Question

The population of California was 29.76 million in 1990 and 33.87 million in 2000. Use the model found in part a (f(x) =29.76(1.013)^t to find the average rate of change from 1990 to 1991 and from 2000 to 2001. (Round your answer to the nearest thousand.)

Answer 1

Answer:

The average rate of change from 1990 to 1991 is $0.387 \:\frac{million}{years}$

The average rate of change from 2000 to 2001 is $0.440 \:\frac{million}{years}$

Step-by-step explanation:

The average rate of change of function f(x) over the interval $a\leq x\leq b$ is given by

$\frac{f(b)-f(a)}{b-a}$

It is a measure of how much the function changed per unit, on average, over that interval.

From the information given:

• The function that models the population t years after 1990 is $f(t) =29.76(1.013)^t$
• The year 1990 is t = 0 and the year 2000 is t = 10.

1. The average rate of change from 1990 to 1991 is:

The interval is $0\leq x\leq 1$

$\frac{29.76(1.013)^1-29.76(1.013)^0}{1-0}\\\\\frac{1.013^1\cdot \:29.76-1\cdot \:29.76}{1-0}\\\\\frac{0.38688}{1-0}\\\\0.387 \:\frac{million}{years}$

2. The average rate of change from 2000 to 2001 is

The interval is $10\leq x\leq 11$

$\frac{29.76(1.013)^{11}-29.76(1.013)^{10}}{11-10}\\\\\frac{0.44022}{11-10}\\\\0.440 \:\frac{million}{years}$