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Question

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m 3 . If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

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Solution

Consider x and y as the length and breadth of the base of the tank. Height of the tank is given as 2m and volume of the tank is given as 8 m 3 .

The formula for the volume of tank is length×breadth×height.

Write the equation for the volume of the tank.

xy2=8 xy=4 y= 4 x

The cost of building base of the tank at the rate of 70 per sq. meter is 70xy.

Write the equation for the cost of building of four walls of the tank at a rate of 45 per sq. meter.

45( x2+x2+y2+y2 )=180x+180y

Consider z as the total cost of building the tank.

Write the equation for the total cost of building the tank.

z=70xy+180x+180y

Differentiate both sides of the equation,

dz dx =0+180 720 x 2 d 2 z d x 2 = 1440 x 3

Now,

dz dx =0 180 720 x 2 =0 180 x 2 =720 x=±2

Length cannot be negative.

At x=2,

d 2 z d x 2 = 1440 8 =180 >0

Thus, z is minimum at x=2.

Write the equation for the minimum cost.

280+180×2+ 720 2 =280+360+360 =1000

Thus, the minimum cost is 1000.


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