College Algebra 9th txtbk

368 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
EXAMPLE 3 Earthquake Intensity
If the energy release of one earthquake is 1,000 times that of another, how much larger is
the Richter scale reading of the larger than the smaller?
SOLUTION
Let
and
be the Richter equations for the smaller and larger earthquakes, respectively. Since the larger
earthquake released 1,000 times as much energy, we can write E 2 1,000E 1 .
M 2 2 Substitute 1,000E 1 for
3 log E 2
E 0
2 3 log 1,000E 1
M 1 2 3 log E 1
E 0
M 2 2 3 log E 2
E 0
E 2 .
Use
E 0
log (MN) log M log N; 1,000E 1
E 0
1,000 E 1
E 0
2 3 alog 1,000 log E 1
E 0
b
2 3 a3 log E 1
E 0
b
log 1,000 log 10 3 3
Distribute.
2
2 3 (3) 2 3 log E 1
E 0
2 M 1
3 log E 1
E 0
is M 1 !
An earthquake with 1,000 times the energy of another has a Richter scale reading of 2 more
than the other.
MATCHED PROBLEM 3
If the energy release of one earthquake is 10,000 times that of another, how much larger is
the Richter scale reading of the larger than the smaller?
ROCKET FLIGHT: The theory of rocket flight uses advanced mathematics and physics to show
that the velocity v of a rocket at burnout (depletion of fuel supply) is given by
v c ln
W t
W b
Rocket equation (3)
where c is the exhaust velocity of the rocket engine, is the takeoff weight (fuel, structure,
and payload), and W b is the burnout weight (structure and payload).
Because of the Earth’s atmospheric resistance, a launch vehicle velocity of at least
9.0 kilometers per second is required to achieve the minimum altitude needed for a stable
orbit. Formula (3) indicates that to increase velocity v, either the weight ratio W tW b must
be increased or the exhaust velocity c must be increased. The weight ratio can be increased
by the use of solid fuels, and the exhaust velocity can be increased by improving the fuels,
solid or liquid.
W t
EXAMPLE 4 Rocket Flight Theory
A typical single-stage, solid-fuel rocket may have a weight ratio W tW b 18.7 and an
exhaust velocity c 2.38 kilometers per second. Would this rocket reach a launch velocity
of 9.0 kilometers per second?