   Chapter 2.3, Problem 50E

Chapter
Section
Textbook Problem

Linear Depreciation of an Asset In computing income tax, businesses are allowed by law to depreciate certain assets such as buildings, machines, furniture, and automobiles over a period of time. Linear depreciation, or the straightline method, is often used for this purpose. Suppose an asset has an initial value of $C and is to be depreciated linearly over n years with a scrap value of$S. Show that the book value of the asset at any time t(0 ≤ t ≤ n) is given by the linear function V ( t ) = C − C − S n t Hint: Find an equation of the straight line that passes through the points (0, C) and (n, S). Then rewrite the equation in the slope-intercept form.

To determine

To show: The book value of the asset at any time t (0tn) is given by the function,

V(t)=CCSnt

Explanation

Proof:

The equation of a straight line passes through point (x1,y1) and have slope m is

(yy1)=m(xx1) (1)

Slope m of a line for two points (x1,y1) and (x2,y2) is,

m=y2y1x2x1 (2)

Suppose an asset has an initial value of $C and is to be depreciated linearly over n years with a scrap value of$S, the book value of the asset is calculated by the use of Linear depreciation or straight line method.

To prove the given function, find the equation of straight line passes through the points (0,C) and (n,S).

Substitute 0 for x1 , C for y1 , n for x2 and S for y2 in equation (2) to evaluate the value of m.

m=SCn0=SCn

Substitute V(t) for y, t for x, SCn for m, 0 for x1 and C for y1 in equation (1) to determine the equation of line passes through point (0,C) and has slope SCn

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