# Slide 1 - cszymanski

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The Shadow Problem & The Cone Problem

A man (6 ft) walks away from a
lamp post (15 ft) at 5 ft/sec.
How fast is his shadow lengthening?
How fast is the shadow's tip moving?

Water is flowing into a cone
(ht = 16cm, r = 4cm) at 2 cm3/min.
How fast is the water level rising
when it is 5,10,15 cm deep?
Sand pouring from a hopper at a steady rate forms a
conical pile whose height is observed to remain
twice the radius of the base of the cone. When the
height of the pile is observed to be 20 feet, the
radius of the base of the pile appears to be
increasing at the rate of a foot every two
minutes. How fast is the sand pouring from the
hopper?

A light is on the ground 40ft from a building.
A man 6ft tall walks from the light towards
the building at 6ft/s. How rapidly is his
shadow on the building becoming shorter
when he is 20ft from the building?
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The Cone Problem

Coffee is dripping from a conical shaped filter into a
cylindrical shaped pot at a rate of 2 cubic inches per
second. They both have the same radius, 5 inches
and height, 14 inches.

(A) When the filter is empty, what is the depth of the coffee
in the pot?
(B) How fast is the depth of the coffee in the pot changing
when the coffee in the pot is 4 inches deep?
(C) How fast is the depth of the coffee in the filter changing
when the coffee in the pot is 4 inches deep?
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The Cone Problem

A coffee maker has the shape of a double cone 20 cm
high. The radii at both the top and the base are 4cm.
Coffee is flowing from the top section into the bottom
se tion at a rate of 4 cm cubed/sec. At what rate is the
level of coffee in the top section falling when the coffee
in the top section is 4cm deep? How fast is the level of
coffe in the bottom section rising at the instant when
the coffee in the bottom of the pot is 4 cm deep?
Thank you so much
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A ball is dropped from a height of 64 feet and at a      The Shadow Problem
horizontal distance of 16 feet from a light that is 64
feet above the ground at the top of a light pole. How
fast is the shadow of the ball moving along the
ground one second after the ball is dropped?
Neglect air resistance so that the distance the
football will have dropped as a function of time will
be s = 16t2 with the football dropped at t = 0.
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A tightrope is stretched 30 feet above the ground
between the Jay and Tee buildings which are 50 feet
apart. A tightrope walker, walking at a constant rate
of 2 feet per second from point A to point B, is
illuminated by a spotlight 70 feet above point A.

a. How fast is the shadow of the tightrope walker’s
feet
moving along the ground when she is midway
between the buildings?
Tee   Jay   b. How far from point A is the tightrope walker when
the shadow of her feet reaches the base of the
Tee Building?
c. How fast is the shadow of the tightrope walker’s
feet
moving up the wall of the Tee Building when
she is 10 feet from point B?
A street light is at the top of a 10 ft tall pole. A
woman 6 ft tall walks away from the pole with
a speed of 6 ft/sec along a straight path.

a. How fast is the tip of her shadow moving
when she is 40 ft from the base of the
pole?

b. How fast is the shadow lengthening?

A man is sipping soda through a straw from a
conical cup, 15 cm deep and 8 cm in diameter at the
top. When the soda is 10 cm deep, he is drinking at
the rate of 20 cm3 /s. How fast is the level of the
soda dropping at that time?
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The celebrated actor, Tyrone Olivier, is exactly 6 feet tall. Tonight he delivers a stirring
soliloquy on stage. For dramatic effect he is illuminated only by a single footlight, which
is at the same level as the floor of the stage. 24 feet behind the footlight is the vertical
backdrop of the stage. As Tyrone Olivier nears the climax of his soliloquy, he begins
walking toward the audience (and toward the footlight) at 2 feet per second. When he
is 8 feet from the footlight, at what rate is his shadow on the backdrop increasing in
height?
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A 60 foot rope is looped through a pulley 36 feet above the ground. A lantern is attached to one
end. The other end is held by a man six feet tall. The man starts walking away from the point on
the ground directly beneath the pulley (and beneath the lantern) at a rate of five feet per second.
He is holding onto the rope head-high. When the man is 40 feet away from the point on the
ground directly beneath the pulley, what is the rate of change of the length of his shadow that is
cast by the lantern?

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