Answer :
Alright, let’s work through the problem step by step:
### Step 1: Initial Given Conditions
- Initial value of the car, [tex]\( V_0 = \$20,000 \)[/tex]
- Annual depreciation rate, [tex]\( r = 18\% \)[/tex] or [tex]\( r = 0.18 \)[/tex]
### Step 2: Convert Annual Depreciation to Quarterly Depreciation
To find the quarterly depreciation factor:
1. First, find the annual depreciation factor:
[tex]\[ 1 - r = 1 - 0.18 = 0.82 \][/tex]
2. Take the fourth root of the annual factor to get the quarterly factor:
[tex]\[ \text{quarterly\_factor} = 0.82^{(1/4)} \approx 0.9516 \quad (\text{rounded to four decimal places}) \][/tex]
### Step 3: Finding the Quarterly Rate of Change
To find the quarterly rate of change, we need to determine how much value the car loses each quarter:
1. Subtract the quarterly factor from 1 to get the quarterly depreciation rate:
[tex]\[ \text{quarterly\_rate} = 1 - 0.9516 \approx 0.0484 \][/tex]
2. Convert this rate to a percentage:
[tex]\[ \text{quarterly\_rate\_percentage} = 0.0484 \times 100 = 4.84\% \][/tex]
### Step 4: Constructing the Function
The value of the car [tex]\( f(t) \)[/tex] after [tex]\( t \)[/tex] years can be expressed by considering its quarterly decrease:
1. Each year has 4 quarters, so after [tex]\( t \)[/tex] years, there are [tex]\( 4t \)[/tex] quarters.
2. The value of the car after [tex]\( 4t \)[/tex] quarters is given by:
[tex]\[ f(t) = V_0 \times (\text{quarterly\_factor})^{4t} \][/tex]
Substituting the values, we get:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \][/tex]
### Final Answer:
#### Function:
The function that represents the value of the car after [tex]\( t \)[/tex] years is:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \][/tex]
#### Quarterly Rate of Change:
The percentage rate of change per quarter, to the nearest hundredth of a percent, is:
[tex]\[ \boxed{4.84\%} \][/tex]
Thus, the complete function and the quarterly rate of change are:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \quad \text{and the quarterly rate is 4.84\%} \][/tex]
### Step 1: Initial Given Conditions
- Initial value of the car, [tex]\( V_0 = \$20,000 \)[/tex]
- Annual depreciation rate, [tex]\( r = 18\% \)[/tex] or [tex]\( r = 0.18 \)[/tex]
### Step 2: Convert Annual Depreciation to Quarterly Depreciation
To find the quarterly depreciation factor:
1. First, find the annual depreciation factor:
[tex]\[ 1 - r = 1 - 0.18 = 0.82 \][/tex]
2. Take the fourth root of the annual factor to get the quarterly factor:
[tex]\[ \text{quarterly\_factor} = 0.82^{(1/4)} \approx 0.9516 \quad (\text{rounded to four decimal places}) \][/tex]
### Step 3: Finding the Quarterly Rate of Change
To find the quarterly rate of change, we need to determine how much value the car loses each quarter:
1. Subtract the quarterly factor from 1 to get the quarterly depreciation rate:
[tex]\[ \text{quarterly\_rate} = 1 - 0.9516 \approx 0.0484 \][/tex]
2. Convert this rate to a percentage:
[tex]\[ \text{quarterly\_rate\_percentage} = 0.0484 \times 100 = 4.84\% \][/tex]
### Step 4: Constructing the Function
The value of the car [tex]\( f(t) \)[/tex] after [tex]\( t \)[/tex] years can be expressed by considering its quarterly decrease:
1. Each year has 4 quarters, so after [tex]\( t \)[/tex] years, there are [tex]\( 4t \)[/tex] quarters.
2. The value of the car after [tex]\( 4t \)[/tex] quarters is given by:
[tex]\[ f(t) = V_0 \times (\text{quarterly\_factor})^{4t} \][/tex]
Substituting the values, we get:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \][/tex]
### Final Answer:
#### Function:
The function that represents the value of the car after [tex]\( t \)[/tex] years is:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \][/tex]
#### Quarterly Rate of Change:
The percentage rate of change per quarter, to the nearest hundredth of a percent, is:
[tex]\[ \boxed{4.84\%} \][/tex]
Thus, the complete function and the quarterly rate of change are:
[tex]\[ f(t) = 20000 \times (0.9516)^{4t} \quad \text{and the quarterly rate is 4.84\%} \][/tex]
