Question

exponential decay functions
tennpo buys a new car for $20,000. the value of the car depreciates by 15% each year. if f(x) represents the value of the car after x years, which function represents the cars value?
$f(x) = 20,000(0.85)^x$
$f(x) = 20,000(0.15)^x$
$f(x) = 20,000(1.15)^x$
$f(x) = 20,000(1.85)^x$

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 rate of decay (as a decimal), and \( x \) is the time.

Step2: Identify Values for the Problem

Here, the initial value \( a = 20,000 \) (the price of the new car), and the decay rate \( r = 15\% = 0.15 \).

Step3: Substitute into the Formula

Substitute \( a = 20,000 \) and \( r = 0.15 \) into the decay formula: \( f(x)=20,000(1 - 0.15)^x \).
Simplify \( 1 - 0.15 = 0.85 \), so the function becomes \( f(x)=20,000(0.85)^x \).

Answer:

\( f(x) = 20,000(0.85)^x \) (the option corresponding to this function)