Q. 305.0( 1 Vote )

# A square tank of

Answer :

Given,

• Capacity of the square tank is 250 cubic metres.

• Cost of the land per square meter Rs.50.

• Cost of digging the whole tank is Rs. (400 × h^{2}).

• Where h is the depth of the tank.

Let us consider,

• Side of the tank is x metres.

• Cost of the digging is; C = 50x^{2} + 400h^{2} ---- (1)

• Volume of the tank is; V = x^{2}h ; 250 =x^{2}h

----- (2)

Substituting (2) in (1),

----- (3)

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. __This is because if the function C(x) has a maximum/minimum at a point c then C’(c) = 0.__

Differentiating the equation (3) with respect to x:

[Since ]

------- (4)

To find the critical point, we need to equate equation (4) to zero.

x^{6} = 10^{6}

x = 10

__Now to check if this critical point will determine the minimum volume of the tank, we need to check with second differential which needs to be positive.__

Consider differentiating the equation (4) with x:

----- (5)

[Since and]

Now let us find the value of

As , so the function C is minimum at x=10

Substituting x in equation (2)

h = 2.5 m

Therefore when the cost for the digging is minimum, when x = 10m and h = 2.5m

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