how do you compare the approximate rate of change of the function over the interval 400 < p < 800 with that…

how do you compare the approximate rate of change of the function over the interval 400 < p < 800 with that of the interval 800 < p < 1600? what do these rates of change mean in the context of the problem? which of the statements below support the answers to the questions above? select all that apply. it took the population 10 years (1 decade) to grow by 1/400,000 people from 400,000 to 800,000 people. the rate of change over the interval 800 < p < 1600 was 800. the rate of change over the interval 400 < p < 800 was less than the rate of change over the interval 800 < p < 1600. it took the population 10 years (1 decade) to grow by 400,000 people from 400,000 to 800,000 people. the rate of change over the interval 800 < p < 1600 was 1/800. the rate of change over the interval 400 < p < 800 and that over the interval 800 < p < 1600 were equal. the rate of change over the interval 400 < p < 800 is 1/400. it took the population 10 years (1 decade) to grow by 1/800,000 people from 800,000 to 1,600,000 people. the rate of change over the interval 400 < p < 800 was 400. it took the population 10 years (1 decade) to grow by 800,000 people from 800,000 to 1,600,000 people. the rate of change over the interval 400 < p < 800 was greater than the rate of change over the interval 800 < p < 1600.

how do you compare the approximate rate of change of the function over the interval 400 < p < 800 with that of the interval 800 < p < 1600? what do these rates of change mean in the context of the problem? which of the statements below support the answers to the questions above? select all that apply. it took the population 10 years (1 decade) to grow by 1/400,000 people from 400,000 to 800,000 people. the rate of change over the interval 800 < p < 1600 was 800. the rate of change over the interval 400 < p < 800 was less than the rate of change over the interval 800 < p < 1600. it took the population 10 years (1 decade) to grow by 400,000 people from 400,000 to 800,000 people. the rate of change over the interval 800 < p < 1600 was 1/800. the rate of change over the interval 400 < p < 800 and that over the interval 800 < p < 1600 were equal. the rate of change over the interval 400 < p < 800 is 1/400. it took the population 10 years (1 decade) to grow by 1/800,000 people from 800,000 to 1,600,000 people. the rate of change over the interval 400 < p < 800 was 400. it took the population 10 years (1 decade) to grow by 800,000 people from 800,000 to 1,600,000 people. the rate of change over the interval 400 < p < 800 was greater than the rate of change over the interval 800 < p < 1600.

Answer

Explanation:

Step1: Recall rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $a<x<b$ is $\frac{f(b)-f(a)}{b - a}$. Here, assume the function is related to population $P$, and if we consider the time - related context (since it mentions decades), let's assume the function is $y$ with respect to $P$. But if we just consider the change in population values, for the interval $400 < P<800$, the rate of change $r_1=\frac{\text{Change in function value}}{\text{Change in }P}$. Similarly for $800 < P<1600$, the rate of change $r_2=\frac{\text{Change in function value}}{\text{Change in }P}$.

Step2: Analyze population change statements

If the population changes from $400000$ to $800000$ in 10 years, the change in population is $800000 - 400000=400000$. For the interval $400 < P<800$ (assuming $P$ is in some units related to population, say thousands), if we consider the rate of change in terms of the change in the function with respect to $P$, without knowing the function, if we just consider the change in population values, the rate of change $r_1$: If the population changes from $400000$ to $800000$ (a change of $400000$) and we assume a linear relationship with respect to $P$ (in thousands), the rate of change $r_1=\frac{400000}{800 - 400}=\frac{400000}{400}= 1000$ (if $P$ is in thousands). But if we consider the statements in terms of the given fractions and whole - number values:

  • If the population changes from $400000$ to $800000$ in 10 years, it grew by $400000$ people in 10 years.
  • For the interval $800 < P<1600$, if the population changes from $800000$ to $1600000$ in 10 years, it grew by $1600000 - 800000 = 800000$ people in 10 years. The rate of change $r_2=\frac{800000}{1600 - 800}=\frac{800000}{800}=1000$ (if $P$ is in thousands). But looking at the statements in terms of the given values:
    • The rate of change over the interval $400 < P<800$: If we assume the function is such that the change in the function value is related to the change in population, and the population changes from $400000$ to $800000$ (a change of $400000$) over an interval of $P$ from $400$ to $800$ (a change of $400$ in $P$ - units), the rate of change is $\frac{400000}{400}=1000$. But if we consider the statements as they are written:
      • It took the population 10 years (1 decade) to grow by $400000$ people from $400000$ to $800000$ people. This is correct.
      • The rate of change over the interval $800 < P<1600$: The population changes from $800000$ to $1600000$ (a change of $800000$) over an interval of $P$ from $800$ to $1600$ (a change of $800$ in $P$ - units), the rate of change is $\frac{800000}{800}=1000$. But considering the statements: It took the population 10 years (1 decade) to grow by $800000$ people from $800000$ to $1600000$ people. This is correct. And the rate of change over the interval $400 < P<800$ and $800 < P<1600$ are equal if we calculate them in the same way (change in population/change in $P$ - value).

Answer:

It took the population 10 years (1 decade) to grow by 400,000 people from 400,000 to 800,000 people. It took the population 10 years (1 decade) to grow by 800,000 people from 800,000 to 1,600,000 people. The rate of change over the interval $400 < P<800$ and that over the interval $800 < P<1600$ were equal.