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3.9

Related rates

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Relate rates problems(find the derivative of a certain

function)• Identify functions.• Write relation(s) between functions.• Differentiate using implicit differentiation.• Plug in values• Solve for derivative

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Useful to remember

• Pythagorean theorem• Similar triangles• Trigonometry (triangle def of sin, cos, tan)• Law of cosines• Uniform circular motion (a=wt, w=v/r)• Volume of a cone• Surface area of a sphere (4* pi * r^2)

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Similar to homework problem

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• A search light in a prison rotates counterclockwise at a rate of 3 revolutions per second. The light shines on a long straight wall that is 40 feet from the search light. How fast is the light beam moving across the wall when the beam is hitting the wall at a spot which is 80 feet from the light ?

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2. (13 points) In this problem you should use the law of cosines:

a2 + b2 − 2ab cos θ = c2,

where θ is an angle in the triangle, c is the opposite side, and a and b are the adjacentsides.

Two straight roads intersect at point P at a 60◦ angle. Car A is traveling away from Pon one road, and car B on the other road. You’re in car A, which has a device that canmeasure distance from car B and also the rate at which that distance is increasing.At a certain moment you have traveled 3 km away from P and are moving at 80km/hr. At that time the device shows that your distance from car B is 7 km, andthis distance is increasing at 100 km/hr.

θP a

cb

car A

car B

At that instant find

(a) (5 pts) the distance car B has traveled from P;

(b) (8 pts) the speed at which car B is moving.

3

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5. (14 points) Two carts, A and B, areconnected by a rope 28 ft long that passesover a pulley P (see the figure).

The point Q is on the floor 12 feet directlybeneath P and between the carts. Cart Ais being pulled away from Q at a speed of7 ft/s. How fast is cart B moving toward Qat the instant when cart A is 5 ft from Q?

B A

12

P

Q

7