Greenwood press

TANGENT
141
A line can be tangent to many types of curves, including geometric shapes and functions.
A line tangent to a
circle at point C.
A line tangent to the function
y = x
3
– 3x
2
+ 2x – 7 at (2,–7)
The slope of the tangent line can predict the
productivity rate at a particular moment in time.

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At this rate, it would seem appropriate to take a break so that workers can rejuve-
nate for the afternoon.
Besides worker productivity, the slope of a tangent line can help determine
the speed of an object, the location where business profits are at a maximum, the
hourly rate for business consulting, the moment when ticket sales for a particu-
lar movie have declined rapidly, and many other applications about rates that can
be modeled with functions (see Rates).
Tangents are also used in applications related to circles. For example, radio
signals will reach a distance from the antenna on the tower to the horizon. The
visible sight to the horizon represents the point of tangency, where no other parts
of the earth can be seen. Since a line tangent to a circle is perpendicular to its
radius, this distance can be determined using the Pythagorean theorem. A radio
antenna that is 200 meters tall can have a signal that reaches a distance of approx-
imately 50 km. Since the radius of the earth is approximately 6,380 km, the equa-
tion
s
2
+ 6, 380
2
= 6, 380.2
2
is used to find the signal radius, 
s, based on the
geometric representation depicted below.
An object released from a circular-motion path will leave in a straight-line
tangent to the circle at the point of release. For example, a discus thrower spin-
ning in a circle will throw the disc out towards the open field in a straight path
after she releases the weight from her hand, as shown below. When a cowboy
spins a lasso in a circular path, and then releases his grip, the rope will travel in
a straight path towards the calf that he is trying to capture.