A tool and die company buys a machine for 175,000 dollars...

A tool and die company buys a machine for 175,000 dollars and it depreciates at a rate of $30 \%$ per year. (In other words, at the end of each year the depreciated value is $70 \%$ of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years.

Key Concepts

Exponential Decay

Exponential decay describes a process where the value of an asset decreases at a rate proportional to its current value over consistent time intervals. This concept is crucial for understanding depreciation models where a fixed percentage of the asset’s value is lost each year, leading to a decrease that can be modeled mathematically using exponential functions.

Depreciation

Depreciation is the accounting method used to allocate the cost of a tangible asset over its useful life. In scenarios like machinery or vehicles, depreciation reflects wear and tear, obsolescence, or loss in value over time. This provides a systematic approach to expense allocation in financial reporting.

Compound Depreciation

Compound depreciation refers to the calculation of asset value reduction where each period's depreciation applies to the remaining balance from the previous period. It results in a value that diminishes exponentially and is calculated by multiplying the initial asset value by the retention factor (1 minus the depreciation rate) raised to the number of periods.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are key to modeling processes where change occurs at a constant percentage rate, such as with compound depreciation or decay, making them essential for solving problems involving percentage reductions applied repeatedly over time.

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