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)dt=1500 mean? Choose 1 answer: (A) The town's population grew by 1500 people between t=3 and t=8. B The town's population grew at a rate of 1500 people per year at t=8. (C) The average rate at which the population grew between t=3 and t=8 is 1500 people per year. (D) At t=8, the town's population was 1500 people.

Understand integral expression: Understand the integral expression. The integral ∫_(3)^(8)r(t)dt represents the total population growth from time t=3 to t=8. The function r(t) is the rate of population growth per year, and the integral sums up this growth rate over the time interval from 3 to 8 years.Interpret the equation: Interpret the equation. The equation 1000 + ∫_(3)^(8)r(t)dt = 1500 means that if we start with a population of 1000 people at time t=3 and add the total population growth from t=3 to t=8, we end up with a population of 1500 people at time t=8.Analyze answer choices: Analyze the answer choices. (A) This choice suggests that the population grew by 1500 people between t=3 and t=8. This is incorrect because the equation states that the population was 1000 at t=3 and became 1500 at t=8, which means the growth was 500 people, not 1500.Continue analyzing choices: Continue analyzing the answer choices. (B) This choice suggests that the population grew at a rate of 1500 people per year at t=8. This is incorrect because the equation does not provide information about the rate of growth at a specific time, only the total growth over a period.Continue analyzing choices: Continue analyzing the answer choices. (C) This choice suggests that the average rate of population growth between t=3 and t=8 is 1500 people per year. This is incorrect because the total growth over the 5-year period is 500 people, not 1500.Continue analyzing choices: Continue analyzing the answer choices. (D) This choice suggests that at t=8, the town's population was 1500 people. This is correct because the equation shows that starting with a population of 1000 at t=3 and adding the growth from t=3 to t=8 results in a population of 1500 at t=8.

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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)dt=1500
mean?
Choose 1 answer:
(A) The town's population grew by 1500 people between
t=3 and
t=8.
B The town's population grew at a rate of 1500 people per year at
t=8.
(C) The average rate at which the population grew between
t=3 and
t=8 is 1500 people per year.
(D) At
t=8, the town's population was 1500 people.

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. $\newline$ What does $1000+\int_{3}^{8} r(t) d t=1500$ mean? $\newline$ Choose $1$ answer: $\newline$ (A) The town's population grew by $1500$ people between $t=3$ and $t=8$ . $\newline$ (B) The town's population grew at a rate of $1500$ people per year at $t=8$ . $\newline$ (C) The average rate at which the population grew between $t=3$ and $t=8$ is $1500$ people per year. $\newline$ (D) At $t=8$ , the town's population was $1500$ people.

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Math Problems

Grade 7

Evaluate two-variable equations: word problems

Full solution

Q. 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.

\newline

What does

1000+\int_{3}^{8} r(t) d t=1500

mean?

\newline

Choose

1

answer:

\newline

(A) The town's population grew by

1500

people between

t=3

and

t=8

\newline

(B) The town's population grew at a rate of

1500

people per year at

t=8

\newline

t=3

and

t=8

1500

people per year.

\newline

(D) At

t=8

, the town's population was

1500

people.

Understand integral expression: Understand the integral expression. $\newline$ The integral $\int_{3}^{8} r(t) \, dt$ represents the total population growth from time $t=3$ to $t=8$ . The function $r(t)$ is the rate of population growth per year, and the integral sums up this growth rate over the time interval from $3$ to $8$ years.
Interpret the equation: Interpret the equation. $\newline$ The equation $1000 + \int_{3}^{8}r(t)dt = 1500$ means that if we start with a population of $1000$ people at time $t=3$ and add the total population growth from $t=3$ to $t=8$ , we end up with a population of $1500$ people at time $t=8$ .
Analyze answer choices: Analyze the answer choices. $\newline$ (A) This choice suggests that the population grew by $1500$ people between $t=3$ and $t=8$ . This is incorrect because the equation states that the population was $1000$ at $t=3$ and became $1500$ at $t=8$ , which means the growth was $500$ people, not $1500$ .
Continue analyzing choices: Continue analyzing the answer choices. $\newline$ (B) This choice suggests that the population grew at a rate of $1500$ people per year at $t=8$ . This is incorrect because the equation does not provide information about the rate of growth at a specific time, only the total growth over a period.
Continue analyzing choices: Continue analyzing the answer choices. $\newline$ (C) This choice suggests that the average rate of population growth between $t=3$ and $t=8$ is $1500$ people per year. This is incorrect because the total growth over the $5$ -year period is $500$ people, not $1500$ .
Continue analyzing choices: Continue analyzing the answer choices. $\newline$ (D) This choice suggests that at $t=8$ , the town's population was $1500$ people. This is correct because the equation shows that starting with a population of $1000$ at $t=3$ and adding the growth from $t=3$ to $t=8$ results in a population of $1500$ at $t=8$ .