Hello, henryhighstudent!

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The transmitter is at: .

The radio is at

When is at the closest, line is tangent to the parabola

. . but *not* at its vertex.

Let line have slope

The equation of is: . .[1]

Intersection of and the parabola: .

If is *tangent* to the parabola, there is *one *intersection point;

. . the discriminant is zero.

Hence, we have: .

. . and we have: .

We see that slope must be negative: .

Substitute into [1].

The equation of is: .

We want the -intercept of

. .

which rationalizes to: .

*HA!* . . . Just noticed the wording of the question.

What is the closest the radio can be **to the hill** . . . ?

is the distance from to

The distance from to the hill is: . units.