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Chapter 0: Problem 9

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. The linear depreciation method, or straight-line 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 \(t\), where \(0 \leq t \leq n\), is given by the linear function $$ V(t)=C-\frac{C-S}{n} t $$ Hint: Find an equation of the straight line that passes through the points \((0, C)\) and \((n, S)\). Then rewrite the equation in the slopeintercept form.

Short Answer

Expert verified
In this problem, we must find the linear function that represents the book value of an asset at any time t, given an initial value of $C$, a scrap value of $S$, and a depreciation period of $n$ years. After determining the slope of the line passing through the points (0, C) and (n, S), we find the line equation and rewrite it in slope-intercept form as \(y=\frac{S-C}{n}x + C\). Finally, we substitute \(V(t)\) for y and t for x, obtaining the function \(V(t) = \frac{S-C}{n}t + C\). This equation represents the book value of the asset at any given time t.

Step by step solution

01

Determine the slope of the line

To find the slope (m) of the line that passes through the given points, we can use the formula: \(m=\frac{y_2-y_1}{x_2-x_1}\). Using the points (0, C) and (n, S): \(m = \frac{S-C}{n-0} = \frac{S-C}{n}\)
02

Use the point-slope form of the line equation

The point-slope form of a line equation is given by: \(y - y_1 = m(x - x_1)\). Using the slope we found in Step 1 and the point (0, C), we can write the equation of the line as: \(y - C = \frac{S-C}{n}(x - 0)\)
03

Rewrite the equation in slope-intercept form

Slope-intercept form is given by: \(y = mx + b\). To convert our equation to this form, we need to isolate y: \(y = \frac{S-C}{n}x + C\)
04

Substitute V(t) for y and t for x

Since we are given that V(t) is the book value of the asset at any time t, and 0 <= t <= n, we can rewrite the equation with V(t) for y and t for x: \(V(t) = \frac{S-C}{n}t + C\) This is the equation for the book value (V) of the asset at any time t, with initial value C, scrap value S, and depreciation period n.

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