# Case study Chapter 6 (Application of Derivative)

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:

(i) If x and y represent the length and breadth of its rectangular base, then the relation between the variables is

(a) x + y = 8                       (b) x . y = 4

(c) x + y = 4                        (d) x/y = 4

(ii) If construction of tank cost ₹ 70 per sq. metre for the base and ₹ 45 per square metre for sides, then making cost ‘C’ expressed as a function of x is

(a)               (b)

(c)            (d)

(iii) The owner of a construction agency is interested in minimizing the cost ‘C’ of  whole tank, for this to happen the value of x should be

(a) 4 m                        (b) 3 m

(c) 1 m                         (d) 2 m

(iv) For minimum cost ‘C’ the value of y should be

(a) 1 m                          (b) 3 m

(c) 2 m                           (d) 2 m

(v) The prsdan of village wants to know minimum cost. The minimum cost is

(a) ₹ 2000                   (b) ₹ 4000

(c) ₹ 11,000                 (d) ₹ 1000

8 = x.y.2

⇒ 2x.y = 8 ⇒ x.y = 4

Since ‘C’ is cost of making tank

∴ C = 70x.y + 45×2(2x + 2y)

⇒ C = 70x.y + 90(2x + 2y)

⇒ C = 70x.y +180(x + y)

⇒ C = 280 + 180(x + 4/x)

For maximum or minimum

⇒ x = 2(length can never be negative)

Now,

Hence to minimize C, x = 2 m

Since, x.y = 4

⇒ y = 4/x

⇒ y = 4/2 = 2 m

Since, C = 280+ 180(x + 4/x)

⇒ C = 280 + 180(2 + 2)

⇒ C = 280 + 180×4

⇒ C = 280 +720 = ₹1000

## 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.

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.