16. a mouse population starts with 2,000 mice...
16. a mouse population starts with 2,000 mice and grows at a rate of 5% per year. the number of mice after t years can be modeled by the equation, p(t)=2000(1.05)^t. what is the average rate of change in the number of mice between the second year and the fifth year, rounded to the nearest whole number? (1) 116 (2) 348 (3) 2205 (4) 2553
Answer
# Explanation:
## Step1: Find $P(2)$
Substitute $t = 2$ into $P(t)=2000(1.05)^t$.
$P(2)=2000\times(1.05)^2=2000\times1.1025 = 2205$
## Step2: Find $P(5)$
Substitute $t = 5$ into $P(t)=2000(1.05)^t$.
$P(5)=2000\times(1.05)^5=2000\times1.27628 = 2552.56$
## Step3: Calculate average rate of change
The formula for average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $b = 5$, $f(t)=P(t)$.
Average rate of change$=\frac{P(5)-P(2)}{5 - 2}=\frac{2552.56-2205}{3}=\frac{347.56}{3}\approx116$
# Answer:
(1) 116