An agricultural department monitored the population of honeybee colonies in two adjacent conservation areas starting in 2018 (). In Area 1, the number of colonies increases by a constant 230 colonies each year. In Area 2, the number of colonies increases by a constant 10% each year. At , both areas had 2,000 colonies. If the model for Area 1 predicts colonies at and the model for Area 2 predicts colonies at , what is the value of ?
Cevap: 28
Cevap
28
To find the value of , the populations projected by each model at must be calculated.
For Area 1, the growth is linear with an initial population of 2,000 and a constant annual increase of 230. The model is . At , the population is .
For Area 2, the growth is exponential with an initial population of 2,000 and a constant annual increase of 10%. The model is . At , the population is .
Subtracting the exponential model output from the linear model output yields .
For Area 1, the growth is linear with an initial population of 2,000 and a constant annual increase of 230. The model is . At , the population is .
For Area 2, the growth is exponential with an initial population of 2,000 and a constant annual increase of 10%. The model is . At , the population is .
Subtracting the exponential model output from the linear model output yields .
Adım Adım Çözüm
Anahtar Kavram
Linear growth increases by a constant amount per unit of time, whereas exponential growth increases by a constant percentage (or multiplies by a constant factor) per unit of time.
Alternatif Yöntem
Instead of setting up equations, the population values can be calculated step-by-step for each year.
Year 1 ():
Area 1:
Area 2:
Year 2 ():
Area 1:
Area 2:
Year 3 ():
Area 1:
Area 2:
Subtracting the two values at Year 3 gives the final result: .
Year 1 ():
Area 1:
Area 2:
Year 2 ():
Area 1:
Area 2:
Year 3 ():
Area 1:
Area 2:
Subtracting the two values at Year 3 gives the final result: .
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