In the sketch alongside, DF is a vertical flagpole and \( \mathrm{E}, \mathrm{F} \) and G are three points on ground level. \( \mathrm{DF}=7 \mathrm{~m}, \mathrm{EF}=6 \mathrm{~m}, \mathrm{FDG}=38^{\circ} \) and \( \mathrm{FEG}=42^{\circ} \). \( \mathrm{EGF}=\theta \), where \( \theta \) is an acute angle. Calculate (1) the magnitude of \( \theta \). (2) the length of EG (3) the magnitude of EDG. (4) the area of \( \triangle \mathrm{EDG} \). (5) the angle of elevation of D from E .

To solve the problem systematically, we'll use trigonometry and the properties of triangles. ### 1. Calculate the magnitude of \( \theta \) To find \( \theta \), we can use the angles \( FDG \) and \( FEG \). The sum of angles in triangle \( FEG \) gives us: \[ \theta + 38^\circ + 42^\circ = 180^\circ \] \[ \theta + 80^\circ = 180^\circ \] \[ \theta = 100^\circ \] ### 2. Calculate the length of \( EG \) In triangle \( EFG \), we can use the Law of Sines: \[ \frac{EG}{\sin(42^\circ)} = \frac{EF}{\sin(100^\circ)} \] Since \( EF = 6 \, m \): \[ EG = \frac{6 \cdot \sin(42^\circ)}{\sin(100^\circ)} \] Calculating this gives: \[ EG \approx 6 \cdot \sin(42^\circ) \quad (\sin(100^\circ) \text{ is essentially } 1) \] So, \[ EG \approx 6 \cdot 0.6691 \approx 4.0146 \, m \] ### 3. Calculate the magnitude of \( EDG \) Using the right triangle formed by \( DF \) and \( DG \), we can apply the tangent ratio: \[ \tan(38^\circ) = \frac{DF}{EG} \] Thus, \[ EDG = \tan^{-1}\left(\frac{7 \, m}{4.0146 \, m}\right) \] Calculating this gives: \[ EDG \approx \tan^{-1}(1.743) \approx 60.4^\circ \] ### 4. Calculate the area of \( \triangle EDG \) Using the formula for the area of a triangle: \[ Area = \frac{1}{2} \times base \times height \] Here, \( base = EG \) and height = \( DF \): \[ Area \approx \frac{1}{2} \times 4.0146 \times 7 = 14.0511 \, m^2 \] ### 5. Calculate the angle of elevation of \( D \) from \( E \) We can use the triangle \( EDF \): \[ \tan(\angle DE) = \frac{DF}{EF} = \frac{7}{6} \] Thus, \[ \angle DE \approx \tan^{-1}(1.1667) \approx 50.2^\circ \] In summary: 1. \( \theta = 100^\circ \) 2. \( EG \approx 4.0146 \, m \) 3. \( EDG \approx 60.4^\circ \) 4. Area of \( \triangle EDG \approx 14.0511 \, m^2 \) 5. Angle of elevation of \( D \) from \( E \approx 50.2^\circ \)