1-A street light is mounted at the top of a 13 ft tall pole. A woman 6 ft tall w

Question

1-A street light is mounted at the top of a 13 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole?

2-The altitude of a triangle is increasing at a rate of 2 centimeters/minute while the area of the triangle is increasing at a rate of 5 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 9 centimeters and the area is 100 square centimeters.

3-A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 7 feet below the level of the pulley. (There is a diagram of this situation with problem 18 on p132 of the text.)

If the rope is pulled through the pulley at a rate of 20 ft/min, at what rate will the boat be approaching the dock when 90 ft of rope is out?

4-At noon, ship A is 30 miles due west of ship B. Ship A is sailing west at 17 mph and ship B is sailing north at 18 mph. How fast (in mph) is the distance between the ships changing at 5 PM?

Explanation / Answer

man pole
Let x be the distance between the man and the pole.
dx/dt=8
Let y be the distance between the man and the tip of his shadow (length of the shadow)
Let x+y = z (the distance between the tip of the shadow and the pole)
By using similar triangles,
z/13 =x / (13-6)
z/13 = x/7
z=(13/7) x
z=(13/7)x
dz/dt = (13/7) dx/dt
dz/dt = (13/7)(13)=169/49=3.4489
Regardless of how far he is from the pole, the tip of his shadow is moving at a rate of 3.4889 ft /s

2.

A = (1/2)bh

when A = 100
100 = (1/2)b(9)
b = 100/(4.5)
b = 22.22


dA/dt = (1/2)b * (dh/dt) + (1/2)h * (db/dt)
2 = (1/2)(22.22)*(1) + (1/2)(9)*(db/dt)
2 = 11.11 + 4.5(db/dt)
db/dt = (2-11.11)/4.5
db/dt = -2.024 cm/s

3.

Let y = 7 ft, the distance the boat is below the pulley.

Let x be the distance from the bottom of the dock to the front of the boat.

Let c be the amount of rope out.
c is the hypotenuse of a right triangle formed between the height of the dock and the distance from the dock t the boat.

y is constant, while x and c are all changing with time, so:

[x(t)]

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