Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.

a. What is the value of the machine after 10 years?

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Identify the initial value of the machine, which is \$2.5 million. This will be our starting amount, denoted as $V_0 = 2.5$ million.

Understand that the machine depreciates by 6.8% each year based on its current value. This means each year the machine retains 100% - 6.8% = 93.2% of its value from the previous year.

Express the value of the machine after $n$ years using the exponential decay formula: $V_n = V_0 \times (1 - r)^n$, where $r$ is the depreciation rate as a decimal (6.8% = 0.068).

Substitute the known values into the formula: $V_{10} = 2.5 \times (1 - 0.068)^{10}$ to represent the value after 10 years.

To find the value after 10 years, calculate the expression $2.5 \times 0.932^{10}$. This will give the depreciated value of the machine after 10 years.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Decay

Exponential decay describes a process where a quantity decreases by a fixed percentage over equal time intervals. In this problem, the machine's value decreases by 6.8% each year, meaning its value after each year is multiplied by (1 - 0.068). This concept helps model the depreciation over multiple years.

Depreciation

Depreciation is the reduction in the value of an asset over time, often due to wear and tear or obsolescence. Here, the machine’s value decreases annually by a fixed percentage, which is a common method for calculating tax deductions and understanding asset worth.

Compound Interest Formula

The compound interest formula, V = P(1 - r)^t, calculates the future value of an amount after repeated percentage decreases or increases. In depreciation, it models how the value diminishes year after year, where P is the initial value, r is the depreciation rate, and t is the number of years.

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.a. What is the value of the machine after 10 years?