places. Perimeter ≈ 41.08 cm The perimeter of the triangle is 41.08 cm. 53. Step 1: Calculate the total time elapsed from Monday 6:00 a.m. to Friday 6:00 a.m. From Monday 6:00 a.m. to Friday 6:00 a.m. is 4 full days. Total hours = 4 days × 24 hours/day = 96 hours. Step