Question

The value of a car that depreciates over time can be modeled by the function Z(t)=18000(0. 9)^{3t+2}. Z(t)=18000(0. 9)

3t+2. Write an equivalent function of the form Z(t)=ab^t

Answer 1

Step-by-step explanation: To write the given function Z(t) = 18000(0.9)^(3t+2) in the form Z(t) = ab^t, we need to simplify the expression.

Let's break it down step by step:

Z(t) = 18000(0.9)^(3t+2)

First, we can rewrite 0.9 as a fraction: 0.9 = 9/10.

Z(t) = 18000(9/10)^(3t+2)

Next, we can simplify the expression inside the parentheses:

(9/10)^(3t+2) = [(9/10)^3]^t * (9/10)^2

Calculating the numerator and denominator of the first term separately:

[(9/10)^3] = (9^3)/(10^3) = 729/1000

Now, we substitute the simplified expression back into Z(t):

Z(t) = 18000 * (729/1000)^t * (9/10)^2

Simplifying further:

Z(t) = 18000 * (729/1000)^t * (81/100)

Finally, we can rewrite the expression as:

Z(t) = (18000 * 81/100) * [(729/1000) * (81/100)]^t

Letting a = 18000 * 81/100 and b = (729/1000) * (81/100), the equivalent function becomes:

Z(t) = ab^t