College Algebra 9th txtbk

SECTION 5–4 Logarithmic Models 369
SOLUTION We can use the rocket equation (3) with c 2.38 and W tW b 18.7:
v c ln W t
W b
2.38 ln 18.7
6.97 kilometers per second
The velocity of the launch vehicle is far short of the 9.0 kilometers per second required to
achieve orbit. This is why multiple-stage launchers are used—the deadweight from a preceding
stage can be jettisoned into the ocean when the next stage takes over.
MATCHED PROBLEM 4
A launch vehicle using liquid fuel, such as a mixture of liquid hydrogen and liquid oxygen,
can produce an exhaust velocity of c 4.7 kilometers per second. However, the weight ratio
W tW b must be low—around 5.5 for some vehicles—because of the increased structural
weight to accommodate the liquid fuel. How much more or less than the 9.0 kilometers per
second required to reach orbit will be achieved by this vehicle?
Z Data Analysis and Regression
Based on the logarithmic graphs we studied in the last section, when a quantity increases
relatively rapidly at first, but then levels off and increases very slowly, it might be a good
candidate to be modeled by a logarithmic function. Most graphing calculators with regression
commands can fit functions of the form y a b ln x to a set of data points using
the same techniques we used earlier for other types of regression.
EXAMPLE 5 Home Ownership Rates
Table 3 Home Ownership
Rates
Home Ownership
Year Rate (%)
1940 43.6
1950 55.0
1960 61.9
1970 62.9
1980 64.4
1990 64.2
2000 67.4
SOLUTIONS
The U.S. Census Bureau published the data in Table 3 on home ownership rates.
A logarithmic model for the data is given by
R 36.7 23.0 ln t
where R is the home ownership rate and t is time in years with t 0 representing 1900.
(A) Use the model to predict the home ownership rate in 2015.
(B) Compare the actual home ownership rate in 1950 to the rate given by the model.
(A) The year 2015 is represented by t 115. Evaluating
R 36.7 23.0 ln t
at t 115 predicts a home ownership rate of 72.4% in 2015.
(B) The year 1950 is represented by t 50. Evaluating
R 36.7 23.0 ln t
at t 50 gives a home ownership rate of 53.3% in 1950. The actual home ownership
rate in 1950 was 55%, approximately 1.7% greater than the number given by
the model.