A farmer depreciates a 120,000 tractor. He estimates that...

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.

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.

Key Concepts

Exponential Decay

Exponential decay is a mathematical process where a quantity decreases by a fixed percentage over equal time intervals. It is used to model scenarios where the rate of decline is proportional to the current amount, such as in the depreciation of assets.

Depreciation

Depreciation refers to the reduction in the value of an asset over time due to factors like wear and tear or obsolescence. It is often modeled using exponential decay when the asset loses a constant percentage of its value each period.

Cost Analysis

Cost analysis is the process of evaluating the total costs associated with an activity or investment by combining various cost components. In this context, it involves adding both fixed costs, like depreciation, and variable costs, such as hourly operating expenses, to estimate overall expenses.

Unit Conversion and Rounding

Unit conversion and rounding are essential for ensuring that numerical results are expressed consistently and appropriately. This involves converting calculated values to the required units and rounding to a specified level of precision to provide a usable and understandable estimate.

Transcript

0:00 Hello.

00:01 In this problem, we're going to be using an exponential function that has been given to us that models a certain situation.

00:08 So in this case, v of t represents the resale value of a tractor in thousands of dollars two years after it has been purchased.

00:14 In part a, we're asked to find the resale value after five years to the nearest thousand.

00:19 So five years would be our t.

00:22 So we're going to be plugging in five for t.

00:26 So we're finding v of five.

00:28 I've done that here with a calculator.

00:29 So v of 5, which comes out to this number.

00:35 We are asked to round that to the nearest thousand, and v of t is already in thousands of dollars.

00:40 So this is representing 39 .3216 ,000.

00:44 So to the nearest thousand, that would be $39 ,000.

00:48 So after five years, the tractor would be worth $39 ,000.

00:57 In part b, we're asked to find the cost of renting the tractor for the first year, including both depreciation and the hourly costs.

01:05 So it's $18 per hour for labor, $36 per hour for fuel, and $22 per hour overhead costs.

01:12 And it runs for 800 hours in that first year.

01:16 So first let's deal with the hourly costs.

01:19 If we look at the total hourly costs, that would be the $18 plus $36 plus $22.

01:25 That would give us the total cost of running it per hour.

01:30 So adding up all of our hourly costs, we get the total running cost of $76 per hour...

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