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You may assume that electricity prices remain constant, and inflation can be ignored. Assume the power generated by the solar panels is the same year-round.
Section A: Population Growth
Figure 1: Exponential population growth graph
The curve shows the rate of increase in population over time. In the given graph, P0 is the y-intercept and represents the population level at time (t) = 0.
Re-arranging the formula to solve for r, we get:
Taking natural log for both sides, we get:
Here, 2011 = 0 and 2015= 4
Here, Pt = 1371 (population in 2015)
P0 = 1344 (population in 2011)
t = time = 4
= log () = log (1.02) = (0.0086) = 0.00215 0.002
Here, Pt = 1311 (population in 2015)
P0 = 1247 (population in 2011)
log () = log (1.051) = (0.0216) = 0.0054 0.005
Hence, the annual growth rate of population for China is 0.002 or 0.2% and for India is 0.005 or 0.5%.
1344 × 1.038731 = 1396.055
= = 1344 × 1.053376 = 1415.737
= = 1247 × 1.099659 = 1371.275
= = 1247 × 1.138828 = 1420.119
Thus, the estimated population for China and India using the general formula of is given as follows:
Table 1: Estimated population (in millions)
Equation (a) for China is given as:
Equation (b) for China is given as:
Equating equation a and b, we get:
Solving the above equation for t:
Putting natural log on both sides, we get:
log (1.078) = 0.003t
= 10.8442720082 10.8 years
Population in 2011 = 1344
Doubling the values and putting in the general equation, we get:
Solving the above equation for t, we get:
log (2) = 0.002t
t = 0.3010/0.002 = 150.51 or 150 years 6 months
Population in 2011 = 1247
log (2) = 0.005t
t = 0.3010/0.005 = 60.20 or 60 years 2 months
Thus, the population of China will take 150.5 years to double in size, while India’s population will double in approximately 60 years 2 months.
Section B: Malthusian Disaster
Pt = ao (1 + r)t
St = b + mt
Where, S = food supply
b = initial quantity or y intercept
m = change in supply
t = time
Population Growth: Pt = 1000 (1+ 0.105)t
Food Supply: St = 2100 + 160t
Solving the above equation for values t = 0, 10, 20, 30,40 and 50, we get:
Table 2: Population and Food Supply at different time intervals
Figure 2: Population and Food Supply Graph
From the figure shown above, it can be seen that rate of growth in population is much higher as compared to increase in food supply. While the equation for food supply is depicted by a straight line, the population equation is a J-shaped curve that shows greater increase in population as compared to change in time.
Population growth = Food supply
1000 (1+ 0.105)t = 2100 + 160t
Solving for t, it is found that the population will start experiencing food shortage after 15 years.
1000 (1+ 0.105)t = 19,000
t log (1.105) = log (19)
t × 0.0433 = 1.27875
t = 1.27875/0.0433 = 29.9
Thus, the year in which the population’s demand for food exceeds 19,000 per year is 30th year.
1000 (1+ 0.101)t = 19,000
t log (1.101) = log (19)
t × 0.041787 = 1.27875
t = 1.27875/0.041787 = 30.60 or 30 years 7 months
Thus, after 30 years and 7 months, the population demand for food will exceed 19,000 tonnes.
Table 3: Quarterly Electricity Bill from Energy Provider
First 1000 kwh
> 1000 kWh
15 years cost
Table 4: Present Values of electricity bill
50% Renewable Energy
Table 5: Quarterly cost of next best alternative
30 years cost
*For calculating costs, it is assumed that solar panels were re-installed after 15 years.
Table 5: Present Values of electricity bill
Current bill without solar energy
Bill with solar energy
PVIF At 5.15%
Table 6: Calculation of present value of electricity bill
Net Inflow = Savings = value of electricity bill without solar energy – electricity bill with solar energy
NPV = Saving – Initial Investment = $136,620.359 - $14,000 = $122,620.359
Table 7: Calculation of present value of electricity bill
Net Inflow = Savings = value of electricity bill without solar energy – electricity bill with solar energy = $299,500.08 - $169,990.28
NPV = Saving – Initial Investment = $129,509.8 - $14,000 = $115,509.8
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