A balloonist is directly above a straight road 1.5 miles long that joins two towns. She finds that the town closer to her is at an angle of depression of $39^{\circ}$ and the farther town is at an angle of depression of $35^{\circ}$. How high above the ground is the balloon?
The balloonist is about miles above the ground. (Round the final answer to two decimal places as needed. Round all inter

Step 1 ：Let's denote the distance to the balloon from the closer town as \(d_1\) and the height of the balloon as \(h\). We have \(\tan(39^{\circ}) = \frac{h}{d_1}\).

Step 2 ：Similarly, let's denote the distance to the balloon from the farther town as \(d_2\) and the height of the balloon as \(h\). We have \(\tan(35^{\circ}) = \frac{h}{d_2}\).

Step 3 ：We also know that \(d_2 = d_1 + 1.5\) miles.

Step 4 ：Solving these equations, we find that \(d_1\) is approximately 13.69 miles.

Step 5 ：Substituting \(d_1\) into the first equation, we find that the height of the balloon \(h\) is approximately 11.09 miles.

Step 6 ：Final Answer: The balloonist is approximately \(\boxed{11.09}\) miles above the ground.