Greenwood press

124
SLOPE

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of liquid will slow down the burning of the wick. Hence, fatter candles do not
need very long wicks, because they will likely form a pool of liquid as they burn.
Pilots of airplanes and hot-air balloons use square roots to estimate viewing
distances. The viewing distance in kilometers, 
d, from an airplane on a clear
day, depending on its altitude in meters, 
a, can be estimated by the equation v =
3.56
√
a. The viewing distance from an airplane to the horizon is perpendicular
to the radius of the earth, forming a right triangle between the airplane, horizon,
and center of the earth (see the figure below). The Pythagorean theorem can be 
used to compare the distances, 
v
2
+ 6380
2
= (6380 +
a
1000
)
2
, using the fact 
that the radius of the earth is 6,380 km. The square-root version of the equation
is approximately equal to this format, since commercial airplanes do not fly
much higher than 10,000 meters. 
Police investigators use square roots at the scenes of auto accidents. They can
estimate the speed of a car by the length of the tire skids and the conditions of the
road. The speed of a car in miles per hour, 
s, that skidded d feet is s =
√
30f d.
The variable 
f describes the coefficient of friction of the road. On dry concrete,
this value is about 0.8, and in wet conditions, 
f is about 0.4. Measuring the length
of the skids will help determine if the offender’s speed was a factor that con-
tributed to the accident.
The period of a pendulum, or the time it takes to move back and forth, can 
be determined by the equation 
t = 2π
l
g
, where 
t is the time in seconds, l is its 
length in meters, and 
g is the acceleration due to gravity (9.8 meters/second
2
).
This equation is actually a combination of a couple of equations, 
g = lw
2
and
w =
2π
t
, that relate to circular motion and the pendulum’s length, period, and an-
gular velocity 
w. Notice that the mass of the object at the end of pendulum is not
included in the equation, because all objects will fall at the same rate, regardless
of their mass. The pendulum equation is useful for clockmakers, because a
grandfather clock is designed so that its pendulum arm takes one second to swing
in one direction, or two seconds to swing back and forth. If 
t = 2 is substituted
into the equation, then the pendulum arm length 
l will be approximately 1 meter
long.
Using square roots can help a person become a better consumer of art. The
best view of a picture is when the angle, 
α, from the bottom of the picture to the
top is greatest, as shown in the following figure. An ideal distance, 
d, to stand