Question

a certain fish grows so that after t years of life its length is given by l(t)=12 - 1/4^t inches. calculate the average rate of growth over the first year of life. suggested tutorial: learn it: find the average rate of change of a function on an interval resources read it

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = L(t)$ over the interval $[a,b]$ is $\frac{L(b)-L(a)}{b - a}$. Here, $a = 0$, $b = 1$, and $L(t)=12-\frac{1}{4^{t}}$.

Step2: Calculate $L(1)$

Substitute $t = 1$ into $L(t)$: $L(1)=12-\frac{1}{4^{1}}=12 - \frac{1}{4}=\frac{48 - 1}{4}=\frac{47}{4}$.

Step3: Calculate $L(0)$

Substitute $t = 0$ into $L(t)$: $L(0)=12-\frac{1}{4^{0}}=12 - 1 = 11$.

Step4: Calculate the average rate of change

$\frac{L(1)-L(0)}{1 - 0}=\frac{\frac{47}{4}-11}{1}=\frac{\frac{47}{4}-\frac{44}{4}}{1}=\frac{\frac{47 - 44}{4}}{1}=\frac{3}{4}$.

Answer:

$\frac{3}{4}$