the population p, in hundreds, of a small mining town in the california gold rush, is given by the function…
the population p, in hundreds, of a small mining town in the california gold rush, is given by the function p = f(t), where t is in years since 1850. use the graph of f(t) to answer the following.\na) when did the population of the town reach zero?\nb) find two different time intervals for which the average rate of change of p with respect to t was zero.\na) the year the population of the town reached zero is
Answer
Explanation:
Step1: Identify population - zero point on graph
Look at the graph of $P = f(t)$. The population $P$ is zero when the curve intersects the $t$-axis.
Step2: Determine the year
Since $t$ is in years since 1850, and the curve intersects the $t$-axis at $t = 6$. So the year is $1850 + 6=1856$.
Step3: Recall average - rate - of - change formula
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. We want $\frac{f(b)-f(a)}{b - a}=0$, which means $f(b)=f(a)$.
Step4: Find intervals with zero average rate of change
From the graph, we can see that for the interval $[0,2]$, $P(0)=20$ and $P(2)=20$. Also for the interval $[6,8]$, $P(6)=0$ and $P(8)=0$.
Answer:
a) 1856 b) $[0,2]$, $[6,8]$