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Space Exploration
Source: Source: NASA
On October 5, 1991 the Magellan spacecraft entered the atmosphere of Venus and transmitted the temperature T in
kelvins (K) versus the altitude h in kilometers (km) until its signal was lost at an altitude of about 34 km. Discounting the
initial erratic signal, the data strongly suggested a linear relationship, so a least squares straight line fit was used on the
linear part of the data to obtain the equation
By setting in this equation, the surface temperature of Venus was estimated at .
EXAMPLE 3 Fitting a Quadratic Curve to Data
According to Newton's second law of motion, a body near the earth's surface falls vertically downward according to the
equation
(12)
where
s = vertical displacement downward relative to some fixed point
= initial displacement at time
= initial velocity at time
g = acceleration of gravity at the earth's surface
Suppose that a laboratory experiment is performed to evaluate g using this equation. A weight is released with unknown
initial displacement and velocity, and at certain times the distances fallen relative to some fixed reference point are
measured. In particular, suppose it is found that at times , and .5 seconds, the weight has fallen
, and 3.73 feet, respectively, from the reference point. Find an approximate value of g using these
data.
Solution
The mathematical problem is to fit a quadratic curve
