1. **Problem statement:** Determine the amplitude of the angle $DGF$ where the plane $DGF$ is given by the equation $$5x + 5y + 2z = 10.$$ 2. **Step 1: Identify the vectors forming the angle $DGF$.** The angle $DGF$ is formed at point $G$ by points $D$ and $F$. To find this angle, we need the vectors $\overrightarrow{GD}$ and $\overrightarrow{GF}$. 3. **Step 2: Find coordinates of points $D$, $G$, and $F$.** From the problem, $G$ is given as $(0,0.5,0)$ (assuming $z=0$ since not specified). The plane $DGF$ is $5x + 5y + 2z = 10$. We need coordinates of $D$ and $F$ on this plane. 4. **Step 3: Find points $D$ and $F$ on the plane.** Assuming $D$ and $F$ lie on the coordinate axes or given points: - Let $D$ be on $x$-axis: set $y=0$, $z=0$ in plane equation: $$5x + 5(0) + 2(0) = 10 \Rightarrow 5x = 10 \Rightarrow x=2.$$ So, $D = (2,0,0)$. - Let $F$ be on $z$-axis: set $x=0$, $y=0$ in plane equation: $$5(0) + 5(0) + 2z = 10 \Rightarrow 2z=10 \Rightarrow z=5.$$ So, $F = (0,0,5)$. 5. **Step 4: Calculate vectors $\overrightarrow{GD}$ and $\overrightarrow{GF}$.** $$\overrightarrow{GD} = D - G = (2 - 0, 0 - 0.5, 0 - 0) = (2, -0.5, 0)$$ $$\overrightarrow{GF} = F - G = (0 - 0, 0 - 0.5, 5 - 0) = (0, -0.5, 5)$$ 6. **Step 5: Use the dot product formula to find the angle $\theta = \angle DGF$.** The dot product formula is: $$\overrightarrow{GD} \cdot \overrightarrow{GF} = |\overrightarrow{GD}| |\overrightarrow{GF}| \cos \theta$$ 7. **Step 6: Calculate dot product and magnitudes.** $$\overrightarrow{GD} \cdot \overrightarrow{GF} = (2)(0) + (-0.5)(-0.5) + (0)(5) = 0 + 0.25 + 0 = 0.25$$ $$|\overrightarrow{GD}| = \sqrt{2^2 + (-0.5)^2 + 0^2} = \sqrt{4 + 0.25} = \sqrt{4.25}$$ $$|\overrightarrow{GF}| = \sqrt{0^2 + (-0.5)^2 + 5^2} = \sqrt{0 + 0.25 + 25} = \sqrt{25.25}$$ 8. **Step 7: Calculate $\cos \theta$.** $$\cos \theta = \frac{0.25}{\sqrt{4.25} \times \sqrt{25.25}} = \frac{0.25}{\sqrt{4.25 \times 25.25}} = \frac{0.25}{\sqrt{107.3125}} = \frac{0.25}{10.36} \approx 0.0241$$ 9. **Step 8: Calculate the angle $\theta$.** $$\theta = \arccos(0.0241) \approx 88.62^\circ$$ **Final answer:** The amplitude of the angle $DGF$ is approximately $88.62^\circ$.