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A 15 ft long ladder rests against a vertical wall.
Its top begins sliding down the wall at a speed of 2 ft/s,
while its base begins sliding along the ground away from the wall.
How fast is the angle between the top of the ladder
and the wall changing at the moment the angle is π/3 radians?
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A boat is pulled into a dock by means of a rope
attached to a crank on the dock.
The rope is tethered to a cleat on the bow of the boat
that is one meter below the pulley.
If the rope is reeled into the crank at a rate of one m/s,
at what rate will the boat be approaching the dock
the moment there only ten meters of rope out?
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A police cruiser is chasing a speeding car.
The cruiser is headed south towards an intersection.
The speeding car has already passed through the intersection, turning left towards the east.
When the police cruiser is still 0.6 miles north of the intersection,
the car is 0.8 miles to the east of it,
and the police radar determines that
the distance between their cruiser and the car is increasing at 20 mph.
If the cruiser is cruising at 60 mph when this measurement is made,
what is the speed of the car?
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An oil well located near the middle of a calm, shallow lake
is leaking oil onto the surface of the lake
at the rate of 16 ft³ per minute.
Since oil and water don’t mix, as the oil leaks onto the surface of the lake
it spreads out into a uniform layer, called an oil slick,
that floats on top of the water.
The slick is circular, centered at the leak, and 0.02 ft thick.
Find the rate at which the radius of the slick is increasing
when the radius is 500 ft, and again when the radius is 700 ft.
Why would you expect the rates of change of the radii to be different,
even though the oil is being added at a constant rate?
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A plane is flying at 500 mph at an altitude of 3 miles in a direction away from you
(i.e. the point on the ground directly beneath the plane is moving away from you).
How fast is the plane’s distance from you increasing
the moment when the plane is flying over a point on the ground four miles from you?
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A 5 ft tall woman is walking on a sidewalk away from a streetlamp.
The bulb of the lamp is 12 ft above the ground,
and she is walking at about 3.5 ft/s.
At what speed is the tip of her shadow moving along the sidewalk?
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You are inflating a spherical balloon at the rate of seven cm³/s.
How fast is its radius increasing at the instant when the radius is four centimeters?
How fast is the surface area increasing at the same time?
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Water is poured into an inverted cone cup at the rate of 10 cm³/s.
The cone has a height of 30 cm and a base radius of 10 cm.
How fast is the water level rising when the water depth,
from the tip of the cone to the surface of the water, is 4 cm?
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A swing consists of a board at the end of a 10 ft long rope.
Think of the board as a point \(P\) at the end of the rope,
and let \(Q\) be the point of attachment at the other end.
Suppose that the swing is directly below \(Q\) at time \(t=0\)
and is being pushed by someone who walks at the speed of 6 ft/s from left to right.
Find (a) the speed the swing is rising after one second
and (b) the angular speed of the rope in radians-per-second after one second.
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The ideal gas law \(pV = T\) relates pressure \(p\) and
volume \(V\) and temperature \(T\). Assume the temperature
\(T = 50\) is fixed and \(\dot V=-5\). Find the rate \(\dot p\) with
which the pressure increases when \(V = 10\) and \(p= 5.\)
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There are cosmological models which see our universe
as a four dimensional sphere which expands in space
time. Assume the volume \(V = \frac 1 2 \pi^2 r^2\) increases at a
rate \(\dot V = 100\pi^2r^2\), What is \(\dot r\)?
Evaluate it for \(r = 47\) (billion light years).
Challenges
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Suppose that the minute hand of a clock is 15 mm long and the hour hand is 12 mm.
How fast is the distance between the hour hand and the minute hand changing at 2 pm?
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A tiny water tank has the shape of a horizontal cylinder with radius 1” and length 2”.
If water is being pumped into the tank at a rate of ⅙ mm³ per hour,
find the rate at which the water level is rising when the water is ½ mm deep.