# Two Related Rate Problems

• Oct 15th 2007, 08:13 PM
ebonyscythe
Two Related Rate Problems
I'm not sure why I'm having trouble with these... can someone give me a start or some help through them?

A man walks a straight path at a speed of 4 ft/sec. A search light is located on the ground 20 ft rom the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?

and

Two people start from the same point. One walks north at 3 mph and the other northeast (45 degrees exactly) at 2 mph. How fast is the distance between them changing after 15 minutes? (Hint: use the law of cosines to relate the variables)
• Oct 16th 2007, 12:23 AM
ticbol
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.