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Suppose our lake began with 1000 fish, and 10% of the fish have surviving offspring each year. Since we start with 1000 fish, Po= 1000. How do we calculate P₁? The new population will be the old population, plus an additional 10%. Symbolically: P₁ = Po + 0.10Po Notice this could be condensed to a shorter form by factoring:

Suppose our lake began with 1000 fish, and 10% of the fish have surviving offspring each year. Since we start with 1000 fish, Po= 1000. How do we calculate P₁? The new population will be the old population, plus an additional 10%. Symbolically: P₁ = Po + 0.10Po Notice this could be condensed to a shorter form by factoring:

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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Suppose that every year, only 10% of the fish in a lake have surviving offspring. If there were 100 fish in the lake last year, there would now be 110 fish. If there were 1000 fish in the lake last year, there would now be 1100 fish. Absent any inhibiting factors, populations of people and animals tend to grow by a percent of the existing population each year. Suppose our lake began with 1000 fish, and 10% of the fish have surviving offspring each year. Since we start with 1000 fish, Po 1000. How do we calculate P₁? The new population will be the old population, plus an additional 10%. Symbolically: = P1 = Po + 0.10Po Notice this could be condensed to a shorter form by factoring: P₁= Po+0.10Po = 1Po + 0.10Po= (1+0.10)Po= 1.10Po While 10% is the growth rate, 1.10 is the growth multiplier. Notice that 1.10 can be thought of as "the original 100% plus an additional 10%" For our fish population, P₁ = 1.10(1000) = 1100 We could then calculate the population in later years. Fill in the blanks. P₂ = 1.10P₁ = 1.10( P3= 1.10P2= 1.10( P4 = 1.10 P3= 1.10P2 = 1.10( Pn = 1.10 = 1.100 General Recursive Form: Pn= + =
Transcribed Image Text:Suppose that every year, only 10% of the fish in a lake have surviving offspring. If there were 100 fish in the lake last year, there would now be 110 fish. If there were 1000 fish in the lake last year, there would now be 1100 fish. Absent any inhibiting factors, populations of people and animals tend to grow by a percent of the existing population each year. Suppose our lake began with 1000 fish, and 10% of the fish have surviving offspring each year. Since we start with 1000 fish, Po 1000. How do we calculate P₁? The new population will be the old population, plus an additional 10%. Symbolically: = P1 = Po + 0.10Po Notice this could be condensed to a shorter form by factoring: P₁= Po+0.10Po = 1Po + 0.10Po= (1+0.10)Po= 1.10Po While 10% is the growth rate, 1.10 is the growth multiplier. Notice that 1.10 can be thought of as "the original 100% plus an additional 10%" For our fish population, P₁ = 1.10(1000) = 1100 We could then calculate the population in later years. Fill in the blanks. P₂ = 1.10P₁ = 1.10( P3= 1.10P2= 1.10( P4 = 1.10 P3= 1.10P2 = 1.10( Pn = 1.10 = 1.100 General Recursive Form: Pn= + =
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