UpStudy Free Solution:

To model the depreciation of the car's value over time using an exponential equation, we need to understand how depreciation works. The car's value decreases by a certain percentage each year.

Given:

- The initial value of the car (\(P\)) is $24,000.

- The car depreciates by 12% per year.

Depreciation means the car retains \(100\% - 12\% = 88\% \) of its value each year. This can be expressed as a multiplier of 0.88.

The general form of an exponential decay equation is:

\[f( x) = P \cdot ( 1 - r) ^ x\]

where:

- \(P\) is the initial amount (initial value of the car),

- \(r\) is the rate of depreciation,

- \(x\) is the number of years.

In this case:

- \(P = 24,000\),

- \(r = 0.12\),

- The multiplier for each year is \(1 - 0.12 = 0.88\).

So the exponential equation modeling the situation is:

\[f( x) = 24,000 \cdot 0.88^ x\]

Therefore, the correct answer is:

B. \(f( x) = 24,000 \cdot 0.88^ x\)

Supplemental Knowledge

Exponential decay is a process where the quantity decreases at a rate proportional to its current value. This type of decay is commonly seen in scenarios such as depreciation of assets, radioactive decay, and cooling of objects.

The general form of an exponential decay function is:

\[f( x) = P \cdot ( 1 - r) ^ x\]

where:

- \(P\) represents the initial value or amount.

- \(r\) is the rate of decay (expressed as a decimal).

- \(x\) is the time period over which the decay occurs.

In this context, depreciation means that each year, the car retains a certain percentage of its previous year's value. If a car depreciates by 12% annually, it retains \(100\% - 12\% = 88\% \) of its value each year. This can be represented by a multiplier of 0.88.

For example:

- After 1 year: \(f( 1) = P \cdot 0.88^ 1 = P \cdot 0.88\)

- After 2 years: \(f( 2) = P \cdot 0.88^ 2\)

- And so on...

Understanding exponential functions can greatly strengthen both mathematical proficiency and problem-solving ability, but can often prove daunting without proper explanation and step-by-step solutions of algebra to make learning intuitive and engaging! UpStudy makes mastery intuitive with detailed explanations and step-by-step solutions across subjects including mathematics, chemistry, physics and biology; its expert tutors are available round-the-clock online if exams or homework assignments become a problem - join millions who have transformed their academic excellence through UpStudy today - your reliable partner in academic excellence!