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

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

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

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

(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

Case study 1: Read the following and answer the question(Case study application of derivative 1 )
A recangular hall is to be developed for a meeting of farmers in an agriculture college to aware them for new technique in cultivation. It is given that the floor has a fixed perimeter P as shown below.

Solution: For solution click here

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

Dr. Ritam residing to Delhi went to see an apartment of 3 BHK in Noida. The window of the house was in the form of a rectangle surmounted by a semicircular opening having a perimeter of the window 10 m as shown in figure.

Solution: For solution click here

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

On the request of villagers, a construction agency designs a tank with the help of an architect. Tank consist of rectangular base with recangular sides, open at the top so that its depth is 2 m and volume is 8 cubic meter as shown below:

Solution: For solution click here

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

These days chinese and Indian troops are engaged in aggressive melee,face-off skirmishes at location near the disputed Pangong Lake in Ladakh.

One day a helicopterof enemy is flying along the curve represented by y = x² + 7. A soldier placed at (3, 7) wants to shoot down the helicopter when it is nearest to him.

Solution: For solution click here