91Ó°ÊÓ

A farmer depreciates a \(\$ 120,000\) tractor. He estimates that the resale value \(V(t)\) (in \(\$ 1000\) ) of the tractor \(t\) years after purchase is \(80 \%\) of its value from the previous year. Therefore, the resale value can be approximated by \(V(t)=120(0.8)^{t}\). a. Find the resale value 5 yr after purchase. Round to the nearest \(\$ 1000\). b. The farmer estimates that the cost to run the tractor is \(\$ 18 / \mathrm{hr}\) in labor, \(\$ 36 / \mathrm{hr}\) in fuel, and \(\$ 22 / \mathrm{hr}\) in overhead costs (for maintenance and repair). Estimate the farmer's cost to run the tractor for the first year if he runs the tractor for a total of \(800 \mathrm{hr}\). Include hourly costs and depreciation.

Step by step solution

01

- Define the depreciation formula

The resale value of the tractor after t years is given by the formula: \[ V(t) = 120(0.8)^t \] Here, \(V(t)\) is the value in thousands of dollars.

02

- Calculate the resale value after 5 years

We need to find the resale value after 5 years, so substitute \(t = 5\) into the depreciation formula: \[ V(5) = 120(0.8)^5 \] Calculate (0.8)^5: \( (0.8)^5 \approx 0.32768 \) Now, multiply by 120: \( 120 \times 0.32768 \approx 39.3216 \) This means the resale value is approximately \( V(5) \approx 39.3216 \) thousand dollars. Rounding to the nearest thousand: \( V(5) \approx 39 \) thousand dollars or \$ 39,000.

03

- Calculate the costs per hour

The farmer estimates the following costs to run the tractor: - Labor: \$ 18/hr - Fuel: \$ 36/hr - Overhead: \$ 22/hr Sum these costs to get the total cost per hour: \( 18 + 36 + 22 = 76 \) dollars per hour.

04

- Calculate the annual operational cost

The total operational cost for 800 hours is: \( 76 \times 800 \) dollars \( 76 \times 800 = 60,800 \) dollars.

05

- Calculate annual depreciation cost

The depreciation cost for the first year can be found by the difference in value from year 0 to year 1. Initial value (year 0): 120,000 dollars Value after 1 year using the formula: \[ V(1) = 120(0.8)^1 = 96 \] thousand dollars or 96,000 dollars. Depreciation cost: \( 120,000 - 96,000 = 24,000 \) dollars.

06

- Calculate total first-year cost

Add the operational cost to the depreciation cost to find the total cost for the first year: \( 60,800 + 24,000 = 84,800 \) dollars. Therefore, the total cost to run the tractor for the first year is \$ 84,800.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Depreciation formula

Depreciation is a method used to allocate the cost of a tangible asset over its useful life. The provided depreciation formula for the tractor is \[ V(t) = 120(0.8)^t \] where \( V(t) \) represents the resale value in thousands of dollars after \( t \) years. In this formula:

• \( 120 \) is the initial value in thousands of dollars
• \( 0.8 \) is the yearly depreciation rate (80% of its value from the previous year)

This formula helps predict how much the tractor will be worth in the future based on how long it has been in use. It's essential to understand the formula because it sets the foundation for calculating other financial aspects related to the asset.

Resale value calculation

To find the resale value of the tractor years after its purchase, we use the depreciation formula. For example, to find the resale value 5 years after purchase, we substitute \( t = 5 \) into the formula:\[ V(5) = 120(0.8)^5 \]Calculating \( (0.8)^5 \):\[ (0.8)^5 \approx 0.32768 \]Multiplying by 120:\[ 120 \times 0.32768 \approx 39.3216 \]Therefore, the resale value after 5 years is approximately \( \$39,000 \). This calculation shows how much value the tractor loses over time and helps in financial planning for selling or replacing the asset.

Operational cost estimation

Estimating the operational cost of running the tractor involves summing up different hourly costs. The farmer has estimated:

• Labor: \( \$18/hr \)
• Fuel: \( \$36/hr \)
• Overhead costs (maintenance and repair): \( \$22/hr \)

Summing these, the total operational cost per hour is:\[ 18 + 36 + 22 = 76 \] dollars/hr. For a total of 800 hours of use in the first year, the overall operational cost becomes:\[ 76 \times 800 = 60,800 \] dollars. Knowing the operational cost helps in budgeting and understanding the financial demands of running the tractor.

Annual depreciation cost

The annual depreciation cost represents the loss in value of the tractor within a year. To find this, we compare the initial value to the value after one year. Initially, the tractor's value is \( \$120,000 \). After one year:\[ V(1) = 120(0.8)^1 = 96 \] thousand dollars, or \( \$96,000 \). The depreciation cost for the first year is then:\[ 120,000 - 96,000 = 24,000 \] dollars. This cost shows how much value the tractor loses in its first year of use and is crucial for understanding overall expenses.

Total cost analysis

To get a complete picture of the cost of running the tractor for the first year, we add the operational cost to the depreciation cost. From previous calculations:

• Operational cost: \( \$60,800 \)
• Depreciation cost: \( \$24,000 \)

Total first-year cost is:\[ 60,800 + 24,000 = 84,800 \] dollars. This total cost analysis is vital in making informed decisions about the viability of using the tractor and helps in financial planning and budgeting for future investments.