Question

a machine is now worth $141,800 and will be depreciated linearly over a 8 - year period, at which time it will be worth $50,840 as scrap. (a) find the rule of depreciation function f. (b) what is the domain of f? (c) what will the machine be worth in 4 years? (a) find the rule of depreciation function f. f(x) = (do not include the $ symbol in your answer )

Explanation:

Step1: Determine the slope of the linear function

The machine's value changes linearly over time. We have two points: at \( x = 0 \) (now), the value \( f(0)=141800 \), and at \( x = 8 \) (after 8 years), the value \( f(8)=50840 \). The slope \( m \) of a linear function \( f(x)=mx + b \) is calculated by \( m=\frac{f(8)-f(0)}{8 - 0} \).
\[
m=\frac{50840 - 141800}{8}=\frac{-90960}{8}=- 11370
\]

Step2: Determine the y - intercept

For a linear function \( f(x)=mx + b \), when \( x = 0 \), \( f(0)=b \). Since \( f(0) = 141800 \), then \( b = 141800 \).

Step3: Write the depreciation function

Using the slope - intercept form of a linear equation \( y=mx + b \), the depreciation function \( f(x) \) is \( f(x)=-11370x + 141800 \), where \( x \) represents the number of years.

Answer:

\( f(x)=-11370x + 141800 \)