hmwrk12e.html

Homework 12: Related Rate Problems

Part A: Basic problems:

Exercise 12.1. A cylindrical tank has radius 20 cm. How fast does the water level in the tank drop when the water is being drained at 25 / ?
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Exercise 12. 2. A ladder 13 m long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 0.6 m/sec. How fast is the top sliding down the wall when the foot of the ladder is 5 m from the wall?

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Exercise 12. 3.
A rotating beacon is located 2 miles out in the water. Let be the point on the shore that is closest to the beacon. As the beacon rotates at / , the beam sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point miles downshore from the point

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Exercise 12. 4 . A baseball diamond is a square ft on a side. A player runs from first base to second base at / . At what rate is the player's distance from third base decreasing when she is half way from first to second base?
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Exercise 12. 5 . Sand is poured onto a surface at / , forming a conical pile whose base radius is always half its altitude. How fast is the altitude of the pile increasing when the pile is cm high?
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Exercise 12. 6. A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point ft higher than the front of the boat. The rope is being pulled through the ring at the rate of / . How fast is the boat approaching the dock when ft of rope are out?
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Exercise 12. 7. A balloon is at a height of meters, and is rising at the constant rate of / . At that instant a bicycle passes beneath it, traveling in a straight line at the constant speed of / . How fast is the distance between them increasing seconds later?
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Exercise 12. 8 . A cone-like vat has square cross-section (rather than circular cross-section, as in the case of the usual cone). The dimensions at the top are by , and the depth is . If water is flowing into the vat at / , how fast is the water level rising when the depth of water (at the deepest point) is ? Note : the volume of any conical shape is 1/3(height)(area of the base) .
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Exercise 12. 9.
The sun is setting at the rate of / / on a day when the sun passed directly overhead. How fast is the shadow of a meter wall lengthening at the moment when the shadow is meters? (See the picture.)

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Exercise 12. 10 . A woman 5 ft tall walks at the rate of / away from a streetlight that is ft above the ground. At what rate is the tip of her shadow moving? At what rate is her shadow lengthening? Notice that in this problem these rates do not depend on the instant in question --- both rates are constant, provided that the woman's speed is constant.
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Exercise 12. 11. A police helicopter is flying at at a constant altitude of mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly mile from the helicopter, and that this distance is decreasing at mph. Find the speed of the car.
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Exercise 12. 12 . The trough shown below is constructed by fastening together three slabs of wood of dimensions by , and then attaching the construction to a wooden wall at each end. The angle was originally , but because of poor construction the sides are collapsing. The trough is full of water. At what rate (in / ) is the water spilling out over the top of the trough if the sides have each fallen to an angle of , and are collapsing at the rate of per second?

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Exercise 12. 13. Suppose that in Example 12.6 the angle between the two roads is [Maple Math] , instead of (that is, the South--North road actually goes in a somewhat northwesterly direction from ). In this problem use the law of cosines: .

(a) Express / in terms of , / , / .
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(b) Notice that the angle is constant (and so is also constant). State a story problem (perhaps involving a bug on a door, rather than a car on a road) in which not only but also are all variables. In that case suppose you know the values of and their rates. Find a formula for / .

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Part B: Additional Practice (do these as soon as possible, certainly before the midterm)

Exercise 12. 14. A light shines from the top of a pole high. An object is dropped from the same height from a point away. How fast is the object's shadow moving on the ground one second later? (Neglect air resistance and take / for the falling object.)
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Exercise 12. 15 . The two blades of a pair of scissors are fastened at the point . Let a denote the distance from to the tip of the blade (the point ). Let denote the angle at the tip of the blade that is formed by the line and the bottom edge of the blade (see the diagram). Suppose that a piece of paper is cut in such a way that the center of the scissors at A is fixed, and the paper is also fixed. As the blades are closed (i.e., the angle in the diagram is decreased), the distance y between and increases, cutting the paper.

(a) Express in terms of , , and .
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(b) Suppose you know the rate / at which the angle between the blades is decreasing. Express / in terms of , and / .
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(c) Suppose that the distance a is , and the angle is . Further suppose that you are closing the scissors at the rate of / . At the instant when , find the rate (in / ) at which the paper is being cut.

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Exercise 12. 16. In the previous problem,

(a) express in terms of / the speed of the point on the scissors that moves fastest (this point is );
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(b) if we have kilometers (i.e., we have scissors of grand proportions) and if radians, find a value of / for which d / is a little faster than the speed of light (
/ ) when the pair of scissors is almost closed;
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(c) would the situation in your answer to part (b) be possible, or would it violate the Theory of Relativity? Explain.
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Exercise 12. 17 . In the situation of Example 12.6, find the rate at which the distance between the cars is increasing or decreasing if

(a) car A is meters north of , car is meters east of , both cars are going at constant speed toward , and the two cars will collide in 10 seconds;
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(b) seconds ago car started from rest at and has been picking up speed at the steady rate of / , and seconds after car started car passed moving east at constant speed 60 / ;
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(c) the functions and are given by the graphs below and you want to find the rate at which the cars are separating at time . Hint : (Estimate derivatives by drawing tangent lines.)

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Exercise 12. 18. In Example 12.6 suppose that instead of car you have a helicopter flying at speed to the east of . Let be the altitude of the helicopter, and let be the distance from car to the helicopter. Then the three-dimensional Pythagorean Theorem states: . Find a formula for / in terms of and the various rates if:

(a) the helicopter is flying at speed / at constant altitude;
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(b) the helicopter is flying at horizontal speed / (i.e., the point on the ground under the helicopter is moving at this speed), and at the same time is gaining altitude at rate / .
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