Question 1206901
To solve this problem, we need to set up a coordinate system where the origin (0,0) is at the center of the bottom of the tower. The hyperbola will have a vertical transverse axis. 

**Step 1: Determine the equation of the hyperbola.**

Given the dimensions of the tower, we can determine the values of a and b in the standard equation of a hyperbola with a vertical transverse axis:

```
(y^2/a^2) - (x^2/b^2) = 1
```

* **a:** Half the distance between the vertices (top and bottom of the tower) = (581 - 393.5) / 2 = 93.75
* **b:** Half the diameter of the smallest part of the tower = 143 / 2 = 71.5

So, the equation of the hyperbola is:

```
(y^2/93.75^2) - (x^2/71.5^2) = 1
```

**Step 2: Find the width at a height of 38 feet.**

At a height of 38 feet, y = 38. We need to solve for x:

```
(38^2/93.75^2) - (x^2/71.5^2) = 1
```

Solving for x, we get:

```
x ≈ ± 53.8
```

The width at a height of 38 feet is 2 * 53.8 = **107.6 feet**.