Chapter 5.4, Problem 33E

a.

To determine

To explain: why ${\displaystyle \underset{0}{\overset{t}{\int }}f(s)ds}$ represent the loss in value of the machine over the period of time t since the last overhaul.

Explanation of Solution

Given information:

Rate of depreciation: $f=f(t)$ , $t$ is the time measured in months since the last overhauled. $A$ is the fixed cost which is incurred every time machine is overhauled.

Formula used: differentiation, fundamental theorem of calculus.

As derivative gives the rate of change of function, in this case rate of depreciation is $F\text{'}(t)=f(t)$ .

By taking integral of both side and applying fundamental theorem of calculus ,

  $F(t)={\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}$

As we defined $f(t)$ as the value of equipment, thus the equation tell the total loss of value at the time t.

b.

To determine

To explain: What does $C$ represent and why would the company want to minimize $C$ .

Explanation of Solution

Given information: $f=f(t)$ , $t$ is the time measured in months since the last overhauled. $A$ is the fixed cost which is incurred every time machine is overhauled.

And $C(t)=\frac{1}{t}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]$

Formula used: differentiation, fundamental theorem of calculus.

Here, $A$ is the fixed cost which is incurred every time machine is overhauled and ${\displaystyle \underset{0}{\overset{t}{\int }}f(s)ds}$ represent the loss in value of the machine over the period of time t. so, $\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]$ represent the total expenditure over some time period $t$ . As $\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]$ is divided by the term $t$ it represent the cost per month as $t$ is measured in month.

Hence,

  $C$ represent average expenditure of the manufacturing of the company per month. The company would like to spend as little as possible to maximize the profit hence company will try to minimize $C$ .

c.

To determine

To show: $C$ has minimum value at the number $t=T$ , where $C(T)=f(T)$

Explanation of Solution

Given information: $f=f(t)$ , $t$ is the time measured in months since the last overhauled. $A$ is the fixed cost which is incurred every time machine is overhauled.

And $C(t)=\frac{1}{t}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]$

Formula used: fundamental theorem of calculus

We have $C(t)=\frac{1}{t}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]$ , now differentiating $C$ using the fundamental theorem of calculus , $C\text{'}(t)=\frac{1}{t^2}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]+\frac{1}{t}f(t)$

To find the extremum point put $C\text{'}(t)=0$

  $\begin{array}{l}\Rightarrow C\text{'}(t)=\frac{1}{t^2}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]+\frac{1}{t}f(t)=0\\ \Rightarrow \frac{1}{t^2}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]=\frac{1}{t}f(t)\\ \Rightarrow f(t)=\frac{1}{t}\left[A+{\displaystyle \underset{o}{\overset{t}{\int }}f(s)ds}\right]=C(t)\end{array}$

Now, if the minimum value is $t=T$ then, $C(T)=f(T)$ .

Hence,

At minimum value is $t=T$ , $C(T)=f(T)$ .