# Case study Chapter 6 (Application of derivative)

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.

(i) If represents the position of helicopter on the curve y = x² +7, when the distance D from soldier placed at S(3, 7) is minimum, then the relation between is

(a)                    (b)

(c)                  (d)

(ii) The distance ‘D’ expressed as a function of is

(a)                 (b)

(c)         (d)

(iii) The soldier at S wants to know when the enemy helicopter is nesrest to soldier, then the value should be

(a) 4                          (b) 3

(c) 8                           (d) 5

(iv) When the enemy helicopter is nearest to soldier, then the value of D should be

(a) 4 units                  (b) 5 units

(c) √5 units                 (d) √7 units

(v) The narest position of helicopter from soldier is

(a) (1, √3)                     (b) (1, 8)

(c) (1, 7)                         (d) (1, √7)

Since, lie on the curve y = x² + 7

⇒  satisfy the equation of curve

⇒

Here, since soldier is at (3, 7).

Since, lie on the curve  y =x² + 7

We have ,

For minimum value of D i.e. D²

[Since have no real roots i.e.real value of is not possible.]

⇒ For minimum points,(distance) or nearest distance

Also

Since,(x_1, y_1) lie on the curve y = x² + 7

For nearest distance D i.e. minimum value of D,

Since,

For minimum value of D

and

∴ Nearest position of helicopter is (1, 8).

## 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.

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.