Case study application of derivative 5 chapter 6 class 12

Case study Chapter 6 (Application of Derivative)

Case study 5:– Indian Railways is the largest rail network in Asia and world’s second largest. No doubt huge amount of money of Indian government paid as fuel cost. Decreasing fuel cost can increase railway profit and hence will improve Indian economy.(Case study application of derivative 5 )

Case study application of derivative 5 — Indian Railways is the largest rail network in Asia and world’s second largest. No doubt

This is the fact that the fuel cost for running a train is proportional to the square of the speed generated in km per hour. The fuel cost is ₹ 48 per hour at speed 16 km per hour and fixed charges amount to ₹ 1200 per hour.

Based on above information answer the question:

(i) (a) If F be the fuel cost per hour and v the speed in km/h. then find the relation between F and v.

(b) If C is the total cost for running the train ‘D’ km, express ‘C’ in the term v.

(ii) (a) Railway department is interested to find most economical speed v of train when total distance covered by train is ‘D’ km. What should be value of v ?

(b) Find the minimum cost for economical speed to cover distance 100 km.

Solution: (i) (a) From question

$F\propto v^2$

$\Rightarrow F = kv^2 -----(i)$

But, it is given that if F = Rs 48/hour, then v = 16 km/h

$\Rightarrow 48 =k.(16)^2$

$\Rightarrow k = \frac{3}{16}$

Putting it in (i), we get

$F = \frac{3}{16}.v^2$

$\Rightarrow 16 F = 3v^2$

(b) Let t be the time taken by train to cover the distance ‘D’.

Then C = 1200 t + Ft

$C = 1200.\frac{D}{v}+\frac{3v^2}{16}.\frac{D}{v}$

$\left[D = tv \Rightarrow t =\frac{D}{v}\text{and} F = \frac{3v^2}{16}\right]$

$C = D.\left(\frac{1200}{v}+\frac{3v}{16}\right)$

(ii) (a) Economical speed means that speed of train for which cost ‘C’ is minimum.

Now

$C =D.\left(\frac{1200}{v}+\frac{3v}{16}\right)$

$\therefore \frac{dC}{dv}=D.\left(-\frac{1200}{v^2}+\frac{3}{16}\right)$

For optimum value

$\frac{dC}{dv}=0$

$\Rightarrow D.\left[-\frac{1200}{v^2}+\frac{3}{16}\right]=0$

$\Rightarrow \frac{1200}{v^2} =\frac{3}{16}$

$\Rightarrow v^2 =\frac{200\times 16}{3}$

$\Rightarrow v = 20\times 4 = 80$ km/h

Now, $\frac{d^2C}{dv^2} = D\left(+\frac{2\times 1200}{v^2}\right)$

$\left(\frac{d^2C}{dv^2}\right)_{(v=80)} = +ve$

∴ For economical speed v = 80 km/h

(b) For economical speed v = 80 km/h

Now,

$C = 100\times \left(\frac{1200}{80}+\frac{3\times 80}{16}\right)$

$= 100\times (15+15)$

= 30×100 = ₹ 3000

Some other Case study problem

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Case study 2:- Read the following and answer the question.(Case study application of derivative 2)

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Case study 3:- Read the following and answer the question:(Case study application of derivative 3)

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Case study 4:- Read the following and answer the question:(Case study application of derivative 4 )

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