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The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 70 years? How fast is the population growing at t = 70?

The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 70 years? How fast is the population growing at t = 70?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 70 years? How fast is the population growing at t = 70? Step 1 dp We are given that the population of a town grows at a rate proportional to the population at time t. In other words, if the population at time t is given by P(t), then for some constant k we have = KP. dt In general with such a proportional relationship, we know that the population function has the form P(t) = cekt. The reason for this is that = KP is a simple separable linear equation. dt dp dp P = cdt In(IP) kt+C P = cekt Knowing the initial population, we can solve for the coefficient c. P(0) = cek(0) = C As the initial population is 500, we know that c =
Transcribed Image Text:The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 70 years? How fast is the population growing at t = 70? Step 1 dp We are given that the population of a town grows at a rate proportional to the population at time t. In other words, if the population at time t is given by P(t), then for some constant k we have = KP. dt In general with such a proportional relationship, we know that the population function has the form P(t) = cekt. The reason for this is that = KP is a simple separable linear equation. dt dp dp P = cdt In(IP) kt+C P = cekt Knowing the initial population, we can solve for the coefficient c. P(0) = cek(0) = C As the initial population is 500, we know that c =
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