Math, asked by hs2561313, 4 days ago

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. prove that for the minimum metal sheet its depth will be half of the width​

Answers

Answered by OoVinayLankeroO
2

Answer:

Let the length, width and height of the open tank be x, x and y with respectively. Then its volume is x

2

y and total surface area is x

2

+4xy.

It is given that the tank can be hold a given quantity of water. This mean that its volume is constant. Let it be V.

V=x

2

y

The cost of the material will be least if the total surface area is least. Let's denote total surface area

S=x

2

+4xy

We have to minimize S subjects to the condition that the volume is constant.

S=x

2

+4xy

S=x

2

+

x

4V

dx

dS

=2x−

x

2

4V

dx

2

d

2

S

=2+

x

3

8V

For maximum or minimum value of S

dx

dS

=0

⇒2x−

x

2

4V

=0

⇒2x

3

=4V

⇒2x

3

=4x

2

y

⇒x=2y

Clearly

dx

2

d

2

x

=2+

x

3

8V

>0 for all x.

Here S is minimum when x=2y depth of tanh is half of width.

Step-by-step explanation:

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