1.If a snowball melts so that its surface area decreases at a rate of , find the
ID: 2836420 • Letter: 1
Question
1.If a snowball melts so that its surface area decreases at a rate of , find the rate at which the diameter decreases when the diameter is .
2.A street light is mounted at the top of a 14 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 4 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 ft from the base of the pole?
3.Two cars start moving from the same point. one travels south at 30 mi/h and the other travels west at 60 mi/h. At what rate is the distance between the cars increasing 5 hours later?
4.The altitude of a triangle is increasing at a rate of centimeters/minute while the area of the triangle is increasing at a rate of square centimeters/minute. At what rate is the base of the triangle changing when the altitude is centimeters and the area is square centimeters?
5.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 14 ft/min, at what rate will the boat be approaching the dock when 100 ft of rope is out?
6.t noon, ship A is 50 miles due west of ship B. Ship A is sailing west at 20 mph and ship B is sailing north at 24 mph. How fast (in mph) is the distance between the ships changing at 6 PM?
7.A particle is moving along the curve . As the particle passes through the point , its -coordinate increases at a rate of units per second. Find the rate of change of the distance from the particle to the origin at this instant.
8.Gravel is being dumped from a conveyor belt at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 17 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by .
Explanation / Answer
1.A = 4 pi r^2 = pi D^2
dA/dt = 2 pi D x dD/dt
1 = 2 x pi x 10 x dD/dt
dD/dt = 1/(20 pi) = 0.016 cm/min
2.Let x be the distance between the bottom of the pole and the woman.
Let y be the length of her shadow.
Better draw a figure hope you do that.
Now by similar triangle property, y/(x+y) = 6/14 = 2/7
By the compodendo-dividendo rule, y/x = 2/5
Or y = (2/5) x
Getting derivatives wrt to time, dy/dt = (2/5) dx/dt
Given dx/dt = 4 ft/sec. So dy/dt = 8/5 ft/sec
3.Let x be position of first car from start point
Let y be position of second car from start point
Let t be the time in hours
x = 30t . . . . . . dx/dt = 30
y = 60t . . . . . . dy/dt = 60
Let s be distance between the two cars
s
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