1. In one of the cities it has been determined that average yearly increase of employment in services
is equal 6.5%, while during last 8 years industrial employment grew with constant rate from 150000 to
225000. Calculate value of the algometry coefficient between services and industrial employment.
2. In one German city it has been identified that during last 3.5 years population decreased from
350000 to 345000. Forecast population 15 years from now, assuming that current average decrease
rate will not change during next 15 years.
3. It has been identified that growth rates of population and service employment in the cities are in
relation 1:1.25. In one of the cities with population 450000, employment in services equals 55000.
What will be the number of employees in services when population will grow to 500000.
4. Find the state of equilibrium nearest to the origin of the coordinate system of the system described
by the differential equation: d(x)/dt= sin(x)/x
Calculate the function that defines behaviour of the system variable close to indicated state of
equilibrium (use the representation of the original function as firs two monomials of appropriate
polynomial series)
5. The value of fixed assets is decreasing in accordance to the exponential decay with the
exponential rate β. After what time, the value of fixed assets decreases by half?
6. Yearly amortization rate of industrial buildings is equal 2.5%. After what time, the initial value of the
buildings will decrease to half of its initial value, if amortization is calculated in accordance of
exponential decay?
7. After what time the value of residential building will decrease to 40% of its initial value (at the date
of finishing construction) id depreciation process follows exponential decay, and yearly depreciation
of residential buildings equals 1.5% ?
8. Evident regularity of the world economic growth has been identified (Kondratiev’s wave). The
cycles are repeating every 50 years. Calculate the constant of imaginary part of exponential growth
coefficient that determines that cycle period is close to 50 years. Identify the unit of this constant.
9. Chart below shows crude oil consumption by China between years 1980-2010. Taking values from
years 1985 and 2010, calculate the forecasted consumption of the crude oil by China in the year
2025, assuming that the trend will not change.
10.
The table presents population of two countries in the years 2000 and 2010. When the population of
both countries will equal and what will be their population assuming that increase of the population of
both countries follows exponential growth.
𝑑𝑃
11. Certain system behaves in accordance to equation: 𝑑𝑡 = 1 − 𝑒 (𝑎−𝑏𝑃) where P is a system state
variable. Find the state of dynamic equilibrium of thi system (value of the P is not changing).
12 (n/a)
13. Graph on right shows the forecast of the change of the
population of selected European countries (in %) between
2005 and 2050. When it may be expected that population
of France exceed the population of Germany if population
of France in 2010 was equal 61mln and population of
Germany in year 2012 was81 million (assume that
populations of both countries change in accordance to
exponential growth.
14. The chart on right shows three
scenarios of world population
forecast to year 2050. Identify the
law that has been used to sketch
the „Low” scenario (bottom curve)
and make the rough estimation of
the coefficient of this law assuming
that between years 1950-1980
population growth can be
considered as close to exponential
law.
15. Graph on right presents new forecast of
world population up to year 2100, done by UN
experts. Identify the law that has been used for
total population forecasting and approximate
factors of this law, assuming that between
1950 –1995 population growths can be
considered as close to exponential growth.
16. Belgium and Nederland are known as closely linked
socioeconomic systems. On the graph one can find the
forecast of national population changes (forecasted change
in % between 2005 and 2050. When it can be expected that sum of the populations of both countries
will increase by 10% relative to population in 2005 if population of Belgium in 2008 was 10 million,
population of Nederland in 2011 was 16 million, and populations of each country are growing in
accordance to exponential growth.
17. It is well observed that between years 2001-2008 export to import
balance of Poland is improving. In the table beside, values of imported
and exported goods are presented (in billions PLN). Find the year
when it can be expected that value of import will equal value of export
if both will grow in accordance to exponential growth.
𝑑𝑄
18. Q is a system variable, and its dynamics is defined as 𝑑𝑡 = 𝑒 (𝑎𝑄+𝑐) + 𝑑 . Find the value of the
system variable when system is in the dynamic equilibrium.
19 (n/a)
20. The chart beside shows the changes of North Africa
population between years1960-2010. Calculate the reverse
forecast of this population in the year 1900 assuming that
during entire time the trend was very stable.
21. Average prices of land on the area of cities are decreasing in the exponential function of the
distance to the centre the city, with the coefficient α where distances measured in [km], In what
distance from the centre of the city one can expect the price decreases by half comparing to the
highest prices?
𝑑𝑃
22. Behaviour of the system is described by following equation 𝑑𝑡 = log⁡[(0.05 ∗ 𝑃) − 6] , where P is a
system variable. Find the dynamic state of equilibrium of this system (value of P when system
reaches equilibrium).
23. Recently research of mobile phone use has been done in Poland. As an outcome the dynamic of
𝑑𝑞
this process has been identified in the form of the equation 𝑑𝑡 = 0.4𝑞 + 𝑎11 𝑞 2 where q denotes the
number of mobile phones per 1000 inhabitants. In the year 2003, number of mobile phones per 1000
inhabitants equalled 400. What should be the value of a11 if it has identified in western European
countries that the level of saturation is three times bigger than ratio in Poland in 2003?
24. During recent years it has been found in polish cities that brut area of residential land use and
brut area of land use for services are growing in accordance to exponential law with coefficients 0.02
and 0.035 accordingly.
In one of the cities in year 2012 the residential land use area was 9 km2 while services land use was
1.6 km2 at the same time. What is expected area of services in this city, when residential area will
grow to 15 km2?