Need a start only? Not the whole solutions?

Okay.

In the 1st Problem, the diagram to visualize the Problem is that of a right triangle, with these:

----one leg = 20ft

----the other leg = x ft ---the distance the man will travel to reach the point on the the path closest to the searchlight

----hypotenuse = unknown

----angle between the hypotenuse and the 20-ft leg = theta.

So,

tan(theta) = x/20

Differentiate both sides with respect to time t,

sec^2(theta) *d(theta)/dt = (1/20)dx/dt

You are given dx/dt = 4 ft/second

At that instant, you can solve for sec(theta).

sec(theta) = hypotenuse / adjacent side

sec(theta) = sqrt(15^2 +20^2) / 20

etc....

d(theta)/dt is in radians per second. That is the rate of rotation of the searchlight.

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In the 2nd Problem,

The figure is a triangle, with these:

---vertical side = 3t miles

---northeasterly side = 2t miles

---third side = x miles

---angle between the 3t and 2t sides = 45 degrees.

As the hint says, use the Law of Cosines to express x in terms of t.

Then differentiate both sides of the equation with respect to time t.

Etc.

Remember, cos(45deg) is a constant.