Page 35 - Advanced engineering mathematics

1.1 Terminology and Separable Equations 15

30. Determine the time it takes to drain a spherical tank P(t) with respect to time should be influenced by
with a radius of 18 feet if it is initially full of water, growth factors (for example, current population) and
which drains through a circular hole with a radius of 3 also factors tending to retard the population (such as
inches in the bottom of the tank. Use k = 0.8. limitations on food and space). He formed a model
by assuming that growth factors can be incorporated
31. A tank shaped like a right circular cone, vertex down,
into a term aP(t) and retarding factors into a term
is 9 feet high and has a diameter of 8 feet. It is initially
2
−bP(t) with a and b as positive constants whose val-
full of water.
ues depend on the particular population. This led to his
(a) Determine the time required to drain the tank
logistic equation
through a circular hole with a diameter of 2 inches
2

at the vertex. Take k = 0.6. P (t) = aP(t) − bP(t) .
(b) Determine the time it takes to drain the tank if it is Note that, when b = 0, this is the exponential model.
inverted and the drain hole is of the same size and Solve the logistic model, subject to the initial
shape as in (a), but now located in the new (flat) condition P(0) = p 0 , to obtain
base.
ap 0 at
32. Determine the rate of change of the depth of water P(t) = a − bp 0 + bp 0 e at e .
in the tank of Problem 31 (vertex at the bottom) if
the drain hole is located in the side of the cone 2 This is the logistic model of population growth.Show
feet above the bottom of the tank. What is the rate of that, unlike exponential growth, the logistic model
change in the depth of the water when the drain hole produces a population function P(t) that is bounded
is located in the bottom of the tank? Is it possible to above and increases asymptotically toward a/b as
determine the location of the drain hole if we are told t →∞. Thus, a logistic model produces a population
the rate of change of the depth and the depth of the function that never grows beyond a certain value.
water in the tank? Can this be done without knowing 34. Continuing Problem 33, a 1920 study by Pearl and
the size of the drain opening? Reed (appearing in the Proceedings of the National
33. (Logistic Model of Population Growth) In 1837, Academy of Sciences) suggested the values
the Dutch biologist Verhulst developed a differential −10
a = 0.03134, b = (1.5887)10
equation to model changes in a population (he was
studying fish populations in the Adriatic Sea). Ver- for the population of the United States. Table 1.1
hulst reasoned that the rate of change of a population gives the census data for the United States in ten year

TABLE 1.1 Census data for Problems 33 and 34, Section 1.1.
Year Population P(t) Percent error Q(t) Percent error
1790 3,929,214
1800 5,308,483
1810 7,239,881
1820 9,638,453
1830 12,886,020
1840 17,069,453
1850 23,191,876
1860 31,443,321
1870 38,558,371
1880 50,189,209
1890 62,979,766
1900 76,212,168
1910 92,228,496
1920 106,021,537
1930 123,202,624
1940 132,164,569
1950 151,325,798
1960 179,323,175
1970 203,302,031
1980 226,547,042

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