Problem 87 - Precalculus: Mathematics for Calculus

Question

A construction company purchases a bulldozer for \(\$ 160,000\). Each year the value of the bulldozer depreciates by \(20 \%\) of its value in the preceding year. Let \(V_{n}\) be the value of the bulldozer in the \(n\) th year. (Let \(n=1\) be the year the bulldozer is purchased.) (a) Find a formula for \(V_{n}\) (b) In what year will the value of the bulldozer be less than \(\$ 100,000 ?\)

Step-by-Step Solution

Answer

(a) The formula is \( V_n = 160,000 \times 0.8^{n-1} \); (b) The value will be less than $100,000 in year 4.

Step 1: Understanding Depreciation Rate

Every year, the value of the bulldozer is reduced by 20% of its previous year's value. This means the bulldozer retains 80% of its value from the previous year after depreciation.

Step 2: Establish the Initial Value

At the first year, which is when the bulldozer is purchased, we have \( V_1 = 160,000 \).

Step 3: Develop a Formula for Value Over Time

Since the value of the bulldozer decreases by 20% each year, the remaining value each year is 80% of the previous year. This can be expressed as the formula \( V_n = 160,000 \times 0.8^{n-1} \) for year \( n \).

Step 4: Determine the Year When Value Goes Below $100,000

Set the formula \( V_n = 160,000 \times 0.8^{n-1} \) to be less than 100,000 and solve for \( n \):\[ 160,000 \times 0.8^{n-1} < 100,000 \] Divide both sides by 160,000: \[ 0.8^{n-1} < \frac{100,000}{160,000} \] which simplifies to:\[ 0.8^{n-1} < 0.625 \]Now take the natural logarithm of both sides:\[ \ln(0.8^{n-1}) < \ln(0.625) \]which simplifies to:\[ (n-1) \cdot \ln(0.8) < \ln(0.625) \]Now solve for \( n-1 \): \[ n-1 > \frac{\ln(0.625)}{\ln(0.8)} \] \[ n-1 > 2.678 \]So, \( n > 3.678 \). Therefore, \( n = 4 \) (since \( n \) must be a whole number).

Key Concepts

Depreciation

Depreciation refers to the process by which an asset loses its value over time. This is commonly due to wear and tear, age, or obsolescence. In accounting and financial planning, depreciation helps determine the decreasing value of an asset each year.
For example, if a piece of machinery like a bulldozer is expected to lose 20% of its value annually, it means that what you paid for this bulldozer decreases in terms of its worth in the market.

• Depreciation allows companies to write off the gradual loss of value of their assets against their profits.
• It is also used to approximate the remaining value of an asset over time for tax purposes or resale."

In our bulldozer example, we began with an initial value of $160,000, and the model reduces this value by 20% each year. The key point to remember is that depreciation continues year over year, and knowing the rate helps in calculating future values.

Exponential Decay

Exponential decay happens when the value of something decreases at a constant rate proportionate to its current value. You'll notice this in many areas, such as radioactive decay, cooling processes, and financial depreciation.
With the bulldozer, its value decreases by a factor of 0.8 every year. This factor is constant, meaning the change each year depends on the previous year's value.

• Exponentially decaying systems can often be described by the formula \( V_n = V_0 \times (1 - r)^n \), where \( V_0 \) is the initial amount, \( r \) is the rate of decay, and \( n \) is the number of periods.
• In our case, \( r = 0.2 \), so \( 1-r = 0.8 \).

This rate effect means that every year, the bulldozer retains only 80% of the previous year’s value. The predictable pattern of change is what makes exponential decay a reliable model for such situations.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world systems with mathematical language. It allows for simulations, calculations, and predictions about how a system behaves under different circumstances.
Our bulldozer scenario is a classic example of modeling the depreciation of an asset over time:

• By using an exponential decay formula, we can predict future values of the bulldozer’s worth.
• The formula \( V_n = 160,000 \times 0.8^{n-1} \) serves as our model for estimating the bulldozer's value each year.

With models like this, businesses can forecast budget needs, maintenance costs, and future investment decisions. Understanding how to create and interpret such models is essential for those making strategic financial decisions based on asset depreciation.