Question: Two support wires are fastened to the top of a TV satellite dish tower from two points on the ground, $A$ and $B$, on either side of the tower. One wire is $18m$ long, and the other is $12m$ long. The angle of elevation of the longer wire is $28^\circ$. How tall is the satellite dish tower?
1. **State the problem:** We need to find the height of the satellite dish tower given two wires fastened from points $A$ and $B$ on the ground to the top of the tower. The longer wire is $18m$ with an angle of elevation of $28^\circ$. The shorter wire is $12m$ but its angle is not given.
2. **Identify the known values:**
- Length of longer wire $= 18m$
- Angle of elevation of longer wire $= 28^\circ$
- Length of shorter wire $= 12m$
- Height of tower $= h$ (unknown)
3. **Use trigonometry:** The height $h$ of the tower is the vertical component of the longer wire. Using the sine function which relates the opposite side (height) to the hypotenuse (wire length):
$$h = 18 \times \sin(28^\circ)$$
4. **Calculate the sine value:**
$$\sin(28^\circ) \approx 0.4695$$
5. **Calculate the height:**
$$h = 18 \times 0.4695 = 8.451$$
6. **Interpretation:** The height of the satellite dish tower is approximately $8.45m$.
**Final answer:**
$$\boxed{8.45\text{ meters}}$$