Question

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? b. After how many years is the value of the machine \(10 \%\) of its original value?

Short answer

Answer: The value of the machine after 10 years will be approximately \$1.24665 million. It will take approximately 35 years for the machine's value to become 10% of its original value.

Step-by-step solution

Compound Depreciation Formula

To find the value of the machine after a specific number of years with ongoing depreciation, we can use the compound depreciation formula. The formula is: Value after n years = Initial Value * (1 - Depreciation Rate) ^ n where n is the number of years. In our case, the initial value is \(\$2.5\) million, and the depreciation rate is \(6.8 \%\), which in decimal form is \(0.068\).

Find Value after 10 Years

Now, we can calculate the value of the machine after 10 years using the formula from Step 1: Value after 10 years = Initial Value * (1 - Depreciation Rate) ^ 10 = 2.5 * (1 - 0.068) ^ 10 ≈ 2.5 * (0.932) ^ 10 ≈ 2.5 * 0.49866 ≈ \$1.24665$ million Thus, the value of the machine after 10 years is approximately \(\$1.24665\) million.

Find Number of Years for 10% Value

We want to find the number of years when the machine's value becomes 10% of its original value. So, we set up the equation: 0.10 * Initial Value = Initial Value * (1 - Depreciation Rate) ^ n Divide both sides by the initial value: 0.10 = (1 - 0.068) ^ n To solve for n, we can use the properties of logarithms. Take the natural logarithm of both sides: ln(0.10) = ln((1 - 0.068) ^ n) Use the logarithm property of exponents: ln(0.10) = n * ln(1 - 0.068) Isolate n on one side: n = ln(0.10) / ln(1 - 0.068) n ≈ 34.38

Round Up to the Nearest Whole Year

As we can't have a fraction of a year, we should round up to the nearest whole year. Therefore, it will take approximately 35 years for the machine's value to become 10% of its original value.

Key concepts

Depreciation Rate

The depreciation rate is a percentage that reflects the amount by which the value of an asset decreases over time. In the context of our exercise, we are dealing with a depreciation rate of 6.8% per year for the die-casting machine. This means each year, the machine loses 6.8% of its value from the previous year. Calculating depreciation involves converting this percentage into a decimal by dividing by 100. So, 6.8% becomes 0.068 in decimal form. Here's a simple glimpse into depreciation rates:

• The higher the rate, the quicker the value decreases over time.
• It is an essential factor for accounting, tax purposes, and determining the sale price of used equipment.

Understanding the depreciation rate helps predict how asset values decline, aiding businesses in planning and financial forecasting.

Machine Value

The machine's value initially starts at \(2.5 million. Over time, as depreciation occurs, this value continues to decrease. As evidenced by our exercise, to calculate machine value after a certain number of years, we use the compound depreciation formula. This formula is:Value after \(n\) years = Initial Value \(\times (1 - \text{Depreciation Rate})^n\)In the example, the machine's value after 10 years becomes approximately \)1.24665 million. This was determined by reducing the machine's value each year based on the 6.8% depreciation rate. Some essentials about machine value include:

• It's important for determining the resale value of assets.
• A lower machine value impacts financial statements and tax obligations.

Calculating machine value accurately is vital for businesses to realize the current worth of their assets.

Natural Logarithm

A natural logarithm is a mathematical concept that is widely used to solve exponential equations, especially those involving compound processes like depreciation or growth. The natural logarithm (ln) represents the logarithm base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828.In the problem solved, we used the natural logarithm to find the number of years it takes for the machine's value to reach 10% of its original value. This involved taking the natural logarithm of both sides of the equation:\[0.10 = (1 - 0.068)^n\]Which transforms into:\[\ln(0.10) = n \cdot \ln(1 - 0.068)\]Then, solve for \(n\) by isolating it as follows:\[n = \frac{\ln(0.10)}{\ln(1 - 0.068)}\]This calculation shows that it takes about 35 years for the machine's value to depreciate to 10% of its original value. Understanding natural logarithms is crucial as it helps break down complex problems involving exponential decay or growth into manageable calculations.