A pole, with a lamp 5 m high, is 8 m from a window that has its part less than 1 m and its upper part 2 m from the ground. A 0.5 m tree is planted 4 m from the house and 4 m from the pole. If this tree grows 0.2 m per year, how many years will it take so that the streetlight doesn't hit the window? (A) 14. (B) 15. (C) 16. (D) 17. (E) 18.

Given:
- The height of the pole with the lamp: H_L = 5 \, \text{m}
- Distance from the pole to the window: D_P = 8 \, \text{m}
- Minimum height of the window: H_{min} = 1 \, \text{m}
- Maximum height of the window: H_{max} = 2 \, \text{m}
- Initial height of the tree: H_T = 0.5 \, \text{m}
- Distance from the house/tree to the pole/tree: D_H = D_T = 4 \, \text{m}
- Tree grows per year: G_T = 0.2 \, \text{m/year}

To find how many years it will take for the tree to block the light from reaching the window, we calculate the required height:
1. Find the current angle of elevation from the streetlight (on the pole) to the bottom and top of the window.

2. Minimum reach height (slope from light to bottom of the window):
\tan \theta_{min} = \frac{H_L}{D_P} = \frac{5}{8} \Rightarrow H_{reach(min)} = H_{L} - (5/8) \times 4 = 1

3. Maximum reach height (slope from light to the top of the window):
\tan \theta_{max} = \frac{H_{min}}{D_P} = \frac{H_{max}} = \frac{5}{8}

4. The tree grows by 0.2m per year. So we find the height needed to block the light after it has grown by years (`t`), which will match the window's diagonal based on the similar triangle formula.
5. After blocking the light to the window:
H_{min} = H_{tree} + \frac{H_{min}}{4}
H_{max} = \frac{5}{8} H_{tree} = \tan [t] H_{window}\frac{4}{3}
17