To solve the problem directly:

(a) Locate Point A so that the total length of the guy wire is minimal:

1. Let the horizontal distance from Point A to the base of the taller tower (50') be $x$. The distance from Point A to the base of the shorter tower (30') is $150 - x$.

2. The total length of the wires is: $L(x) = \sqrt{x^2 + 50^2} + \sqrt{(150 - x)^2 + 30^2}.$

3. Differentiate $L(x)$ with respect to $x$ and solve $\frac{dL}{dx} = 0$.

   After solving: $x \approx 87.5 \, \text{ft}.$

   Therefore, Point A is approximately 87.5 ft from the base of the 50' tower.

(b) Show that the length is minimized when the angles at A are equal:

This can be shown by symmetry: for minimum length, the angles $\theta_1$ and $\theta_2$ (formed by the wires and the ground) must be equal. For $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$, this leads to the derived condition that the angles are equal at $x \approx 87.5$.

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