Angel is trying to find the height of a radio antenna on the roof of a local building. He stands at a horizontal distance of 21 meters from the building. The angle of elevation from his eyes to the roof (point $A$ ) is $38^{\circ}$, and the angle of elevation from his eyes to the top of the antenna (point $B$ ) is $46^{\circ}$. If his eyes are 1.66 meters from the ground, find the height of the antenna (the distance from point $A$ to point $B$ ). Round your answer to the nearest tenth of a meter if necessary.

Step 1 ：Let \(h_A\) be the height of the building from the ground to point \(A\), and \(h_B\) be the height of the building from the ground to point \(B\). We can use the tangent function to find \(h_A\) and \(h_B\):

Step 2 ：\[\tan(38^\circ) = \frac{h_A - 1.66}{21}\]

Step 3 ：\[\tan(46^\circ) = \frac{h_B - 1.66}{21}\]

Step 4 ：Solve these equations for \(h_A\) and \(h_B\):

Step 5 ：\[h_A \approx 18.07\]

Step 6 ：\[h_B \approx 23.41\]

Step 7 ：Find the height of the antenna by subtracting \(h_A\) from \(h_B\):

Step 8 ：\[\text{antenna height} = h_B - h_A \approx 5.34\]

Step 9 ：\[\boxed{\text{The height of the antenna is approximately 5.3 meters}}\]