The population of a town grows at a rate of $r(t)$ people per year (where $t$ is time in years). At $t=3$, the town's population was 1000 people.

What does $1000+\int_{3}^{8} r(t) d t=1500$ mean?

Question

The population of a town grows at a rate of $r(t)$ people per year (where $t$ is time in years). At $t=3$, the town's population was 1000 people.

What does $1000+\int_{3}^{8} r(t) d t=1500$ mean?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Interpret the given equation: $1000 + \int_{3}^{8} r(t) dt = 1500$. ‖ ‖ step 2 ⋮ Recognize that $1000$ represents the initial population of the town at $t=3$. ‖ ‖ step 3 ⋮ Understand that $\int_{3}^{8} r(t) dt$ represents the total population growth from year $3$ to year $8$. ‖ ‖ step 4 ⋮ Acknowledge that $1500$ is the population of the town at $t=8$. ‖ ‖ step 5 ⋮ Calculate the total population growth by subtracting the initial population from the population at $t=8$: $1500 - 1000$. ‖ ‖ step 6 ⋮ Conclude that the total population growth over the 5 years is $500$ people. ‖ ∻Answer∻ ⚹ The equation $1000 + \int_{3}^{8} r(t) dt = 1500$ means that the population of the town grew by 500 people over the 5 years from year 3 to year 8. ⚹ ∻Key Concept∻ ⚹Interpreting Integral Equations in Context⚹ ∻Explanation∻ ⚹The integral $\int_{a}^{b} r(t) dt$ in the context of population growth represents the total change in population from time $a$ to time $b$. The equation given combines this change with the initial population to find the population at the later time.⚹◊What is the interpretation of the expression $1000+\int_{3}^{8} r(t) dt=1500$ in the context of population growth?⍭ Generate me a similar question◊

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