Example
A spotlight is on the ground 20 ft away from a wall and a 6 ft tall person is
walking toward the wall at a rate of
5
2
ft/sec.
How fast is the height of the shadow changing when the person is 8 ft from the
wall?
Solution:
The variables marked in the following picture
†
,
are:
•
x
(
t
) = changing distance of the person from the spotlight at time
t
,
•
z
(
t
) = changing distance of the person from the wall at time
t
(
z
(
t
) = 20

x
(
t
)), and
•
y
(
t
) = changing height of the shadow on the wall.
We are
given
that
z
0
(
t
) =

5
2
(negative because the distance function
z
(
t
) is decreasing with time); we are also given that
at a particular instant
t
?
in time,
z
(
t
?
) = 8
,
so
x
(
t
?
) = 20

8 = 12
.
The problem is then asking for the rate of change of the height of the shadow at the instant
t
?
;
that is, we
want
y
0
(
t
?
)
.
To determine properties about
y
0
(
t
) from the known rate
z
0
(
t
), it is necessary to relate the
quantities
y
(
t
) and
z
(
t
). This can be done by noting that there are two similar right triangles
in the picture given above:
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•
The
larger triangle
has vertices at the spotlight, the base of the wall, and at the upper
tip of the shadow on the wall; this triangle has sides of length 20,
y
(
t
), with hypotenuse
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 Fall '10
 KIHYUNHYUN
 Calculus, triangle, 8 FT, 6 ft, 2 ft

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