Question

fill in the blank 17 points
the function $v(t)=32,000 \cdot 0.96^{t}$ represents the value of a new car as a function of time.
the a-value is ________ which represents the ________
the b-value is ________ meaning that the function ______ by ________ %

Explanation:

Step1: Identify initial value (a-value)

The exponential function form is $V(t)=a \cdot b^t$, where $a$ is the initial value at $t=0$. Substitute $t=0$:
$V(0)=32,000 \cdot 0.96^0 = 32,000 \cdot 1 = 32,000$
This is the initial value of the car.

Step2: Identify growth/decay factor (b-value)

In $V(t)=32,000 \cdot 0.96^t$, $b=0.96$. Since $0

Step3: Calculate decay percentage

Subtract the decay factor from 1 to find the percentage decrease:
$1 - 0.96 = 0.04 = 4\%$

Answer:

The a-value is $\boldsymbol{32000}$ which represents the initial value of the new car
The b-value is $\boldsymbol{0.96}$ meaning that the function decreases by $\boldsymbol{4}$%