Questions: Find C and a so that f(x)=Ca^x models the situation described. State what the variable x represents in your formula. In 2000 a house was worth 240,000, and its value decreases by 9% each year thereafter. C= (Type an integer or a decimal.) a= (Type an integer or a decimal.) What is the correct interpretation of x ? A. The variable x represents the rate at which the worth of the house decreases. B. The variable x represents the initial worth of the house. C. The variable x represents the worth of the house t years after 2000. D. The variable x represents the number of years after 2000.

Find C and a so that f(x)=Ca^x models the situation described. State what the variable x represents in your formula.
In 2000 a house was worth 240,000, and its value decreases by 9% each year thereafter.
C= (Type an integer or a decimal.)
a= (Type an integer or a decimal.)

What is the correct interpretation of x ?
A. The variable x represents the rate at which the worth of the house decreases.
B. The variable x represents the initial worth of the house.
C. The variable x represents the worth of the house t years after 2000.
D. The variable x represents the number of years after 2000.

Transcript text: Find C and a so that $\mathrm{f}(\mathrm{x})=\mathrm{Ca}^{\mathrm{x}}$ models the situation described. State what the variable x represents in your formula. In 2000 a house was worth $\$ 240,000$, and its value decreases by $9 \%$ each year thereafter. $C=$ $\square$ (Type an integer or a decimal.) $a=$ $\square$ (Type an integer or a decimal.) What is the correct interpretation of $x$ ? A. The variable $x$ represents the rate at which the worth of the house decreases. B. The variable $x$ represents the initial worth of the house. C. The variable x represents the worth of the house $t$ years after 2000. D. The variable x represents the number of years after 2000.

Solution

Solution Steps

To model the situation, we need to identify the initial value of the house and the rate at which it decreases. The initial value, $C$, is the worth of the house in 2000, which is $240,000. The rate of decrease is 9% per year, so the base of the exponential function, $a$, will be $1 - 0.09 = 0.91$. The variable $x$ represents the number of years after 2000, which corresponds to option D.

Step 1: Identify the Initial Value

The initial value of the house in the year 2000 is given as $ C = 240000 $.

Step 2: Determine the Rate of Decrease

The house's value decreases by $ 9\% $ each year. Therefore, the remaining value after one year is $ 1 - 0.09 = 0.91 $. Thus, we have $ a = 0.91 $.

Step 3: Interpret the Variable $ x $

The variable $ x $ represents the number of years after 2000. This corresponds to option D.