Question

the population of a small town behaves according to the function $f(x) = 7000(0.98)^x$. at what rate is the population decaying?
○ the population is decaying at a rate of 98%.
● the population is decaying at a rate of 2%.
○ this does not model a decay.
○ the population is decaying at a rate of 70,000 per period.

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential decay function is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the decay rate (expressed as a decimal), and \( x \) is the time variable.

Step2: Compare with given function

The given function is \( f(x) = 7000(0.98)^x \). We can rewrite \( 0.98 \) as \( 1 - 0.02 \). So comparing with \( f(x) = a(1 - r)^x \), we have \( 1 - r = 0.98 \), which means \( r = 1 - 0.98 = 0.02 \). To express this as a percentage, we multiply by 100, so \( r = 2\% \). This shows the population is decaying at a rate of 2%.

Answer:

The population is decaying at a rate of 2% (the second option: "The population is decaying at a rate of 2%").