The variation in size from year to year of a particular population can be
modelled by an exponential model with annual proportionate growth rate
0.1246. The size of the population at the start of the initial year is 360.
Choose the TWO options that give, as predicted by the model,
(a) the population size after 10 years;
(b) the number of years for the population size to reach at least 2500.
Options
A 15 B 17 C 20 D 24
E 849 F 1035 G 1165 H 1310
population=360(1+r)^10=360*1.1246^10=1165
check that.
thanks
To find the population size after a certain number of years using the exponential growth model, we can use the formula:
P(t) = P0 * e^(rt)
Where:
P(t) = Population size at time t
P0 = Initial population size
r = Annual proportionate growth rate
t = Number of years
(a) To find the population size after 10 years, we can substitute the given values into the formula:
P(10) = 360 * e^(0.1246 * 10)
Calculating this value:
P(10) ≈ 360 * e^(1.246)
Using the exponential function, e^x, we can calculate the value inside the parentheses:
e^(1.246) ≈ 3.4745
Multiplying this value by 360:
P(10) ≈ 360 * 3.4745 ≈ 1248.42
Therefore, the population size after 10 years, predicted by the model, is approximately 1248.42.
(b) To find the number of years for the population size to reach at least 2500, we need to solve the equation:
2500 = 360 * e^(0.1246t)
Dividing both sides of the equation by 360:
6.94 ≈ e^(0.1246t)
To isolate t, we can use the natural logarithm function (ln):
ln(6.94) ≈ ln(e^(0.1246t))
Using the logarithmic rule, ln(e^x) = x, we get:
ln(6.94) ≈ 0.1246t
Dividing both sides of the equation by 0.1246:
t ≈ ln(6.94) / 0.1246 ≈ 17.69
Therefore, it takes approximately 17.69 years for the population size to reach at least 2500.
From the given options, the correct answers are:
(a) Population size after 10 years: Option E 849
(b) Number of years to reach at least 2500: Option B 17